# Survey of Recent Applications of the Chaotic Lozi Map

René Lozi

► To cite this version:

René Lozi. Survey of Recent Applications of the Chaotic Lozi Map. Algorithms, 2023, 16 (10), pp.491. 10.3390/a16100491. hal-04533393

HAL Id: hal-04533393

https://hal.science/hal-04533393

Submitted on 4 Apr 2024

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
<figure><img src="image_1.png" /></figure>

HAL
open science

# Survey of Recent Applications of the Chaotic Lozi Map

René Lozi

► To cite this version:

René Lozi. Survey of Recent Applications of the Chaotic Lozi Map. Algorithms, 2023, 16 (10), pp.491. 10.3390/a16100491. hal-04533393

HAL Id: hal-04533393

https://hal.science/hal-04533393v1

Submitted on 4 Apr 2024

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_2.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# Survey of Recent Applications of the Chaotic Lozi Map

René Lozi ID

LJAD, CNRS, Université Côte d'Azur, F-06000 Nice, France; Rene.LOZI@univ-cotedazur.fr

**Abstract:** Since its original publication in 1978, Lozi's chaotic map has been thoroughly explored and continues to be. Hundreds of publications have analyzed its particular structure and applied its properties in many fields (e.g., improvement of physical devices, electrical components such as memristors, cryptography, optimization, evolutionary algorithms, synchronization, control, secure communications, AI with swarm intelligence, chimeras, solitary states, etc.) through algorithms such as the COLM algorithm (Chaotic Optimization algorithm based on Lozi Map), Particle Swarm Optimization (PSO), and Differential Evolution (DE). In this article, we present a survey based on dozens of articles on the use of this map in algorithms aimed at real applications or applications exploring new directions of dynamical systems such as chimeras and solitary states.

**Keywords:** Lozi map; fractional Lozi map; chaotic attractor; applications; control of chaos; memristor; particle swarm optimization (PSO); differential evolution (DE); self-organizing migrating algorithm (SOMA); chimera; synchronization; fuzzy synchronization; solitary states; cryptography; chaotic benchmark

## 1. Introduction

Among the most popular discrete chaotic attractors (Belykh map, Lorenz map, Hénon map, etc.), Lozi's chaotic map has been thoroughly explored since its publication in 1978, and continues to be today. Hundreds of publications have analyzed its particular structure and applied its properties in many fields, including cryptography, electronic devices such as memristors, optimization, synchronization, secure communications, AI with swarm intelligence, chimeras, solitary states, and more.

Zeraoulia [1] wrote a collection of results published between 1997 to 2013, including 700 references. However, new results in theory and applications are continually being published in many different journals and books, and I am frequently asked about them. This is why I have decided to gather all the new results on applications of this map in the current survey in order to render them easily accessible. All these results are the responsibility of their original authors, not mine.

In Section 2, we recall the origin of this map, its initial definition and classical chaotic properties in the dissipative case, and its dynamics features, including fixed points, invariant manifolds, basin of attraction, etc. I briefly mention a number of new results on advanced properties concerning other statistical, hyperbolic, ergodic, and topological properties (e.g., bifurcation, symbolic dynamics and the kneading sequence, topological entropy, Hausdorff dimension). The chaotic properties in the conservative case are evoked as well. The most recent definitions of the fractional, variable fractional, and complex fractional Lozi map are described. However, the aim of this study is to highlight the applications of this map, not the theoretical results; thus, the presentation of the latter has been reduced to a minimum in order to keep this article to a reasonable length. Interested readers can refer to [2] for an overview of the generalizations of this map.

In Section 3, the use of the Lozi map in electrical devices, memristors, and computers is developed. Section 4 describes the numerous applications of this map in the domain of optimization. Particle Swarm Optimization (PSO), Differential Evolution (DE), and the Self-Organizing Migrating Algorithm (SOMA) are among of the many new algorithms using the

<figure><img src="image_2.png" /></figure>

### check for updates

**Citation:** Lozi, R. Survey of Recent Applications of the Chaotic Lozi Map. *Algorithms* 2023, 16, 491. https://doi.org/10.3390/a16100491

Academic Editor: Jesper Jansson

Received: 6 September 2023

Revised: 16 October 2023

Accepted: 19 October 2023

Published: 22 October 2023

<figure><img src="image_3.png" /></figure>

**Copyright:** © 2023 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
# Survey of Recent Applications of the Chaotic Lozi Map

René Lozi ID

LJAD, CNRS, Université Côte d'Azur, F-06000 Nice, France; Rene.LOZI@univ-cotedazur.fr

**Abstract:** Since its original publication in 1978, Lozi's chaotic map has been thoroughly explored and continues to be. Hundreds of publications have analyzed its particular structure and applied its properties in many fields (e.g., improvement of physical devices, electrical components such as memristors, cryptography, optimization, evolutionary algorithms, synchronization, control, secure communications, AI with swarm intelligence, chimeras, solitary states, etc.) through algorithms such as the COLM algorithm (Chaotic Optimization algorithm based on Lozi Map), Particle Swarm Optimization (PSO), and Differential Evolution (DE). In this article, we present a survey based on dozens of articles on the use of this map in algorithms aimed at real applications or applications exploring new directions of dynamical systems such as chimeras and solitary states.

**Keywords:** Lozi map; fractional Lozi map; chaotic attractor; applications; control of chaos; memristor; particle swarm optimization (PSO); differential evolution (DE); self-organizing migrating algorithm (SOMA); chimera; synchronization; fuzzy synchronization; solitary states; cryptography; chaotic benchmark

## 1. Introduction

Among the most popular discrete chaotic attractors (Belykh map, Lorenz map, Hénon map, etc.), Lozi's chaotic map has been thoroughly explored since its publication in 1978, and continues to be today. Hundreds of publications have analyzed its particular structure and applied its properties in many fields, including cryptography, electronic devices such as memristors, optimization, synchronization, secure communications, AI with swarm intelligence, chimeras, solitary states, and more.

Zeraoulia [1] wrote a collection of results published between 1997 to 2013, including 700 references. However, new results in theory and applications are continually being published in many different journals and books, and I am frequently asked about them. This is why I have decided to gather all the new results on applications of this map in the current survey in order to render them easily accessible. All these results are the responsibility of their original authors, not mine.

In Section 2, we recall the origin of this map, its initial definition and classical chaotic properties in the dissipative case, and its dynamics features, including fixed points, invariant manifolds, basin of attraction, etc. I briefly mention a number of new results on advanced properties concerning other statistical, hyperbolic, ergodic, and topological properties (e.g., bifurcation, symbolic dynamics and the kneading sequence, topological entropy, Hausdorff dimension). The chaotic properties in the conservative case are evoked as well. The most recent definitions of the fractional, variable fractional, and complex fractional Lozi map are described. However, the aim of this study is to highlight the applications of this map, not the theoretical results; thus, the presentation of the latter has been reduced to a minimum in order to keep this article to a reasonable length. Interested readers can refer to [2] for an overview of the generalizations of this map.

In Section 3, the use of the Lozi map in electrical devices, memristors, and computers is developed. Section 4 describes the numerous applications of this map in the domain of optimization. Particle Swarm Optimization (PSO), Differential Evolution (DE), and the Self-Organizing Migrating Algorithm (SOMA) are among of the many new algorithms using the

<figure><img src="image_3.png" /></figure>

check for
updates

**Citation:** Lozi, R. Survey of Recent Applications of the Chaotic Lozi Map. *Algorithms* 2023, 16, 491. https://doi.org/10.3390/a16100491

Academic Editor: Jesper Jansson

Received: 6 September 2023

Revised: 16 October 2023

Accepted: 19 October 2023

Published: 22 October 2023

<figure><img src="image_4.png" /></figure>

**Copyright:** © 2023 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
Lozi map as a core function. Non-evolutionnary optimization algorithms are described as well, including the Improved Lozi map-based Chaotic Optimization Algorithm (ILCOA).

Applications in control theory are analyzed in Section 5, while applications in cryptography are addressed in Section 6. The coupling of several copies of this map in networks and their synchronization allows for the existence of new theoretical objects (e.g., chimeras, solitary states, and cellular neural networks), as is shown in Section 7. Finally, miscellaneous applications, such as the use of the Lozi map as a benchmark in several studies in the domains of economics and in education, are considered in Section 8, and a brief conclusion is presented in Section 9.

## 2. The Lozi Chaotic Map

### 2.1. Origin

On 15 June 1977, 46 years ago, I discovered this piecewise linear chaotic map of the plane. I remember that the idea came to me around 11 am during the defense of the Ph.D. thesis of one of my colleagues in the department of mathematics of the University of Nice (France) [1] (p. xxv).

A week before, I had attended the International Conference on Mathematical Problems in Theoretical Physics in Roma. During the inaugural talk on 6 June, David Ruelle conjectured that, for the Hénon chaotic map, the theoretical entropy should be equal to the characteristic exponent [3]. As explained in [4], this is how I learned of the first example of a chaotic and strange attractor in the plane: the Hénon attractor (see Figure 1). Surprisingly, I had never before met Michel Hénon, who worked a few kilometers from the math department of the university at the Nice Observatory.

<figure><img src="image_4.png" /><figcaption>Figure 1. Hénon map for the parameter value $a = 1.2$, $b = 0.42$, initial value $x_0 = 0$, $y_0 = 0$.</figcaption></figure>

Hénon used an IBM-7040 computer to numerically simulate the Lorenz map [5]. He found it difficult to highlight its inner nature due to the very high dissipativity of this map, for which the rate of volume contraction is provided by the Lie derivative of the Lorenz equations. Solving such equations for the parameters chosen by Lorenz provides $V(t) = V(0)e^{-\frac{t}{3}}$. Hence, volumes are reduced by a factor $10^6$ after one time unit. Hénon, who was a specialist in the motion of planets, had good experience with Poincaré's map, which inspired him the way to build the metaphoric model in [6]:

$$ \begin{cases} x_{n+1} = 1 - ax_n^2 + y_n, \\ y_{n+1} = bx_n. \end{cases} \qquad (1) $$
Lozi map as a core function. Non-evolutionnary optimization algorithms are described as well, including the Improved Lozi map-based Chaotic Optimization Algorithm (ILCOA).

Applications in control theory are analyzed in Section 5, while applications in cryptography are addressed in Section 6. The coupling of several copies of this map in networks and their synchronization allows for the existence of new theoretical objects (e.g., chimeras, solitary states, and cellular neural networks), as is shown in Section 7. Finally, miscellaneous applications, such as the use of the Lozi map as a benchmark in several studies in the domains of economics and in education, are considered in Section 8, and a brief conclusion is presented in Section 9.

## 2. The Lozi Chaotic Map

### 2.1. Origin

On 15 June 1977, 46 years ago, I discovered this piecewise linear chaotic map of the plane. I remember that the idea came to me around 11 am during the defense of the Ph.D. thesis of one of my colleagues in the department of mathematics of the University of Nice (France) [1] (p. xxv).

A week before, I had attended the International Conference on Mathematical Problems in Theoretical Physics in Roma. During the inaugural talk on 6 June, David Ruelle conjectured that, for the Hénon chaotic map, the theoretical entropy should be equal to the characteristic exponent [3]. As explained in [4], this is how I learned of the first example of a chaotic and strange attractor in the plane: the Hénon attractor (see Figure 1). Surprisingly, I had never before met Michel Hénon, who worked a few kilometers from the math department of the university at the Nice Observatory.

<figure><img src="image_5.png" /><figcaption>Figure 1. Hénon map for the parameter value $a = 1.2$, $b = 0.42$, initial value $x_0 = 0$, $y_0 = 0$.</figcaption></figure>

Hénon used an IBM-7040 computer to numerically simulate the Lorenz map [5]. He found it difficult to highlight its inner nature due to the very high dissipativity of this map, for which the rate of volume contraction is provided by the Lie derivative of the Lorenz equations. Solving such equations for the parameters chosen by Lorenz provides $V(t) = V(0)e^{-\frac{t}{3}}$. Hence, volumes are reduced by a factor $10^6$ after one time unit. Hénon, who was a specialist in the motion of planets, had good experience with Poincaré's map, which inspired him the way to build the metaphoric model in [6]:

$$ \begin{cases} x_{n+1} = 1 - ax_n^2 + y_n, \\ y_{n+1} = bx_n. \end{cases} \qquad (1) $$
This model can be represented by the iterates of any initial point $(x_0, y_0)^T$ belonging to the basin of attraction using the map $\mathcal{H}_{a,b}: \mathbb{R}^2 \to \mathbb{R}^2$

$$ \mathcal{H}_{a,b}\left(\begin{array}{c} x \\ y \end{array}\right)=\left(\begin{array}{c} 1-ax^{2}+y, \\ bx. \end{array}\right). \qquad (2) $$

The contracting properties of this map are only determined by the parameter $b$. When $b = 0.3$, the contraction in one iteration is mild enough that the sheaves of the attractors are distinguishable on a graphic. Finally, Hénon observed the astonishing fractal structure of the attractor on the image drawn by the computer, which amazed the research community.

At that time, I was a young researcher trying to understand bifurcation theory in the field of numerical analysis [7]. This area was not very well known in France. I was working on the discretization of nonlinear problems using the finite element method, in which nonlinear functions are approximated by piecewise linear functions.

During the Roma conference, I tried without success to apply the spirit of the method of finite elements to the Hénon attractor. Back in Nice after this conference, I eventually decided, using paper and pencil, to change the square function of the Hénon attractor, which is U shaped, into the absolute value function, which has a V shape, implying a folding property (the folding property is important for horseshoes, a main ingredient of chaos, as highlighted by Stephen Smale [8]).

I was owner of a small desktop computer (an HP 9820) linked to an HP 9862 plotter. Immediately after the defense I tested this change of function and found a similar attractor of the Hénon attractor, with straight lines instead of curves for the map [9] (see Figure 2).

$$ \mathcal{L}_{a,b}\left(\begin{array}{c} x \\ y \end{array}\right)=\left(\begin{array}{c} 1-a|x|+y, \\ bx. \end{array}\right) \qquad (3) $$

<figure><img src="image_5.png" /><figcaption>Figure 2. Original Lozi map in dimension 2 for the parameter value $a = 1.62, b = 0.56$, initial value $x_0 = 0, y_0 = 0$.</figcaption></figure>

## 2.2. Classic Chaotic Properties of the Dissipative Map ($|b| < 1$)

The Lozi map is called “dissipative” when $|b| < 1$, which means that the image of any subset $\Delta$ by $f$ has a measure which is less than the measure of $\Delta: \text{measure}(f(\Delta)) < \text{measure}(\Delta)$.

### 2.2.1. Domain of Existence of Strange Attractor

In 1979, during the International Conference on Nonlinear Dynamics, patronized by the New York Academy of Sciences from 17–21 December, Michal Misiurewicz, in a plenary talk, presented a rigorous proof that for a set of parameters $M$ this map has a strange attractor, coining on this occasion the name “Lozi map” [10]. This happened only two years
This model can be represented by the iterates of any initial point $(x_0, y_0)^T$ belonging to the basin of attraction using the map $\mathcal{H}_{a,b}: \mathbb{R}^2 \to \mathbb{R}^2$

$$ \mathcal{H}_{a,b}\left(\begin{array}{c} x \\ y \end{array}\right)=\left(\begin{array}{c} 1-ax^{2}+y, \\ bx. \end{array}\right). \qquad (2) $$

The contracting properties of this map are only determined by the parameter $b$. When $b = 0.3$, the contraction in one iteration is mild enough that the sheaves of the attractors are distinguishable on a graphic. Finally, Hénon observed the astonishing fractal structure of the attractor on the image drawn by the computer, which amazed the research community.

At that time, I was a young researcher trying to understand bifurcation theory in the field of numerical analysis [7]. This area was not very well known in France. I was working on the discretization of nonlinear problems using the finite element method, in which nonlinear functions are approximated by piecewise linear functions.

During the Roma conference, I tried without success to apply the spirit of the method of finite elements to the Hénon attractor. Back in Nice after this conference, I eventually decided, using paper and pencil, to change the square function of the Hénon attractor, which is U shaped, into the absolute value function, which has a V shape, implying a folding property (the folding property is important for horseshoes, a main ingredient of chaos, as highlighted by Stephen Smale [8]).

I was owner of a small desktop computer (an HP 9820) linked to an HP 9862 plotter. Immediately after the defense I tested this change of function and found a similar attractor of the Hénon attractor, with straight lines instead of curves for the map [9] (see Figure 2).

$$ \mathcal{L}_{a,b}\left(\begin{array}{c} x \\ y \end{array}\right)=\left(\begin{array}{c} 1-a|x|+y, \\ bx. \end{array}\right) \qquad (3) $$

<figure><img src="image_6.png" /><figcaption>Figure 2. Original Lozi map in dimension 2 for the parameter value $a = 1.62, b = 0.56$, initial value $x_0 = 0, y_0 = 0$.</figcaption></figure>

## 2.2. Classic Chaotic Properties of the Dissipative Map ($|b| < 1$)

The Lozi map is called “dissipative” when $|b| < 1$, which means that the image of any subset $\Delta$ by $f$ has a measure which is less than the measure of $\Delta: measure(f(\Delta)) < measure(\Delta)$.

### 2.2.1. Domain of Existence of Strange Attractor

In 1979, during the International Conference on Nonlinear Dynamics, patronized by the New York Academy of Sciences from 17–21 December, Michal Misiurewicz, in a plenary talk, presented a rigorous proof that for a set of parameters $M$ this map has a strange attractor, coining on this occasion the name “Lozi map” [10]. This happened only two years
after my discovery of this new chaotic attractor, and only one year after its publication [9]. This set in the plane of parameters (a, b) is defined by

$$ \mathcal{M} = \left\{ (a, b) \mid b > 0, a\sqrt{2} < b + 2, b < \frac{a^2 - 1}{2a + 1}, 2a + b < 4 \right\} \quad (4) $$

(see Figure 3).

<figure><img src="image_6.png" /><figcaption>Figure 3. The set $\mathcal{M}$ of the plane of parameters in which the Lozi map has a strange attractor, following the first proof of Misiurewicz (4). Pink lines: $b = 0$ and $b = 1$, blue line: $b = a-1$, green line: $b = 4-2a$, brown line: $b = a\sqrt{2}-2$, red curve: $b = \frac{a^2-1}{2a+1}$.</figcaption></figure>

Later, Misiurewicz and Stimac [11] slightly enlarged this set of parameter $\mathcal{M}$ (defined only for $b > 0$, and as $\mathcal{M}^+$ in Figure 4) in the upper edge of the triangle by removing the condition $b < \frac{a^2-1}{2a+1}$.

$$ \mathcal{M}^+ = \left\{ (a, b) \mid b > 0, a\sqrt{2} < b + 2, 2a + b < 4 \right\} \quad (5) $$

The case $b < 0$ was studied by Cao and Liu [12], and recently by Kucharski [13], who found a quasi-symmetric set of parameters $\mathcal{M}^-$ (Figure 4). However, the right-hand boundary of $\mathcal{M}^-$ is not completely symmetric to the right-hand boundary of $\mathcal{M}^+$.

$$ \mathcal{M}^- = \left\{ (a, b) \mid -1 < b < 0, a\sqrt{2} > b + 2, b > \frac{-8a + 3a^2 + \sqrt{-16a^3 + 9a^4}}{8} \right\} \quad (6) $$

### 2.2.2. Fixed Points, Invariant Manifolds, and Basin of Attraction

It is straightforward to explicitly compute the fixed points and periodic orbits of any order of $\mathcal{L}_{a,b}$ (3) due to the piecewise linearity of the absolute value.

When $1 - a < b < a + 1$ (for $a > 0$, the regions $\mathcal{M}^-$ and $\mathcal{M}^+$ are both in the cone defined by these inequalities), there exist two fixed points:

$$ \mathcal{P}^+ = \left( \frac{1}{a+1-b'}, \frac{b}{a+1-b} \right), \quad \mathcal{P}^- = \left( \frac{1}{1-a-b'}, \frac{b}{1-a-b} \right). $$

$\mathcal{P}^+$ belongs to the quadrant $\{x \ge 0, y \ge 0\}$ and $\mathcal{P}^-$ to the quadrant $\{x < 0, y < 0\}$.

For $a > |b| + 1$, that is, in the domain of the parameter space where both $\mathcal{P}^+$ and $\mathcal{P}^-$ coexist, it is easy to compute the local stability of these points by evaluating the corresponding eigenvalues of the Jacobian matrix of $\mathcal{L}_{a,b}$. It can be shown that both points are saddles.
after my discovery of this new chaotic attractor, and only one year after its publication [9]. This set in the plane of parameters (a, b) is defined by

$$ \mathcal{M} = \left\{ (a, b) \mid b > 0, a\sqrt{2} < b + 2, b < \frac{a^2 - 1}{2a + 1}, 2a + b < 4 \right\} \quad (4) $$

(see Figure 3).

<figure><img src="image_7.png" /><figcaption>Figure 3. The set $\mathcal{M}$ of the plane of parameters in which the Lozi map has a strange attractor, following the first proof of Misiurewicz (4). Pink lines: $b = 0$ and $b = 1$, blue line: $b = a-1$, green line: $b = 4-2a$, brown line: $b = a\sqrt{2}-2$, red curve: $b = \frac{a^2-1}{2a+1}$.</figcaption></figure>

Later, Misiurewicz and Stimac [11] slightly enlarged this set of parameter $\mathcal{M}$ (defined only for $b > 0$, and as $\mathcal{M}^+$ in Figure 4) in the upper edge of the triangle by removing the condition $b < \frac{a^2-1}{2a+1}$.

$$ \mathcal{M}^+ = \left\{ (a, b) \mid b > 0, a\sqrt{2} < b + 2, 2a + b < 4 \right\} \quad (5) $$

The case $b < 0$ was studied by Cao and Liu [12], and recently by Kucharski [13], who found a quasi-symmetric set of parameters $\mathcal{M}^-$ (Figure 4). However, the right-hand boundary of $\mathcal{M}^-$ is not completely symmetric to the right-hand boundary of $\mathcal{M}^+$.

$$ \mathcal{M}^- = \left\{ (a, b) \mid -1 < b < 0, a\sqrt{2} > b + 2, b > \frac{-8a + 3a^2 + \sqrt{-16a^3 + 9a^4}}{8} \right\} \quad (6) $$

### 2.2.2. Fixed Points, Invariant Manifolds, and Basin of Attraction

It is straightforward to explicitly compute the fixed points and periodic orbits of any order of $\mathcal{L}_{a,b}$ (3) due to the piecewise linearity of the absolute value.

When $1 - a < b < a + 1$ (for $a > 0$, the regions $\mathcal{M}^-$ and $\mathcal{M}^+$ are both in the cone defined by these inequalities), there exist two fixed points:

$$ \mathcal{P}^+ = \left( \frac{1}{a+1-b'}, \frac{b}{a+1-b} \right), \quad \mathcal{P}^- = \left( \frac{1}{1-a-b'}, \frac{b}{1-a-b} \right). $$

$\mathcal{P}^+$ belongs to the quadrant $\{x \ge 0, y \ge 0\}$ and $\mathcal{P}^-$ to the quadrant $\{x < 0, y < 0\}$.

For $a > |b| + 1$, that is, in the domain of the parameter space where both $\mathcal{P}^+$ and $\mathcal{P}^-$ coexist, it is easy to compute the local stability of these points by evaluating the corresponding eigenvalues of the Jacobian matrix of $\mathcal{L}_{a,b}$. It can be shown that both points are saddles.
<figure><img src="image_7.png" /><figcaption>Figure 4. The sets $M^+$ and $M^-$ for the plane of parameters where the Lozi map has a strange attractor in the second proof of Misiurewicz and Stimac (5) and the proof of Kucharski (6). Blue lines: $b = a-1$ and $b = 1-a$, red lines: $b = a\sqrt{2}-2$ and $b = 2-a\sqrt{2}$, cyan line: $b = 4-2a$, dark green line: $b = \frac{-8a+3a^2+\sqrt{-16a^3+9a^4}}{8}$.</figcaption></figure>

Moreover, Misiurewicz [10] showed that the chaotic attractor of this map belongs to the unstable invariant manifold of $\mathcal{P}^+$. More exactly, the chaotic (and strange) attractor $\tilde{\mathcal{S}}$ can be constructed from the successive forward iterations of a trapping triangular region $T$,

$$ \tilde{\mathcal{S}} = \bigcap_{n=0}^{\infty} \mathcal{L}^n(T), $$

with $T$ being the triangle with vertices at the points $I$, $\mathcal{L}(I)$, and $\mathcal{L}^2(I)$, where $I$ is the point given by the intersection of the unstable manifold of the fixed point $\mathcal{P}^+$ with the horizontal axis

$$ I = \left( \frac{2 + a + \sqrt{a^2 + 4b}}{2(1 + a - b)}, 0 \right). $$

The basin of attraction of the attractor is modeled by parts of the stable manifold of the fixed point $\mathcal{P}^-$, as shown by Baptista et al. [14]. They considered the point $X$ intersection of this stable manifold with the vertical axis. Straightforward computation provides

$$ X = \left( 0, \frac{b(2 - a + \sqrt{a^2 + 4b})}{(a - \sqrt{a^2 + 4b})(a + b - 1)} \right). $$

They considered certain point $U$ belonging on the horizontal axis as well, first defined by Ishii [15], for which the expression is

$$ U = \left( \frac{b(2 - a + \sqrt{a^2 + 4b}(a(1 + b) + (1 - b)\sqrt{a^2 + 4b})}{(1 - a - b)(a - \sqrt{a^2 + 4b})(a - \sqrt{a^2 + 4b})(2b(b - 1) + a^2(1 + 2b) + a(1 - 2b)\sqrt{a^2 + 4b})}, 0 \right). $$

Baptista et al. showed that the basin of attraction is bounded by a polygonal line entirely characterized by the points $U$ and $X$ along with their successive preimages.

### 2.3. Advanced Dynamical Properties of the Dissipative Map ($|b| < 1$)

An open region $B$ in the parameter space for which the Lozi map exhibits periodic points of least period $n$ was rigorously determined for all $n > 13$ by Boroński, Kucharski and Ou [16] as follows:

$$ \mathcal{P}_{\text{str}} = \{(a, b) | b > 0, b < l_{\text{str}}(a) \text{ when } a < a_0, \text{ and } b < l_{\text{tan}}(a) \text{ when } a > a_0\} \quad (7) $$
<figure><img src="image_8.png" /><figcaption>Figure 4. The sets $M^+$ and $M^-$ for the plane of parameters where the Lozi map has a strange attractor in the second proof of Misiurewicz and Stimac (5) and the proof of Kucharski (6). Blue lines: $b = a-1$ and $b = 1-a$, red lines: $b = a\sqrt{2}-2$ and $b = 2-a\sqrt{2}$, cyan line: $b = 4-2a$, dark green line: $b = \frac{-8a+3a^2+\sqrt{-16a^3+9a^4}}{8}$.</figcaption></figure>

Moreover, Misiurewicz [10] showed that the chaotic attractor of this map belongs to the unstable invariant manifold of $\mathcal{P}^+$. More exactly, the chaotic (and strange) attractor $\tilde{\mathcal{S}}$ can be constructed from the successive forward iterations of a trapping triangular region $T$,

$$ \tilde{\mathcal{S}} = \bigcap_{n=0}^{\infty} \mathcal{L}^n(T), $$

with $T$ being the triangle with vertices at the points $I$, $\mathcal{L}(I)$, and $\mathcal{L}^2(I)$, where $I$ is the point given by the intersection of the unstable manifold of the fixed point $\mathcal{P}^+$ with the horizontal axis

$$ I = \left( \frac{2 + a + \sqrt{a^2 + 4b}}{2(1 + a - b)}, 0 \right). $$

The basin of attraction of the attractor is modeled by parts of the stable manifold of the fixed point $\mathcal{P}^-$, as shown by Baptista et al. [14]. They considered the point $X$ intersection of this stable manifold with the vertical axis. Straightforward computation provides

$$ X = \left( 0, \frac{b(2 - a + \sqrt{a^2 + 4b})}{(a - \sqrt{a^2 + 4b})(a + b - 1)} \right). $$

They considered certain point $U$ belonging on the horizontal axis as well, first defined by Ishii [15], for which the expression is

$$ U = \left( \frac{b(2 - a + \sqrt{a^2 + 4b}(a(1 + b) + (1 - b)\sqrt{a^2 + 4b})}{(1 - a - b)(a - \sqrt{a^2 + 4b})(a - \sqrt{a^2 + 4b})(2b(b - 1) + a^2(1 + 2b) + a(1 - 2b)\sqrt{a^2 + 4b})}, 0 \right). $$

Baptista et al. showed that the basin of attraction is bounded by a polygonal line entirely characterized by the points $U$ and $X$ along with their successive preimages.

### 2.3. Advanced Dynamical Properties of the Dissipative Map ($|b| < 1$)

An open region $B$ in the parameter space for which the Lozi map exhibits periodic points of least period $n$ was rigorously determined for all $n > 13$ by Boroński, Kucharski and Ou [16] as follows:

$$ \mathcal{P}_{\text{str}} = \{(a, b) | b > 0, b < l_{\text{str}}(a) \text{ when } a < a_0, \text{ and } b < l_{\text{tan}}(a) \text{ when } a > a_0\} \quad (7) $$
and

$$P_3(a, b) = \{(a, b) | 0 \le b \le 1, a > l_{hyp}(b), \text{ and } a > l_3(b)\} \quad (8)$$

where $a_0 = \frac{2}{7}(2+3\sqrt{2}) \approx 1.78$, $l_{str}(a) = -2+\sqrt{2}a$, $l_{tan}(a) = \frac{1}{8}(8a-3a^2-\sqrt{9a^4-16a^3})$, $l_{hyp}(b) = b+1$, and $l_3(b) = \frac{1}{2}[1-b+\sqrt{(1-b)^2+4(1-b+b^2)}]$.

**Theorem 1** ([16]). For all $(a, b) \in \mathcal{B} = P_{str} \cap P_3$ the Lozi map $\mathcal{L}_{a,b}$ has a periodic point of least period $n$ for $n = 1, 2, 3$ and all $n > 13$.

A survey of all the papers describing the other dynamical or statistical properties of the Lozi map cannot be provided in this article because they are too numerous. Readers may refer to [1] for a compendium of results published between 1997 to 2013. Below, only a selection of particular and more recent results are pointed out.

Several new results on bifurcations have been recently published since 2013. Botella-Soler et al. [17] showed that this map presents what they call *bisecting bifurcations*, that is, those which are mediated by an infinite set of neutrally stable periodic orbits (see Figures 5 and 6).

<figure><img src="image_8.png" /><figcaption>Figure 5. Bifurcation diagram for $b = 0.1$. Notice that the bisecting bifurcations take place at $a = 1 \pm b$.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Figure 6. Bifurcation diagram for $a = 1.5$. Notice that the bisecting bifurcations take place at $b = \pm(a-1)$.</figcaption></figure>

Sushko et al. [18] investigated the bifurcation structure of the parameter plane in the vicinity of the curve related to a center bifurcation of the fixed point. A distinguishing property of the Lozi map is that it is conservative (see Section 2.4) at the parameter value
and

$$ \mathcal{P}_3(a,b) = \{(a,b) | 0 \le b \le 1, a > l_{hyp}(b), \text{ and } a > l_3(b)\} \quad (8) $$

where $a_0 = \frac{2}{7}(2+3\sqrt{2}) \approx 1.78$, $l_{str}(a) = -2+\sqrt{2}a$, $l_{tan}(a) = \frac{1}{8}(8a-3a^2-\sqrt{9a^4-16a^3})$, $l_{hyp}(b) = b+1$, and $l_3(b) = \frac{1}{2}[1-b+\sqrt{(1-b)^2+4(1-b+b^2)}]$.

**Theorem 1** ([16]). For all $(a, b) \in \mathcal{B} = \mathcal{P}_{str} \cap \mathcal{P}_3$ the Lozi map $\mathcal{L}_{a,b}$ has a periodic point of least period $n$ for $n = 1, 2, 3$ and all $n > 13$.

A survey of all the papers describing the other dynamical or statistical properties of the Lozi map cannot be provided in this article because they are too numerous. Readers may refer to [1] for a compendium of results published between 1997 to 2013. Below, only a selection of particular and more recent results are pointed out.

Several new results on bifurcations have been recently published since 2013. Botella-Soler et al. [17] showed that this map presents what they call *bisecting bifurcations*, that is, those which are mediated by an infinite set of neutrally stable periodic orbits (see Figures 5 and 6).

<figure><img src="image_9.png" /><figcaption>Figure 5. Bifurcation diagram for $b = 0.1$. Notice that the bisecting bifurcations take place at $a = 1 \pm b$.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Figure 6. Bifurcation diagram for $a = 1.5$. Notice that the bisecting bifurcations take place at $b = \pm(a-1)$.</figcaption></figure>

Sushko et al. [18] investigated the bifurcation structure of the parameter plane in the vicinity of the curve related to a center bifurcation of the fixed point. A distinguishing property of the Lozi map is that it is conservative (see Section 2.4) at the parameter value
corresponding to this bifurcation. As a result, the bifurcation structure close to the center bifurcation curve is quite complicated. In particular, an attracting fixed point (focus) can coexist with various attracting cycles as well as with chaotic attractors, and the number of coexisting attractors increases as the parameter point approaches the center bifurcation curve. Their study contributed to the border collision bifurcation theory as well, as the Lozi map is a particular case of the 2D border collision normal form (2D-BCNF).

The following 2D-BCNF family of difference equations $(x, y) \rightarrow h(x, y)$ was presented by Glendinning and Simpson [19] as a canonical example:

$$h(x, y) = \begin{cases} F_L \begin{bmatrix} x \\ y \end{bmatrix} + \begin{bmatrix} \mu \\ 0 \end{bmatrix}, & x \le 0, \\ F_R \begin{bmatrix} x \\ y \end{bmatrix} + \begin{bmatrix} \mu \\ 0 \end{bmatrix}, & x \ge 0, \end{cases} \qquad (9)$$

where the parameter $\mu$ controls the border-collision bifurcation and with

$$F_L = \begin{bmatrix} \pi_L & 1 \\ -\nu_L & 0 \end{bmatrix}, \quad F_R = \begin{bmatrix} \pi_R & 1 \\ -\nu_R & 0 \end{bmatrix}.$$

In their paper, they restricted their attention to the parameter values $\pi_R \in \mathbb{R}$, $\pi_L > 0$, $\nu_L > 0$, $\nu_R > 0$, $\mu = 1$, for which $f$ is invertible and orientation-preserving.

The role of $\mu$ is to control the border-collision bifurcation. In view of a linear rescaling, it is only necessary to consider it for the values $\{-1, 0, 1\}$ (here, it is 1). The condition $\pi_L > 0$ is needed for the definition of the induced map. If $\pi_L = -\pi_R$ and $\nu_L = \nu_L$, then the 2D BCNF reduces to the Lozi map.

In addition bifurcation studies, Collet and Levy [20] considered ergodic properties of the Lozi map, which they considered as an intermediate stage between the Axiom A dynamical systems and more complicated systems such as the Hénon map. They constructed its Bowen-Ruelle measure and derived several of its properties, which are similar to those of an axiom A system.

A proof of the existence of Sinai-Bowen-Ruelle measures (SBR measures) for this map was provided by Rychlik [21], who additionally proved that the number of SBR measures is finite.

The geometric structure of the chaotic attractor was explored by Cao and Liu [22], who proved the following:

**Proposition 1.** If the parameters $(a, b)$ satisfy the Misiurewicz conditions (4), then the strange attractor $\Lambda_{a,b}$ possesses the following properties:

- The union of the transversal homoclinic points and weak transversal homoclinic points is dense in $\Lambda_{a,b}$

- All periodic points are hyperbolic

- The set of periodic points forms a dense set in $\Lambda_{a,b}$

- Any two hyperbolic points forms a transversal heteroclinic cycle or a weak transversal heteroclinic cycle

Other statistical, hyperbolic, ergodic, and topological properties of the Lozi attractor can be found in Afraimovich et al. [23].

The symbolic dynamics of this map have been extensively studied. In 1991, Zheng [24] described details of this dynamics. To this end, they assigned two families of symbolic sequences for two groups of lines in the phase plane, allowing the order of symbolic sequences to be defined, and derived the ordering rules. Misiurewicz and Stimac [11], in a more detailed study, proved that the set of kneading sequences they introduced determines its symbolic dynamics. They defined two other equivalent approaches as well. The important works of Ishii [15,25], Sand [26], and de Carvalho and Hall [27] in this field of research can be mentioned here as well.
corresponding to this bifurcation. As a result, the bifurcation structure close to the center bifurcation curve is quite complicated. In particular, an attracting fixed point (focus) can coexist with various attracting cycles as well as with chaotic attractors, and the number of coexisting attractors increases as the parameter point approaches the center bifurcation curve. Their study contributed to the border collision bifurcation theory as well, as the Lozi map is a particular case of the 2D border collision normal form (2D-BCNF).

The following 2D-BCNF family of difference equations $(x, y) \rightarrow h(x, y)$ was presented by Glendinning and Simpson [19] as a canonical example:

$$h(x, y) = \begin{cases} F_L \begin{bmatrix} x \\ y \end{bmatrix} + \begin{bmatrix} \mu \\ 0 \end{bmatrix}, & x \le 0, \\ F_R \begin{bmatrix} x \\ y \end{bmatrix} + \begin{bmatrix} \mu \\ 0 \end{bmatrix}, & x \ge 0, \end{cases} \quad (9)$$

where the parameter $\mu$ controls the border-collision bifurcation and with

$$F_L = \begin{bmatrix} \pi_L & 1 \\ -\nu_L & 0 \end{bmatrix}, \quad F_R = \begin{bmatrix} \pi_R & 1 \\ -\nu_R & 0 \end{bmatrix}.$$

In their paper, they restricted their attention to the parameter values $\pi_R \in \mathbb{R}$, $\pi_L > 0$, $\nu_L > 0$, $\nu_R > 0$, $\mu = 1$, for which $f$ is invertible and orientation-preserving.

The role of $\mu$ is to control the border-collision bifurcation. In view of a linear rescaling, it is only necessary to consider it for the values $\{-1, 0, 1\}$ (here, it is 1). The condition $\pi_L > 0$ is needed for the definition of the induced map. If $\pi_L = -\pi_R$ and $\nu_L = \nu_L$, then the 2D BCNF reduces to the Lozi map.

In addition bifurcation studies, Collet and Levy [20] considered ergodic properties of the Lozi map, which they considered as an intermediate stage between the Axiom A dynamical systems and more complicated systems such as the Hénon map. They constructed its Bowen-Ruelle measure and derived several of its properties, which are similar to those of an axiom A system.

A proof of the existence of Sinai-Bowen-Ruelle measures (SBR measures) for this map was provided by Rychlik [21], who additionally proved that the number of SBR measures is finite.

The geometric structure of the chaotic attractor was explored by Cao and Liu [22], who proved the following:

**Proposition 1.** If the parameters $(a, b)$ satisfy the Misiurewicz conditions (4), then the strange attractor $\Lambda_{a,b}$ possesses the following properties:

- The union of the transversal homoclinic points and weak transversal homoclinic points is dense in $\Lambda_{a,b}$

- All periodic points are hyperbolic

- The set of periodic points forms a dense set in $\Lambda_{a,b}$

- Any two hyperbolic points forms a transversal heteroclinic cycle or a weak transversal heteroclinic cycle

Other statistical, hyperbolic, ergodic, and topological properties of the Lozi attractor can be found in Afraimovich et al. [23].

The symbolic dynamics of this map have been extensively studied. In 1991, Zheng [24] described details of this dynamics. To this end, they assigned two families of symbolic sequences for two groups of lines in the phase plane, allowing the order of symbolic sequences to be defined, and derived the ordering rules. Misiurewicz and Stimac [11], in a more detailed study, proved that the set of kneading sequences they introduced determines its symbolic dynamics. They defined two other equivalent approaches as well. The important works of Ishii [15,25], Sand [26], and de Carvalho and Hall [27] in this field of research can be mentioned here as well.
In [15], Ishii constructed a kneading theory à la Milnor-Thurston and showed that topological properties of the dynamics of the Lozi map are determined by its pruning front and primary pruned region only. This allowed a solution to the first tangency problem for the Lozi family to be found. In addition, the boundary of the set of all horseshoes in the parameter space was shown to be algebraic. In [25], the partial monotonicity of the topological entropy and of bifurcations near horseshoes was proved as an application of this result. Upper and lower bounds for the Hausdorff dimension of the Lozi attractor were provided in terms of parameters. In [27], recent results on pruning theory were provided, concentrated on prunings of the horseshoe. The monotonicity of the Lozi family when the Jacobian determinant is close to zero was proved in [26]. The main ingredients of the proof therein are the “pruning pair method” and a detailed analysis of the parameter dependence of the kneading invariant of the tent-map family.

Wormell [28] defined a new property of chaotic attractors, namely, conditional mixing, and conjectured:

**Conjecture 1.** For generic Lozi parameters (a, b) and almost all Lebesgue $x_0 \in R$, the Lozi map has conditional mixing with respect to level curves $x = x_0$, that is, the measures $\rho(\cdot|x_0)$.

Strong and direct numerical evidence has been presented in favor of this conjecture. To obtain such a result rigorously, novel algorithms were developed in [28] that quantify the associated numerical error.

Borinski and Stimac [29] showed that Hénon-like and Lozi-like maps (see [2] for a definition of such maps) on their strange attractors are conjugate to natural extensions of maps on metric trees with dense set of branch points. In consequence, these trees very well approximate the topology of the attractors, and the maps on them give good models of the dynamics.

To conclude this section, I refer to the general survey of Anischenko et al. [30], which analyzed many aspects of the properties and use of the Lozi map in several of the contexts developed in this article.

## 2.4. Chaotic Properties of the Conservative Map ($|b| = 1$)

In the conservative case (called the area-preserving case) $measure(f(\Delta)) = measure(\Delta)$, there is no attractor.

In addition to the analysis of Sushko et al. [18] (see Section 2.3), this conservative map has been studied by Lopesino et al. in [31], who proved that when $a > 4$, a chaotic saddle exists in the square

$$S_q = \{(x, y) \in \mathbb{R}^2 | |x| \le R, |y| \le R\}$$

with

$$R = R(a) = \frac{a}{4(a-2)}. \qquad (10)$$

The conservative case was studied by Li et al. [32], who highlighted that it can generate initial values-related coexisting infinite orbits. Its moving orbits are extremely dependent on its initial values, and present periodic, quasi-periodic, and chaotic orbits with different types and topologies. That means that the emergence of extreme multistability appears in the area-preserving Lozi map.

As an example, several such orbits are plotted in Figure 7. Li et al. noted that the coexistence of double or multiple attractors has been found in the Hénon map, the M-dimensional nonlinear hyperchaotic model [33], and multistage DC/DC switching converters [34]. Two types of simple 2D hyperchaotic maps with sine trigonometric nonlinearity and constant controllers were shown to generate initially boosted infinite attractors along a phase line [35,36]. Recently, a simple two-dimensional Sine map was presented to obtain infinitely many initially boosted attractors along a phase plane [37]. However, the authors emphasized that all these newly presented discrete maps only exhibit coexisting attractors with different positions. It can be remarked that coexisting infinite attractors with different
In [15], Ishii constructed a kneading theory à la Milnor-Thurston and showed that topological properties of the dynamics of the Lozi map are determined by its pruning front and primary pruned region only. This allowed a solution to the first tangency problem for the Lozi family to be found. In addition, the boundary of the set of all horseshoes in the parameter space was shown to be algebraic. In [25], the partial monotonicity of the topological entropy and of bifurcations near horseshoes was proved as an application of this result. Upper and lower bounds for the Hausdorff dimension of the Lozi attractor were provided in terms of parameters. In [27], recent results on pruning theory were provided, concentrated on prunings of the horseshoe. The monotonicity of the Lozi family when the Jacobian determinant is close to zero was proved in [26]. The main ingredients of the proof therein are the “pruning pair method” and a detailed analysis of the parameter dependence of the kneading invariant of the tent-map family.

Wormell [28] defined a new property of chaotic attractors, namely, conditional mixing, and conjectured:

**Conjecture 1.** For generic Lozi parameters (a, b) and almost all Lebesgue $x_0 \in R$, the Lozi map has conditional mixing with respect to level curves $x = x_0$, that is, the measures $\rho(\cdot|x_0)$.

Strong and direct numerical evidence has been presented in favor of this conjecture. To obtain such a result rigorously, novel algorithms were developed in [28] that quantify the associated numerical error.

Borinski and Stimac [29] showed that Hénon-like and Lozi-like maps (see [2] for a definition of such maps) on their strange attractors are conjugate to natural extensions of maps on metric trees with dense set of branch points. In consequence, these trees very well approximate the topology of the attractors, and the maps on them give good models of the dynamics.

To conclude this section, I refer to the general survey of Anischenko et al. [30], which analyzed many aspects of the properties and use of the Lozi map in several of the contexts developed in this article.

## 2.4. Chaotic Properties of the Conservative Map ($|b| = 1$)

In the conservative case (called the area-preserving case) $measure(f(\Delta)) = measure(\Delta)$, there is no attractor.

In addition to the analysis of Sushko et al. [18] (see Section 2.3), this conservative map has been studied by Lopesino et al. in [31], who proved that when $a > 4$, a chaotic saddle exists in the square

$$S_q = \{(x, y) \in \mathbb{R}^2 | |x| \le R, |y| \le R\}$$

with

$$R = R(a) = \frac{a}{4(a - 2)}. \qquad (10)$$

The conservative case was studied by Li et al. [32], who highlighted that it can generate initial values-related coexisting infinite orbits. Its moving orbits are extremely dependent on its initial values, and present periodic, quasi-periodic, and chaotic orbits with different types and topologies. That means that the emergence of extreme multistability appears in the area-preserving Lozi map.

As an example, several such orbits are plotted in Figure 7. Li et al. noted that the coexistence of double or multiple attractors has been found in the Hénon map, the M-dimensional nonlinear hyperchaotic model [33], and multistage DC/DC switching converters [34]. Two types of simple 2D hyperchaotic maps with sine trigonometric nonlinearity and constant controllers were shown to generate initially boosted infinite attractors along a phase line [35,36]. Recently, a simple two-dimensional Sine map was presented to obtain infinitely many initially boosted attractors along a phase plane [37]. However, the authors emphasized that all these newly presented discrete maps only exhibit coexisting attractors with different positions. It can be remarked that coexisting infinite attractors with different
topologies and different positions in the discrete maps, such as those in the area-preserving Lozi Map, have rarely been reported.

<figure><img src="image_10.png" /><figcaption>Figure 7. Quasi-periodic and chaotic orbits of the area-preserving map (3) in the plane (x, y) when a = 1.1 and b = -1. Initial values: green (most extended chaotic orbit): x₀ = 2.1, y₀ = 2.0; red (four-ellipse quasiperiodic orbit inside the green orbit): x₀ = 2.5, y₀ = 1.0; blue (32-ellipse quasiperiodic orbit outside the green orbit): x₀ = 0.87, y₀ = 4.36; magenta (chaotic orbit): x₀ = 0.78, y₀ = 5.05; red (nine-ellipse quasiperiodic orbit inside the magenta chaotic orbit): x₀ = 1.2, y₀ = 6.5; light blue (nine-ellipse quasiperiodic orbit inside the nine red ellipses): x₀ = 2.0, y₀ = 6.0.</figcaption></figure>

Li and Gu [38] studied the hierarchical structure of a chaotic sea and the existence condition and delimitation method of island chains for this map. They found the cause of the formation of two types of island chains associated with the elliptic period. The type of island chain depends on the rotation number of the linear matrix that defines the composite mapping (corresponding period).

## 2.5. Fractional Lozi Map

During the last four decades, the Lozi map has been generalized in many ways (for a survey, see [2]). Only its generalizations in the recent paradigm of fractional mappings are reported here, as this is a natural extension of the theory of fractal ordinary differential equations; in this context, the fractional Lozi map is used for electrical devices called memristors (see Section 3.3.2).

### 2.5.1. Classic Fractional Lozi Map

The first published example of fractional map used a constant fractional order v. Using the Caputo-like delta difference

$$ 
\begin{aligned}
{}^c\Delta_a^v X(t) &= \Delta_a^{-(n-v)} \Delta^n X(t), \\
&= \frac{1}{\Gamma(n-v)} \sum_{s=a}^{t-(n-v)} (t-s-1)^{(n-v-s)} \Delta^n X(t),
\end{aligned}
$$

Khennaoui et al. [39] defined the fractal Lozi map as follows:

$$ 
\left\{ 
\begin{aligned}
{}^c\Delta_a^v x(t) &= \alpha|x(t-1+v)| + y(t-1+v) + 1 - x(t-1+v), \\
{}^c\Delta_a^v y(t) &= \beta x(t-1+v) - y(t-1+v),
\end{aligned}
\right. 
\qquad (11) $$

for 0 < v ≤ 1 and t ∈ $\mathbb{N}_{a+1-v} = \{a+1-v, a+2-v, \dots\}$. In this example the fractional order of the two fractional differences is identical. This system is commonly referred to as a
topologies and different positions in the discrete maps, such as those in the area-preserving Lozi Map, have rarely been reported.

<figure><img src="image_11.png" /><figcaption>Figure 7. Quasi-periodic and chaotic orbits of the area-preserving map (3) in the plane (x, y) when a = 1.1 and b = -1. Initial values: green (most extended chaotic orbit): x₀ = 2.1, y₀ = 2.0; red (four-ellipse quasiperiodic orbit inside the green orbit): x₀ = 2.5, y₀ = 1.0; blue (32-ellipse quasiperiodic orbit outside the green orbit): x₀ = 0.87, y₀ = 4.36; magenta (chaotic orbit): x₀ = 0.78, y₀ = 5.05; red (nine-ellipse quasiperiodic orbit inside the magenta chaotic orbit): x₀ = 1.2, y₀ = 6.5; light blue (nine-ellipse quasiperiodic orbit inside the nine red ellipses): x₀ = 2.0, y₀ = 6.0.</figcaption></figure>

Li and Gu [38] studied the hierarchical structure of a chaotic sea and the existence condition and delimitation method of island chains for this map. They found the cause of the formation of two types of island chains associated with the elliptic period. The type of island chain depends on the rotation number of the linear matrix that defines the composite mapping (corresponding period).

## 2.5. Fractional Lozi Map

During the last four decades, the Lozi map has been generalized in many ways (for a survey, see [2]). Only its generalizations in the recent paradigm of fractional mappings are reported here, as this is a natural extension of the theory of fractal ordinary differential equations; in this context, the fractional Lozi map is used for electrical devices called memristors (see Section 3.3.2).

### 2.5.1. Classic Fractional Lozi Map

The first published example of fractional map used a constant fractional order $v$. Using the Caputo-like delta difference

$$ \begin{aligned} {}^{c}\Delta_{a}^{v} X(t) &= \Delta_{a}^{-(n-v)} \Delta^{n} X(t), \\ &= \frac{1}{\Gamma(n-v)} \sum_{s=a}^{t-(n-v)} (t-s-1)^{(n-v-s)} \Delta^{n} X(t), \end{aligned} $$

Khennaoui et al. [39] defined the fractal Lozi map as follows:

$$ \left\{ \begin{aligned} {}^{c}\Delta_{a}^{v}x(t) &= \alpha|x(t-1+v)| + y(t-1+v) + 1 - x(t-1+v), \\ {}^{c}\Delta_{a}^{v}y(t) &= \beta x(t-1+v) - y(t-1+v), \end{aligned} \right. \qquad (11) $$

for $0 < v \le 1$ and $t \in \mathbb{N}_{a+1-v} = \{a+1-v, a+2-v, \dots\}$. In this example the fractional order of the two fractional differences is identical. This system is commonly referred to as a
commensurate system. The following equivalent discrete integral equation of such a map can be obtained:

$$ \left\{ \begin{aligned} x(t) &= x(a) + \frac{1}{\Gamma(\nu)} \sum_{s=a+1}^{t-\nu} (t-s-1)^{(\nu-1)} [-\alpha|x(t-1+\nu)| + y(t-1+\nu) + 1 - x(t-1+\nu)], \\ y(t) &= y(a) + \frac{1}{\Gamma(\nu)} \sum_{s=a-\nu}^{t-\nu} (t-s-1)^{(\nu-1)} [\beta x(t-1+\nu) - y(t-1+\nu)], \end{aligned} \right. \quad (12) $$

where $\frac{(t-s-1)^{(\nu-1)}}{\Gamma(\nu)}$ is the discrete kernel function

$$ \frac{(t-s-1)^{(\nu-1)}}{\Gamma(\nu)} = \frac{\Gamma(t-s)}{\Gamma(\nu)\Gamma(t-s-\nu+1)} $$

and $a = 0$ yields the numerical formula

$$ \left\{ \begin{aligned} x(n) &= x(0) + \frac{1}{\Gamma(\nu)} \sum_{j=1}^{n} \frac{\Gamma(n-j+\nu)}{\Gamma(n-j+1)} [-\alpha|x(j-1)| + y(j-1) + 1 - x(j-1)], \\ y(n) &= y(0) + \frac{1}{\Gamma(\nu)} \sum_{j=1}^{n} \frac{\Gamma(n-j+\nu)}{\Gamma(n-j+1)} [\beta x(j-1) - y(j-1)]. \end{aligned} \right. \quad (13) $$

Extended numerical simulations of this fractional map have shown that the value of the fractional order $\nu$ modifies the general shape of the bifurcation diagram (compared to the non-fractional map) as well as the length of the chaotic interval. If $\nu = 0.98$, the bifurcation diagram appears similar the corresponding integer diagram, with the exception of a small widening in the interval where chaos is observed. For smaller values of $\nu$, when $0 \le \alpha \le 0.5$ the orbit no longer goes to a fixed point, unlike the non-fractional map. In fact, as $n$ increases it can be observed that the trajectory becomes unbounded. A major difference between the bifurcation diagram of the integer and fractional maps concerns the interval over which chaos is observed, which becomes slightly smaller as $\nu$ decreases.

Ibrahim and Baleanu [40] used linear interpolation of quadratic and absolute value functions to define and study a combined Hénon–Lozi fractional map.

### 2.5.2. Variable Fractional Lozi Map

Several authors, including Al-Saidi et al. [41], have extended the definition of a fractional map to the case where the order of derivation is variable. To this end, they considered the function $\zeta(\nu)$ instead of $\nu$ in Section 2.5.1:

$$ \begin{aligned} {}^{c}\Delta_{a}^{\zeta(\nu)} X(t) &= \Delta_{a}^{-(n-\zeta(\nu))} \Delta^{n} X(t), \\ &= \frac{1}{\Gamma(n-\zeta(\nu))} \sum_{s=a}^{t-(n-\zeta(\nu))} (t-s-1)^{(n-\zeta(\nu)-s)} \Delta^{n} X(t) \end{aligned} $$

Hence, the variable fractional Lozi map (VFLM) is defined as

$$ \left\{ \begin{aligned} {}^{c}\Delta_{a}^{\zeta(\nu)} x(t) &= \alpha |x(t-1+\zeta(\nu))| + y(t-1+\zeta(\nu)) + 1 - x(t-1+\zeta(\nu)), \\ {}^{c}\Delta_{a}^{\zeta(\nu)} y(t) &= \beta x(t-1+\zeta(\nu)) - y(t-1+\zeta(\nu)), \end{aligned} \right. \quad (14) $$

and for numerical values, when $a = 0$ we obtain

$$ \left\{ \begin{aligned} x(n) &= x(0) + \frac{1}{\Gamma(\zeta(\nu))} \sum_{j=1}^{n} \frac{\Gamma(n-j+\zeta(\nu))}{\Gamma(n-j+1)} [-\alpha|x(j-1)| + y(j-1) + 1 - x(j-1)], \\ y(n) &= y(0) + \frac{1}{\Gamma(\zeta(\nu))} \sum_{j=1}^{n} \frac{\Gamma(n-j+\zeta(\nu))}{\Gamma(n-j+1)} [\beta x(j-1) - y(j-1)]. \end{aligned} \right. \quad (15) $$

Al-Saidi et al. studied the partial dynamics of such VFLMs. Additionally, they looked into the prerequisites that the variable fractional dynamic systems must meet in order to be stable and asymptotically stable. To obtain stable and asymptotically stable zero solutions, they imposed several essential requirements on the VFLM. To stabilize the system, they suggested combining these maps with control rules.
commensurate system. The following equivalent discrete integral equation of such a map can be obtained:

$$ \left\{ \begin{aligned} x(t) &= x(a) + \frac{1}{\Gamma(\nu)} \sum_{s=a+1}^{t-\nu} (t-s-1)^{(\nu-1)} [-\alpha|x(t-1+\nu)| + y(t-1+\nu) + 1 - x(t-1+\nu)], \\ y(t) &= y(a) + \frac{1}{\Gamma(\nu)} \sum_{s=a-\nu}^{t-\nu} (t-s-1)^{(\nu-1)} [\beta x(t-1+\nu) - y(t-1+\nu)], \end{aligned} \right. \quad (12) $$

where $\frac{(t-s-1)^{(\nu-1)}}{\Gamma(\nu)}$ is the discrete kernel function

$$ \frac{(t-s-1)^{(\nu-1)}}{\Gamma(\nu)} = \frac{\Gamma(t-s)}{\Gamma(\nu)\Gamma(t-s-\nu+1)} $$

and $a = 0$ yields the numerical formula

$$ \left\{ \begin{aligned} x(n) &= x(0) + \frac{1}{\Gamma(\nu)} \sum_{j=1}^{n} \frac{\Gamma(n-j+\nu)}{\Gamma(n-j+1)} [-\alpha|x(j-1)| + y(j-1) + 1 - x(j-1)], \\ y(n) &= y(0) + \frac{1}{\Gamma(\nu)} \sum_{j=1}^{n} \frac{\Gamma(n-j+\nu)}{\Gamma(n-j+1)} [\beta x(j-1) - y(j-1)]. \end{aligned} \right. \quad (13) $$

Extended numerical simulations of this fractional map have shown that the value of the fractional order $\nu$ modifies the general shape of the bifurcation diagram (compared to the non-fractional map) as well as the length of the chaotic interval. If $\nu = 0.98$, the bifurcation diagram appears similar the corresponding integer diagram, with the exception of a small widening in the interval where chaos is observed. For smaller values of $\nu$, when $0 \le \alpha \le 0.5$ the orbit no longer goes to a fixed point, unlike the non-fractional map. In fact, as $n$ increases it can be observed that the trajectory becomes unbounded. A major difference between the bifurcation diagram of the integer and fractional maps concerns the interval over which chaos is observed, which becomes slightly smaller as $\nu$ decreases.

Ibrahim and Baleanu [40] used linear interpolation of quadratic and absolute value functions to define and study a combined Hénon–Lozi fractional map.

### 2.5.2. Variable Fractional Lozi Map

Several authors, including Al-Saidi et al. [41], have extended the definition of a fractional map to the case where the order of derivation is variable. To this end, they considered the function $\zeta(\nu)$ instead of $\nu$ in Section 2.5.1:

$$ \begin{aligned} {}^{c}\Delta_{a}^{\zeta(\nu)} X(t) &= \Delta_{a}^{-(n-\zeta(\nu))} \Delta^{n} X(t), \\ &= \frac{1}{\Gamma(n-\zeta(\nu))} \sum_{s=a}^{t-(n-\zeta(\nu))} (t-s-1)^{(n-\zeta(\nu)-s)} \Delta^{n} X(t) \end{aligned} $$

Hence, the variable fractional Lozi map (VFLM) is defined as

$$ \left\{ \begin{aligned} {}^{c}\Delta_{a}^{\zeta(\nu)} x(t) &= \alpha|x(t-1+\zeta(\nu))| + y(t-1+\zeta(\nu)) + 1 - x(t-1+\zeta(\nu)), \\ {}^{c}\Delta_{a}^{\zeta(\nu)} y(t) &= \beta x(t-1+\zeta(\nu)) - y(t-1+\zeta(\nu)), \end{aligned} \right. \quad (14) $$

and for numerical values, when $a = 0$ we obtain

$$ \left\{ \begin{aligned} x(n) &= x(0) + \frac{1}{\Gamma(\zeta(\nu))} \sum_{j=1}^{n} \frac{\Gamma(n-j+\zeta(\nu))}{\Gamma(n-j+1)} [-\alpha|x(j-1)| + y(j-1) + 1 - x(j-1)], \\ y(n) &= y(0) + \frac{1}{\Gamma(\zeta(\nu))} \sum_{j=1}^{n} \frac{\Gamma(n-j+\zeta(\nu))}{\Gamma(n-j+1)} [\beta x(j-1) - y(j-1)]. \end{aligned} \right. \quad (15) $$

Al-Saidi et al. studied the partial dynamics of such VFLMs. Additionally, they looked into the prerequisites that the variable fractional dynamic systems must meet in order to be stable and asymptotically stable. To obtain stable and asymptotically stable zero solutions, they imposed several essential requirements on the VFLM. To stabilize the system, they suggested combining these maps with control rules.
### 2.5.3. Complex Fractional Map

Other authors, such as Joshi et al. [42], have considered the definition of a fractional map (13) for the case where the order of derivation $v$ is a complex number with $0 < Re(v) < 1$. They used $L_1$ to denote the model defined by (13) and $L_2$ to denote the one defined by

$$x(n) = x(0) + \frac{1}{\Gamma(v)} \sum_{j=1}^{n} \frac{\Gamma(n-j+v)}{\Gamma(n-j+1)} [-\alpha|x(j-1)| + x(j-1) + 1 - bx(j-2)]. \quad (16)$$

To systematically investigate the impact of complex order on fractional maps, they set $v = v_0e^{ir\pi/2}$ with $0 < v_0 < 1$, which reduces to the real fractional order for $r = 0$. They studied the bifurcation diagram numerically, comparing it to the bifurcation diagram of the Hénon map.

In the case of the Hénon map, they did not observe chaos for $r \neq 0$, indicating that the chaotic attractor is destroyed when complex order was introduced. They checked their results using a number of different initial conditions, finding that the bifurcation diagram does not change, which highlights that multistability is not very pronounced for the fractional order.

They found, however, that the situation is different for the Lozi map. In this case, the chaos does not disappear. They computed the bifurcation diagrams of the Lozi map for $v_0 = 0.8$ and $r = 0, 0.01, 0.1, 0.5$ for model $L_1$.

The range of variable values indicates that there are parameter zones that are chaotic, or at least periodic with a very large period. They performed other tests as well, such as finding the largest Lyapunov exponent for models $L_1$ and $L_2$ to confirm the presence of chaos (the divergence of trajectories is exponential in the $L_1$ formulation, and the calculation of the Lyapunov exponent is justified). They found that the largest exponents are 0.264 for model $L_1$ and 0.326 for model $L_2$ for the parameters mentioned above. These results confirm that the Lozi map shows chaos for the difference equation of the complex fractional order.

Moreover, they noted that extremely strange bifurcations are observed in model $L_1$, showing bifurcation diagrams for $v = 0.4 + 0.3i$ and $v = 0.4 + 0.5i$ (see Figure 7a,b of [42]) which clearly indicate the possibility of very large periods and a rich bifurcation structure that is not generally seen in integer-order systems.

In addition, they noted that the system has memory; the complex fractional ordered Lozi map shows multistability, and the asymptotic attractor depends on the initial conditions.

Finally, they studied control (see Section 5.2.1) and synchronization (see Section 7.1.6) for this map.

## 3. The Use of the Lozi Map in Electrical Devices, Memristors, and Computers

The first paradigm change in the study of the Lozi map came in 1987, nine years after the initial publication. Instead of focusing on the theoretical study of its mathematical properties or on finding generalizations, electrical engineers used it to build new devices. Subsequently and continuing up to the present day, this map has been continuously involved in the field of electronics, electrical devices, multiple access communications, memristors, and computer software.

### 3.1. *Chaos from Switched-Capacitor Circuits*

Prior to 1990, computers were not as efficient as they are today, and many experimenters used analog electrical systems to explore the behavior of chaotic maps. In 1987, Rodriguez-Vasquez et al. [43] presented a special-purpose analog computer made of switched capacitor circuits for analyzing chaos and bifurcation phenomena in nonlinear discrete dynamical systems modeled by discrete maps. They published results for four maps: the logistic map, a piece-wise linear map

$$x_{n+1} = V - (|x_n| - 0.4)[1 + \operatorname{sgn}(|x_n| - 0.4)], \quad (17)$$
### 2.5.3. Complex Fractional Map

Other authors, such as Joshi et al. [42], have considered the definition of a fractional map (13) for the case where the order of derivation $v$ is a complex number with $0 < Re(v) < 1$. They used $L_1$ to denote the model defined by (13) and $L_2$ to denote the one defined by

$$x(n) = x(0) + \frac{1}{\Gamma(v)} \sum_{j=1}^{n} \frac{\Gamma(n-j+v)}{\Gamma(n-j+1)} [-\alpha|x(j-1)| + x(j-1) + 1 - bx(j-2)]. \quad (16)$$

To systematically investigate the impact of complex order on fractional maps, they set $v = v_0e^{i r \pi / 2}$ with $0 < v_0 < 1$, which reduces to the real fractional order for $r = 0$. They studied the bifurcation diagram numerically, comparing it to the bifurcation diagram of the Hénon map.

In the case of the Hénon map, they did not observe chaos for $r \neq 0$, indicating that the chaotic attractor is destroyed when complex order was introduced. They checked their results using a number of different initial conditions, finding that the bifurcation diagram does not change, which highlights that multistability is not very pronounced for the fractional order.

They found, however, that the situation is different for the Lozi map. In this case, the chaos does not disappear. They computed the bifurcation diagrams of the Lozi map for $v_0 = 0.8$ and $r = 0, 0.01, 0.1, 0.5$ for model $L_1$.

The range of variable values indicates that there are parameter zones that are chaotic, or at least periodic with a very large period. They performed other tests as well, such as finding the largest Lyapunov exponent for models $L_1$ and $L_2$ to confirm the presence of chaos (the divergence of trajectories is exponential in the $L_1$ formulation, and the calculation of the Lyapunov exponent is justified). They found that the largest exponents are 0.264 for model $L_1$ and 0.326 for model $L_2$ for the parameters mentioned above. These results confirm that the Lozi map shows chaos for the difference equation of the complex fractional order.

Moreover, they noted that extremely strange bifurcations are observed in model $L_1$, showing bifurcation diagrams for $v = 0.4 + 0.3i$ and $v = 0.4 + 0.5i$ (see Figure 7a,b of [42]) which clearly indicate the possibility of very large periods and a rich bifurcation structure that is not generally seen in integer-order systems.

In addition, they noted that the system has memory; the complex fractional ordered Lozi map shows multistability, and the asymptotic attractor depends on the initial conditions.

Finally, they studied control (see Section 5.2.1) and synchronization (see Section 7.1.6) for this map.

## 3. The Use of the Lozi Map in Electrical Devices, Memristors, and Computers

The first paradigm change in the study of the Lozi map came in 1987, nine years after the initial publication. Instead of focusing on the theoretical study of its mathematical properties or on finding generalizations, electrical engineers used it to build new devices. Subsequently and continuing up to the present day, this map has been continuously involved in the field of electronics, electrical devices, multiple access communications, memristors, and computer software.

### 3.1. *Chaos from Switched-Capacitor Circuits*

Prior to 1990, computers were not as efficient as they are today, and many experimenters used analog electrical systems to explore the behavior of chaotic maps. In 1987, Rodriguez-Vasquez et al. [43] presented a special-purpose analog computer made of switched capacitor circuits for analyzing chaos and bifurcation phenomena in nonlinear discrete dynamical systems modeled by discrete maps. They published results for four maps: the logistic map, a piece-wise linear map

$$x_{n+1} = V - (|x_n| - 0.4)[1 + \operatorname{sgn}(|x_n| - 0.4)], \quad (17)$$
where $x \in [-4,4]$ for $0 \le V \le 4$, the Hénon map (2), and the Lozi map (3) in the form

$$ \begin{cases} x_{n+1} = 3 - 1.8|x_n| + y_n, \\ y_{n+1} = 0.25x_n. \end{cases} \qquad (18) $$

For (18), they built a rather complicated circuit realization (see Figure 25 of [43]) and compared the attractor measured from this circuit with the corresponding numerical simulation, finding good agreement between them. Even if this example is not, strictly speaking, an application of the Lozi map for electrical purposes, it constitutes one of the first examples of solid realization. The same year, Matsumoto [44] considered a generalization of this map (similar to, though less general, than the one studied by Glendinning and Simpson [19]) and built an electric circuit to study its attractors.

## 3.2. Multiple Access Communication Based on Control of Special Chaotic Trajectories

In 2000, Dmitriev et al. [45] discussed the principle of multiple access in satellite communication systems or cellular telephony based on the fine structure of chaotic attractors and using control of special chaotic trajectories. They demonstrated the experimental verification of the proposed approach for asynchronous packet data transmission. In their approach, they considered that a chaotic attractor can be treated as a number of countable sets of special trajectories consisting of unstable periodic orbits (UPO) and transitions between these orbits. The instability of the periodic orbits and transient trajectories between them gives rise to irregular chaotic behavior. They used the set of the unstable “skeleton” periodic trajectories, constituting the structure of the strange attractor (or a part of this set), as a “reservoir” of potential codes for multi-user communication systems. They observed that the multitude of codes from a certain “reservoir” for communications is practically infinite, i.e., the number of users that can be provided with individual code sets is unlimited.

The initial information carrier (in the absence of the transmitted message) is the chaotic oscillations produced by the dynamic system. The “user” codes are the fragments of the entire complex motion; thus, each user’s “own” code does not disturb the system behavior, and alternating the “owned” code for a certain user and others’ orbits does not change the structure of the total signal in the channel. The signal in the channel is the “information for everybody”, containing pieces of chaotic trajectory, “owned” unstable cycles (orbits) and codes belonging to “others”. This means that the proposed method for the transmitted message formation additionally has the useful property of confidentiality. As both the position and duration of the code packages in the channel can be arbitrary, a sequential asynchronous stream of cycle-codes is realized.

As an example of the realization of their method, they considered twenty period-16 (UPO) of the Lozi map (3) for $a = 1.7$ and $b = 0.5$. They displayed the switching between them in the Figure 2 of [45] and showed from this diagram that the forming of all successive cycles (repeating ten times) is practically instantaneous. Improving on their initial method, they remarked that unstable periodic orbits can be utilized for encoding the entire transmitted information as well as to attribute it to this or that group of users, i.e., playing the role of “chaotic markers”. The idea of using the system of unstable periodic orbits as markers was applied to the problem of asynchronous packet transmission of data from several users through a single common communication channel.

They concluded that the generation and control of UPO may be realized in the relatively high-frequency bands provided by modern digital methods.

They later carried out a similar study with the Lozi map (3) for $a = 1.4$ and $b = 0.3$ in [46].

## 3.3. Memristors

In 1971, L.O. Chua predicted the existence of a missing fourth passive circuit element in addition to the three classical ones (resistors, inductors, and capacitor) [47]. He called these new elements ‘memristors’, referring to a resistor with memory. Memristors are characterized by a nonlinear constitutive relationship between the charge $q$ and flux $\phi$.
where $x \in [-4,4]$ for $0 \le V \le 4$, the Hénon map (2), and the Lozi map (3) in the form

$$ \begin{cases} x_{n+1} = 3 - 1.8|x_n| + y_n, \\ y_{n+1} = 0.25x_n. \end{cases} \qquad (18) $$

For (18), they built a rather complicated circuit realization (see Figure 25 of [43]) and compared the attractor measured from this circuit with the corresponding numerical simulation, finding good agreement between them. Even if this example is not, strictly speaking, an application of the Lozi map for electrical purposes, it constitutes one of the first examples of solid realization. The same year, Matsumoto [44] considered a generalization of this map (similar to, though less general, than the one studied by Glendinning and Simpson [19]) and built an electric circuit to study its attractors.

## 3.2. Multiple Access Communication Based on Control of Special Chaotic Trajectories

In 2000, Dmitriev et al. [45] discussed the principle of multiple access in satellite communication systems or cellular telephony based on the fine structure of chaotic attractors and using control of special chaotic trajectories. They demonstrated the experimental verification of the proposed approach for asynchronous packet data transmission. In their approach, they considered that a chaotic attractor can be treated as a number of countable sets of special trajectories consisting of unstable periodic orbits (UPO) and transitions between these orbits. The instability of the periodic orbits and transient trajectories between them gives rise to irregular chaotic behavior. They used the set of the unstable “skeleton” periodic trajectories, constituting the structure of the strange attractor (or a part of this set), as a “reservoir” of potential codes for multi-user communication systems. They observed that the multitude of codes from a certain “reservoir” for communications is practically infinite, i.e., the number of users that can be provided with individual code sets is unlimited.

The initial information carrier (in the absence of the transmitted message) is the chaotic oscillations produced by the dynamic system. The “user” codes are the fragments of the entire complex motion; thus, each user’s “own” code does not disturb the system behavior, and alternating the “owned” code for a certain user and others’ orbits does not change the structure of the total signal in the channel. The signal in the channel is the “information for everybody”, containing pieces of chaotic trajectory, “owned” unstable cycles (orbits) and codes belonging to “others”. This means that the proposed method for the transmitted message formation additionally has the useful property of confidentiality. As both the position and duration of the code packages in the channel can be arbitrary, a sequential asynchronous stream of cycle-codes is realized.

As an example of the realization of their method, they considered twenty period-16 (UPO) of the Lozi map (3) for $a = 1.7$ and $b = 0.5$. They displayed the switching between them in the Figure 2 of [45] and showed from this diagram that the forming of all successive cycles (repeating ten times) is practically instantaneous. Improving on their initial method, they remarked that unstable periodic orbits can be utilized for encoding the entire transmitted information as well as to attribute it to this or that group of users, i.e., playing the role of “chaotic markers”. The idea of using the system of unstable periodic orbits as markers was applied to the problem of asynchronous packet transmission of data from several users through a single common communication channel.

They concluded that the generation and control of UPO may be realized in the relatively high-frequency bands provided by modern digital methods.

They later carried out a similar study with the Lozi map (3) for $a = 1.4$ and $b = 0.3$ in [46].

## 3.3. Memristors

In 1971, L.O. Chua predicted the existence of a missing fourth passive circuit element in addition to the three classical ones (resistors, inductors, and capacitor) [47]. He called these new elements ‘memristors’, referring to a resistor with memory. Memristors are characterized by a nonlinear constitutive relationship between the charge $q$ and flux $\phi$.
A physical device was not reported until 2008, when a physical model of a two-terminal *hp* device behaving as a memristor was announced [48]. This sparked intense research interest, with thousands of papers being published to date. A generalized version of Ohm’s law for theorizing about such devices was published ten years ago [49].

### 3.3.1. Memristor-Based Lozi Map with Hidden Hyperchaos

Nowadays, the discrete memristor model is a well known research hotspot. Many researchers have devoted themselves to the analysis of chaotic phenomena in discrete memristors. Recently, hidden attractors have been discovered in discrete memristor-based maps [50]. Wang et al. [51] included a discrete-time memristor to create a memristive Lozi map. This new 3D memristor-based Lozi map was established by coupling a discrete memristor to the original 2D Lozi map (3):

$$ egin{cases} x_{n+1} = 1 - a|x_n| + y_n, \\ y_{n+1} = bx_n + ky_n \sin(z_n), \\ z_{n+1} = y_n + z_n, \end{cases} \qquad (19) $$

where *k* is the real-valued control parameter coupling gain between the discrete-time memristor and the Lozi map.

Because there are no fixed points and hyperchaos can emerge, the memristor-based Lozi map is a hidden hyperchaotic map.

For certain specific control parameters, the 3D memristor-based Lozi map can show heterogeneous and homogeneous hidden multistability. It should be noted that heterogeneous hidden multistability implies the coexisting behavior of multiple hidden attractors of different stability types, while homogeneous hidden multistability indicates the coexisting behavior of multiple hidden attractors of the same stability type that only differ in dynamic intervals. In addition to the coexistence of these heterogeneous hidden attractors, the memristor-based Lozi map is very likely to produce the coexistence of homogeneous hidden hyperchaotic attractors, i.e., homogeneous hidden multistability. Therefore, the homogeneous hidden hyperchaotic attractors from the 3D memristor-based Lozi map can be robustly controlled by the memristor’s initial conditions.

Additionally, Wang et al. implemented this memristor in a digital circuit based on a high-performance microcontroller. They physically obtained an image of the hyperchaotic hidden attractors using a digital oscilloscope. Eventually, a digital platform was exploited and its experimental phase portraits were obtained to confirm the numerical portraits.

### 3.3.2. 3D Fractional Memristor-Based Lozi Map

Another method to obtain a memristor-based Lozi map is to consider the fractional counterpart of this map (see Section 2.5). Khennaoui et al. [52] used the Caputo-like delta difference to obtain the fractional map corresponding to (19):

$$ egin{cases} \Delta^{\alpha_1} x_n = 1 - a|x_n| + y_n - x_n, \\ \Delta^{\alpha_2} y_n = bx_n + ky_n \sin(z_n) - y_n, \\ \Delta^{\alpha_3} z_n = y_n, \end{cases} \qquad (20) $$

in which $\Delta^{\alpha_i}$ ($i = 1, 2, 3$) stands for the Grunwald–Letnikov difference operator.

Obviously, when $\alpha_1 = \alpha_2 = \alpha_3 = 1$, the fractional map (20) corresponds to the first-order map (19).

This map exhibits hidden dynamic evolutions and multistability. The description of the fractional discrete memristor-based Lozi map was investigated according to its Lyapunov exponents, phase portraits, and bifurcation diagrams, demonstrating several kind of coexisting multiple hidden attractor behaviours that can be generated by the proposed model.
A physical device was not reported until 2008, when a physical model of a two-terminal *hp* device behaving as a memristor was announced [48]. This sparked intense research interest, with thousands of papers being published to date. A generalized version of Ohm’s law for theorizing about such devices was published ten years ago [49].

### 3.3.1. Memristor-Based Lozi Map with Hidden Hyperchaos

Nowadays, the discrete memristor model is a well known research hotspot. Many researchers have devoted themselves to the analysis of chaotic phenomena in discrete memristors. Recently, hidden attractors have been discovered in discrete memristor-based maps [50]. Wang et al. [51] included a discrete-time memristor to create a memristive Lozi map. This new 3D memristor-based Lozi map was established by coupling a discrete memristor to the original 2D Lozi map (3):

$$ egin{cases} x_{n+1} = 1 - a|x_n| + y_n, \\ y_{n+1} = bx_n + ky_n \sin(z_n), \\ z_{n+1} = y_n + z_n, \end{cases} \qquad (19) $$

where *k* is the real-valued control parameter coupling gain between the discrete-time memristor and the Lozi map.

Because there are no fixed points and hyperchaos can emerge, the memristor-based Lozi map is a hidden hyperchaotic map.

For certain specific control parameters, the 3D memristor-based Lozi map can show heterogeneous and homogeneous hidden multistability. It should be noted that heterogeneous hidden multistability implies the coexisting behavior of multiple hidden attractors of different stability types, while homogeneous hidden multistability indicates the coexisting behavior of multiple hidden attractors of the same stability type that only differ in dynamic intervals. In addition to the coexistence of these heterogeneous hidden attractors, the memristor-based Lozi map is very likely to produce the coexistence of homogeneous hidden hyperchaotic attractors, i.e., homogeneous hidden multistability. Therefore, the homogeneous hidden hyperchaotic attractors from the 3D memristor-based Lozi map can be robustly controlled by the memristor’s initial conditions.

Additionally, Wang et al. implemented this memristor in a digital circuit based on a high-performance microcontroller. They physically obtained an image of the hyperchaotic hidden attractors using a digital oscilloscope. Eventually, a digital platform was exploited and its experimental phase portraits were obtained to confirm the numerical portraits.

### 3.3.2. 3D Fractional Memristor-Based Lozi Map

Another method to obtain a memristor-based Lozi map is to consider the fractional counterpart of this map (see Section 2.5). Khennaoui et al. [52] used the Caputo-like delta difference to obtain the fractional map corresponding to (19):

$$ egin{cases} \Delta^{\alpha_1} x_n = 1 - a|x_n| + y_n - x_n, \\ \Delta^{\alpha_2} y_n = bx_n + ky_n \sin(z_n) - y_n, \\ \Delta^{\alpha_3} z_n = y_n, \end{cases} \qquad (20) $$

in which $\Delta^{\alpha_i}$ ($i = 1, 2, 3$) stands for the Grunwald–Letnikov difference operator.

Obviously, when $\alpha_1 = \alpha_2 = \alpha_3 = 1$, the fractional map (20) corresponds to the first-order map (19).

This map exhibits hidden dynamic evolutions and multistability. The description of the fractional discrete memristor-based Lozi map was investigated according to its Lyapunov exponents, phase portraits, and bifurcation diagrams, demonstrating several kind of coexisting multiple hidden attractor behaviours that can be generated by the proposed model.
Using the classical discretization scheme, the iterative equation of the fractional order memristor-based Lozi map with no equilibrium point (20) is provided by

$$ 
\left\{
\begin{aligned}
x_{n+1} &= 1 - a|x_n| + y_n + (\alpha_1 - 1)x_n - \sum_{p_1=1}^{n} \beta_{p_1} x_{n-p_1}, \\
y_{n+1} &= bx_n + ky_n \sin(z_n) + (\alpha_2 - 1)y_n - \sum_{p_2=1}^{n} \beta_{p_2} y_{n-p_2}, \\
z_{n+1} &= y_n + \alpha_3 z_n - \sum_{p_3=1}^{n} \beta_{p_3} z_{n-p_3},
\end{aligned}
\right.
\quad (21)
$$

where $\beta_{p_1}, \beta_{p_2}$, and $\beta_{p_3}$ are the binomial coefficients:

$$ \beta_{p_j=0} = -\alpha_1, \beta_{p_j} = \left(1 - \frac{1 + \alpha_j}{p+1}\right) \beta_{p_{j-1}}, \quad j = 1, 2, 3. \quad (22) $$

It can be shown that the value of $\beta_p$ decreases when the number of iterations $p$ increases, regardless of the value of the fractional order. Thus, the easiest way to numerically implement the iteration Formula (21) is to use a finite truncation of the memory length $M$ to approximate the fractional discrete memristor map (20). The short memory model $M$ is provided by

$$ 
\left\{
\begin{aligned}
x_{n+1} &= 1 - a|x_n| + y_n + (\alpha_1 - 1)x_n - \sum_{p_1=1}^{M} \beta_{p_1} x_{n-p_1}, \\
y_{n+1} &= bx_n + ky_n \sin(z_n) + (\alpha_2 - 1)y_n - \sum_{p_2=1}^{M} \beta_{p_2} y_{n-p_2}, \\
z_{n+1} &= y_n + \alpha_3 z_n - \sum_{p_3=1}^{M} \beta_{p_3} z_{n-p_3}.
\end{aligned}
\right.
\quad (23)
$$

The authors used a hardware platform based on a microcontroller (Arduino Uno board) to realize such a fractional memristive Lozi map for which the initial conditions can be changed easily. They showed that this map exhibits the initial condition-dependent dynamical phenomena of multistability and confirmed this through numerical simulations.

### 3.4. Chaotic Interleaver for Turbo Codes

In [53], Sahnoune and Berkani presented a new method to design a deterministic interleaver with random-like behavior based on the Lozi map. The interleaver is a fundamental part of turbo code design and plays a critical role in the performance of turbo coding. The basic role of the interleaver is to construct a random code and spread out burst errors. The interleaver provides “scrambled” information data to the second component encoder for constructing random codes, and decorrelates the inputs to the two component decoders; hence, the convergence of the iterative decoding algorithm improves.

An interleaver is a vector $\pi$, where $\pi(n)$ is the position in the information sequence that is interleaved to position $n$. Most proposed deterministic interleavers are linear interleavers with a designed index function provided by

$$ \pi(n) = kn + u \mod N, \quad 0 \le n \le N, \quad (24) $$

where $k$ and $n$ are fixed integers and $k$ is prime relative to $N$.

The chaotic interleaver based on the Lozi map (with $a = 1.7$ and $b = 0.5$) is defined as follows:

* Step 1: the integer sequence $\{n\}_{n=1}^N \in \mathbb{Z}$ is mapped to a real number sequence $\{x(n)\}_{n=1}^N \in \mathbb{R}$ using (3).

* Step 2: $\{x(n)\}_{n=1}^N$ is mapped to another integer sequence $\{s(n)\}_{n=1}^N \in \mathbb{Z}$ by designing $s(n)$ as the position of index of $x(n)$ in the sort ascending of $\{x(n)\}_{n=1}^N$.

* Step 3: the chaotic interleaver indexes are provided by $\pi(\{s(n)\}_{n=1}^N) = \{n\}_{n=1}^N$.

Sahnoune and Berkani concluded that, compared with a random interleaver and dithered Golden interleaver, a chaotic interleaver has low latency, low complexity of implementation, and improves the security of the encoded data.
Using the classical discretization scheme, the iterative equation of the fractional order memristor-based Lozi map with no equilibrium point (20) is provided by

$$ egin{cases} x_{n+1} = 1 - a|x_n| + y_n + (\alpha_1 - 1)x_n - \sum_{p_1=1}^{n} \beta_{p_1} x_{n-p_1}, \\ y_{n+1} = bx_n + ky_n \sin(z_n) + (\alpha_2 - 1)y_n - \sum_{p_2=1}^{n} \beta_{p_2} y_{n-p_2}, \\ z_{n+1} = y_n + \alpha_3 z_n - \sum_{p_3=1}^{n} \beta_{p_3} z_{n-p_3}, \end{cases} \quad (21) $$

where $\beta_{p_1}, \beta_{p_2}$, and $\beta_{p_3}$ are the binomial coefficients:

$$ \beta_{p_j=0} = -\alpha_1, \beta_{p_j} = \left(1 - \frac{1 + \alpha_j}{p+1}\right) \beta_{p_{j-1}}, \quad j = 1, 2, 3. \quad (22) $$

It can be shown that the value of $\beta_p$ decreases when the number of iterations $p$ increases, regardless of the value of the fractional order. Thus, the easiest way to numerically implement the iteration Formula (21) is to use a finite truncation of the memory length $M$ to approximate the fractional discrete memristor map (20). The short memory model $M$ is provided by

$$ egin{cases} x_{n+1} = 1 - a|x_n| + y_n + (\alpha_1 - 1)x_n - \sum_{p_1=1}^{M} \beta_{p_1} x_{n-p_1}, \\ y_{n+1} = bx_n + ky_n \sin(z_n) + (\alpha_2 - 1)y_n - \sum_{p_2=1}^{M} \beta_{p_2} y_{n-p_2}, \\ z_{n+1} = y_n + \alpha_3 z_n - \sum_{p_3=1}^{M} \beta_{p_3} z_{n-p_3}. \end{cases} \quad (23) $$

The authors used a hardware platform based on a microcontroller (Arduino Uno board) to realize such a fractional memristive Lozi map for which the initial conditions can be changed easily. They showed that this map exhibits the initial condition-dependent dynamical phenomena of multistability and confirmed this through numerical simulations.

### 3.4. Chaotic Interleaver for Turbo Codes

In [53], Sahnoune and Berkani presented a new method to design a deterministic interleaver with random-like behavior based on the Lozi map. The interleaver is a fundamental part of turbo code design and plays a critical role in the performance of turbo coding. The basic role of the interleaver is to construct a random code and spread out burst errors. The interleaver provides "scrambled" information data to the second component encoder for constructing random codes, and decorrelates the inputs to the two component decoders; hence, the convergence of the iterative decoding algorithm improves.

An interleaver is a vector $\pi$, where $\pi(n)$ is the position in the information sequence that is interleaved to position $n$. Most proposed deterministic interleavers are linear interleavers with a designed index function provided by

$$ \pi(n) = kn + u \quad \text{mod}N, \quad 0 \le n \le N, \quad (24) $$

where $k$ and $n$ are fixed integers and $k$ is prime relative to $N$.

The chaotic interleaver based on the Lozi map (with $a = 1.7$ and $b = 0.5$) is defined as follows:

* Step 1: the integer sequence $\{n\}_{n=1}^N \in \mathbb{Z}$ is mapped to a real number sequence $\{x(n)\}_{n=1}^N \in \mathbb{R}$ using (3).

* Step 2: $\{x(n)\}_{n=1}^N$ is mapped to another integer sequence $\{s(n)\}_{n=1}^N \in \mathbb{Z}$ by designing $s(n)$ as the position of index of $x(n)$ in the sort ascending of $\{x(n)\}_{n=1}^N$.

* Step 3: the chaotic interleaver indexes are provided by $\pi(\{s(n)\}_{n=1}^N) = \{n\}_{n=1}^N$.

Sahnoune and Berkani concluded that, compared with a random interleaver and dithered Golden interleaver, a chaotic interleaver has low latency, low complexity of implementation, and improves the security of the encoded data.
Urrea et al. [54] compared the performances of a chaotic interleaver with short block lengths to a new chaotic interleaver based on the Duffing map, which showed slightly better performance.

## 4. Applications in Optimization

Most engineering problems can be defined as optimization problems, e.g., finding an optimal trajectory for a robot arm, the optimal thickness of steel in pressure vessels, the optimal set of parameters for controllers, optimal relations or fuzzy sets in fuzzy models, etc. Solutions to such problems are usually difficult to find, and their parameters usually include variables of different types, such as floating point or integer variables.

Applications of the Lozi map in the now-flourishing field of optimization took longer to appear than applications in electrical devices. The main reason for this is due to a paradigm shift in optimization algorithms; instead of using deterministic algorithms such as gradient or steepest descent methods, which are not efficient in high-dimensional problem optimization involving hundred or thousand of variables, heuristic algorithms based on an imitation of Darwin's theory of the evolution of species were introduced a few decades ago. Such algorithms require easy access to random or chaotic numbers. This is why interest has only recently focused on chaotic attractors.

Thirty years after the publication of the Lozi map, Araujo and Coelho used it as a core function for Particle Swarm Optimization (PSO) algorithms [55]. Their article was accepted on October 2007. Optimization algorithms based on chaos theory are methodologies for searching optimal solutions that differ from any of the existing traditional stochastic optimization techniques. Due to the wandering of chaos, it can carry out overall searches in the solution space at higher velocities compared to stochastic ergodic searches, which rely on probability-based computation. This remark appeared in the pioneering work of Caponetto et al. [56], who, four years before Araujo and Coelho, found that chaotic sequences improved the performance of evolutionary algorithms.

In fact, there are many kinds of genetic and evolutionary algorithms used for optimization; see [57] for a survey. In addition to PSO, Differential Evolution (DE) is a main branch of research into metaheuristic algorithms, and a few researchers have used the Lozi map [58]. Among the less conventional approaches for enhancing the performance of metaheuristics is the incorporation of chaos as a source of stochasticity. As a key operation in metaheuristics, stochasticity is typically used for three basic operators: selection, crossover, and mutation. It is possible to implement chaotic dynamics in any of these.

It should be noted that three years before Araujo and Coelho's landmark publication, Zelinka et al. proposed a slightly different algorithm called SOMA (Self-Organizing Migrating Algorithm) using chaotic attractors, which can be classified among the metaheuristic algorithms [59].

Finally, the Lozi map is widely used in non-evolutionary chaotic optimization algorithms (see Section 4.5) such as the Improved Lozi map-based Chaotic Optimization Algorithm (ILCOA).

In this section, we explain how this chaotic map is used in these algorithms.

### 4.1. Particle Swarm Optimization (PSO)

Particle Swarm Optimization is a population-based evolutionary algorithm inspired by simulations of social behaviour instead of the survival of the fittest individual. PSO is randomly initialized with a population of individuals, which correspond to potential solutions. Unlike the most evolutionary algorithms, each particle in PSO is associated with a randomized velocity that "flows" through the problem space. In PSO, a uniform probability distribution is used to generate random numbers. However, the use of other probability distributions may provide improved fine-tuning or even allow the algorithm to escape from local optima, such as the distributions presented in [60]. In the meantime, the use of chaotic sequences has been proposed to generate random numbers for updating the velocity equation.
Urrea et al. [54] compared the performances of a chaotic interleaver with short block lengths to a new chaotic interleaver based on the Duffing map, which showed slightly better performance.

## 4. Applications in Optimization

Most engineering problems can be defined as optimization problems, e.g., finding an optimal trajectory for a robot arm, the optimal thickness of steel in pressure vessels, the optimal set of parameters for controllers, optimal relations or fuzzy sets in fuzzy models, etc. Solutions to such problems are usually difficult to find, and their parameters usually include variables of different types, such as floating point or integer variables.

Applications of the Lozi map in the now-flourishing field of optimization took longer to appear than applications in electrical devices. The main reason for this is due to a paradigm shift in optimization algorithms; instead of using deterministic algorithms such as gradient or steepest descent methods, which are not efficient in high-dimensional problem optimization involving hundred or thousand of variables, heuristic algorithms based on an imitation of Darwin's theory of the evolution of species were introduced a few decades ago. Such algorithms require easy access to random or chaotic numbers. This is why interest has only recently focused on chaotic attractors.

Thirty years after the publication of the Lozi map, Araujo and Coelho used it as a core function for Particle Swarm Optimization (PSO) algorithms [55]. Their article was accepted on October 2007. Optimization algorithms based on chaos theory are methodologies for searching optimal solutions that differ from any of the existing traditional stochastic optimization techniques. Due to the wandering of chaos, it can carry out overall searches in the solution space at higher velocities compared to stochastic ergodic searches, which rely on probability-based computation. This remark appeared in the pioneering work of Caponetto et al. [56], who, four years before Araujo and Coelho, found that chaotic sequences improved the performance of evolutionary algorithms.

In fact, there are many kinds of genetic and evolutionary algorithms used for optimization; see [57] for a survey. In addition to PSO, Differential Evolution (DE) is a main branch of research into metaheuristic algorithms, and a few researchers have used the Lozi map [58]. Among the less conventional approaches for enhancing the performance of metaheuristics is the incorporation of chaos as a source of stochasticity. As a key operation in metaheuristics, stochasticity is typically used for three basic operators: selection, crossover, and mutation. It is possible to implement chaotic dynamics in any of these.

It should be noted that three years before Araujo and Coelho's landmark publication, Zelinka et al. proposed a slightly different algorithm called SOMA (Self-Organizing Migrating Algorithm) using chaotic attractors, which can be classified among the metaheuristic algorithms [59].

Finally, the Lozi map is widely used in non-evolutionary chaotic optimization algorithms (see Section 4.5) such as the Improved Lozi map-based Chaotic Optimization Algorithm (ILCOA).

In this section, we explain how this chaotic map is used in these algorithms.

### 4.1. Particle Swarm Optimization (PSO)

Particle Swarm Optimization is a population-based evolutionary algorithm inspired by simulations of social behaviour instead of the survival of the fittest individual. PSO is randomly initialized with a population of individuals, which correspond to potential solutions. Unlike the most evolutionary algorithms, each particle in PSO is associated with a randomized velocity that "flows" through the problem space. In PSO, a uniform probability distribution is used to generate random numbers. However, the use of other probability distributions may provide improved fine-tuning or even allow the algorithm to escape from local optima, such as the distributions presented in [60]. In the meantime, the use of chaotic sequences has been proposed to generate random numbers for updating the velocity equation.
### 4.1.1. Thermal Vacuum System

In [55], the PSO approach was intertwined with Lozi map chaotic sequences to obtain a Takagi-Sugeno (TS) fuzzy model for representing dynamical behaviors. This proposed chaotic PSO (CPSO) approach was employed to optimize the premises in the IF-THEN rules of the TS fuzzy model, and were utilized in an experimental application, specifically, a thermal vacuum system employed for space environmental emulation and satellite qualification.

The essential idea of this model is to compute a global nonlinear model by partitioning the input space into fuzzy areas and approximating each area through a linear model.

The TS models consist of linguistic IF-THEN rules that can be represented by the following general form:

$$ egin{cases} R^{(j)} : \text{IF } (z_1 \text{ is } A_1^j) \text{ AND } \dots \text{ AND } (z_m \text{ is } A_m^j), \\ \text{THEN } y_j = b_0^j + b_1^j x_1^j + b_{q_j}^j x_{q_j}^j. \end{cases} \tag{25} $$

The IF statements define the premise part, while the THEN functions constitute the subsequent part of the fuzzy system; here, $A_i^j$ represents the labels of the fuzzy sets and $z = [z_1, z_2, \dots, z_m]^T$ with $i = 1, \dots, m$ is the input vector of premise $p$.

PSO is an optimization tool based on a population in which the position of each member particle is a potential solution to an analyzed problem. Each particle is associated with a randomized velocity that moves throughout the problem space. The coordinates of each particle are tracked in the problem space, and are associated with the best solution (fitness) achieved thus far. This value is called *pbest*. Another 'best' value that is tracked by the global version of the particle swarm optimizer is the overall best value. Its location, called *gbest*, can be obtained by any particle in the population. The past best position and the entire best overall position of the group are employed to minimize (maximize) the solution. At each time step, the PSO algorithm changes the velocity (acceleration) of each particle moving toward its *pbest* and *gbest* locations. Acceleration is weighted by random terms, with separate random numbers being generated for acceleration toward the *pbest* and *gbest* locations (see Figure 8).

<figure><img src="image_11.png" /><figcaption>Figure 8. Geometric core of Particle Swarm Optimization (PSO) algorithm.</figcaption></figure>

Araujo and Coelho defined the following three CPSO approaches based on the Lozi map, considering that the velocity equation of each particle is updated at each step $t$ by

$$ v_i(t+1) = K[v_i(t) + c_1 u d_i(t)(p_i(t) - x_i(t)) + c_2 u d_i(t)(p_g(t) - x_i(t))]. \tag{26} $$
### 4.1.1. Thermal Vacuum System

In [55], the PSO approach was intertwined with Lozi map chaotic sequences to obtain a Takagi-Sugeno (TS) fuzzy model for representing dynamical behaviors. This proposed chaotic PSO (CPSO) approach was employed to optimize the premises in the IF-THEN rules of the TS fuzzy model, and were utilized in an experimental application, specifically, a thermal vacuum system employed for space environmental emulation and satellite qualification.

The essential idea of this model is to compute a global nonlinear model by partitioning the input space into fuzzy areas and approximating each area through a linear model.

The TS models consist of linguistic IF-THEN rules that can be represented by the following general form:

$$ \left\{ \begin{array}{l} R^{(j)} : \text{IF } (z_1 \text{ is } A_1^j) \text{ AND } \dots \text{ AND } (z_m \text{ is } A_m^j), \\ \text{THEN } y_j = b_0^j + b_1^j x_1^j + b_{q_j}^j x_{q_j}^j. \end{array} \right. \qquad (25) $$

The IF statements define the premise part, while the THEN functions constitute the subsequent part of the fuzzy system; here, $A_i^j$ represents the labels of the fuzzy sets and $z = [z_1, z_2, \dots, z_m]^T$ with $i = 1, \dots, m$ is the input vector of premise $p$.

PSO is an optimization tool based on a population in which the position of each member particle is a potential solution to an analyzed problem. Each particle is associated with a randomized velocity that moves throughout the problem space. The coordinates of each particle are tracked in the problem space, and are associated with the best solution (fitness) achieved thus far. This value is called *pbest*. Another 'best' value that is tracked by the global version of the particle swarm optimizer is the overall best value. Its location, called *gbest*, can be obtained by any particle in the population. The past best position and the entire best overall position of the group are employed to minimize (maximize) the solution. At each time step, the PSO algorithm changes the velocity (acceleration) of each particle moving toward its *pbest* and *gbest* locations. Acceleration is weighted by random terms, with separate random numbers being generated for acceleration toward the *pbest* and *gbest* locations (see Figure 8).

<figure><img src="image_12.png" /><figcaption>Figure 8. Geometric core of Particle Swarm Optimization (PSO) algorithm.</figcaption></figure>

Araujo and Coelho defined the following three CPSO approaches based on the Lozi map, considering that the velocity equation of each particle is updated at each step $t$ by

$$ v_i(t+1) = K[v_i(t) + c_1 u d_i(t)(p_i(t) - x_i(t)) + c_2 u d_i(t)(p_g(t) - x_i(t))]. \qquad (26) $$
Approach 1 (CPSO1): the parameter udᵢ(t) is modified to

$$v_i(t + 1) = K[v_i(t) + c_1Y_i(t)(p_i(t) - x_i(t)) + c_2ud_i(t)(p_g(t) - x_i(t))]. \quad (27)$$

Approach 2 (CPSO2): the parameter udᵢ(t) is modified to

$$v_i(t + 1) = K[v_i(t) + c_1ud_i(t)(p_i(t) - x_i(t)) + c_2Y_i(t)(p_g(t) - x_i(t))]. \quad (28)$$

Approach 3 (CPSO3): the parameter udᵢ(t) is modified to

$$v_i(t + 1) = K[v_i(t) + c_1Y_i(t)(p_i(t) - x_i(t)) + c_2Y'_i(t)(p_g(t) - x_i(t))], \quad (29)$$

where Yᵢ(t) are obtained from the chaotic attractor (3) with the parameter values a = 1.7 and b = 0.5 by scaling the variable y ∈ [-0.6418, 0.6716] within 0 and 1 at each iteration, as follows:

$$Y_i(t) = \frac{y_i(t) - \sigma}{\delta - \sigma}, \quad (30)$$

with [σ, δ] = (-0.6418, 0.6716). The Y'ᵢ(t) are obtained in the same way as Yᵢ(t), though with different initial values in (3). The constriction coefficient K is usually set to K = 0.729.

After their launch and during their operational life, satellites are exposed to difficult physical conditions due to sunshine, albedo, earth radiation, and earthshine infrared in alternative shadow/eclipse conditions. In order to reproduce these challenging environmental conditions that satellites experience during their operational life as closely as possible, it is possible to use thermal vacuum chambers that consist of a chamber, a 'shroud' (set of pipes) which heats or cools off the environment, and auxiliary equipment. In the thermal vacuum system used by the Brazilian National Institute for Space Research (INPE), the original controller was designed to control the temperature on the shroud [61].

The three proposed approaches shown here have been tested thoroughly. The authors succeeded in eliciting a TS fuzzy model for this nonlinear and time-delay application, with the best algorithm obtained using COSP3.

### 4.1.2. Pid Controller Using PSO

A few years after the initial use of chaotic attractor properties in PSO by Araujo and Coelho, a team led by Ivan Zelinka from the Technical University of Ostrava (VSB-TU, Faculty of Electrical Engineering and Computer Sciences) extensively explored a quasi-similar method in a series of articles. They used a slightly different velocity equation, shown in Equation (26), introducing an inertia weight modification to the PSO strategy based on two control parameters, w_start and w_end:

$$v_i(t + 1) = w \cdot v_i(t) + c_1 \cdot \text{Rand} \cdot (p_{\text{Best}} - x_i(t)) + c_2 \cdot \text{Rand} \cdot (g_{\text{Best}} - x_i(t)) \quad (31)$$

with (32)

$$w = w_{\text{start}} - \frac{(w_{\text{start}} - w_{\text{end}}) \cdot t}{n} \quad (32)$$

where t stands for the current generation number and n for the total number of generations.

Within the Chaos PSO algorithm, the basic inbuilt computer random generator was replaced by a chaotic generator (in this case, using the Lozi map).

First, Pluhacek et al. [62] considered a Partial Integral Derivative (PID) controller for a Direct Current (DC) motor system in order to obtain optimal settings. A DC motor is any of a class of rotary electrical motors that converts direct current electrical energy into mechanical energy. Proportional Integral Derivative (PID) control is the most common control algorithm used by industry, and has been universally accepted in industrial control.

The popularity of PID controllers can be attributed to their robust performance in a wide range of operating conditions and their functional simplicity, which allows engineers to operate them in a simple and straightforward manner.
Approach 1 (CPSO1): the parameter udᵢ(t) is modified to

$$v_i(t + 1) = K[v_i(t) + c_1Y_i(t)(p_i(t) - x_i(t)) + c_2ud_i(t)(p_g(t) - x_i(t))]. \quad (27)$$

Approach 2 (CPSO2): the parameter udᵢ(t) is modified to

$$v_i(t + 1) = K[v_i(t) + c_1ud_i(t)(p_i(t) - x_i(t)) + c_2Y_i(t)(p_g(t) - x_i(t))]. \quad (28)$$

Approach 3 (CPSO3): the parameter udᵢ(t) is modified to

$$v_i(t + 1) = K[v_i(t) + c_1Y_i(t)(p_i(t) - x_i(t)) + c_2Y'_i(t)(p_g(t) - x_i(t))], \quad (29)$$

where Yᵢ(t) are obtained from the chaotic attractor (3) with the parameter values a = 1.7 and b = 0.5 by scaling the variable y ∈ [-0.6418, 0.6716] within 0 and 1 at each iteration, as follows:

$$Y_i(t) = \frac{y_i(t) - \sigma}{\delta - \sigma}, \quad (30)$$

with [σ, δ] = (-0.6418, 0.6716). The Y'ᵢ(t) are obtained in the same way as Yᵢ(t), though with different initial values in (3). The constriction coefficient K is usually set to K = 0.729.

After their launch and during their operational life, satellites are exposed to difficult physical conditions due to sunshine, albedo, earth radiation, and earthshine infrared in alternative shadow/eclipse conditions. In order to reproduce these challenging environmental conditions that satellites experience during their operational life as closely as possible, it is possible to use thermal vacuum chambers that consist of a chamber, a 'shroud' (set of pipes) which heats or cools off the environment, and auxiliary equipment. In the thermal vacuum system used by the Brazilian National Institute for Space Research (INPE), the original controller was designed to control the temperature on the shroud [61].

The three proposed approaches shown here have been tested thoroughly. The authors succeeded in eliciting a TS fuzzy model for this nonlinear and time-delay application, with the best algorithm obtained using COSP3.

### 4.1.2. Pid Controller Using PSO

A few years after the initial use of chaotic attractor properties in PSO by Araujo and Coelho, a team led by Ivan Zelinka from the Technical University of Ostrava (VSB-TU, Faculty of Electrical Engineering and Computer Sciences) extensively explored a quasi-similar method in a series of articles. They used a slightly different velocity equation, shown in Equation (26), introducing an inertia weight modification to the PSO strategy based on two control parameters, w_start and w_end:

$$v_i(t + 1) = w \cdot v_i(t) + c_1 \cdot \text{Rand} \cdot (p_{\text{Best}} - x_i(t)) + c_2 \cdot \text{Rand} \cdot (g_{\text{Best}} - x_i(t)) \quad (31)$$

with (32)

$$w = w_{\text{start}} - \frac{(w_{\text{start}} - w_{\text{end}}) \cdot t}{n} \quad (32)$$

where t stands for the current generation number and n for the total number of generations.

Within the Chaos PSO algorithm, the basic inbuilt computer random generator was replaced by a chaotic generator (in this case, using the Lozi map).

First, Pluhacek et al. [62] considered a Partial Integral Derivative (PID) controller for a Direct Current (DC) motor system in order to obtain optimal settings. A DC motor is any of a class of rotary electrical motors that converts direct current electrical energy into mechanical energy. Proportional Integral Derivative (PID) control is the most common control algorithm used by industry, and has been universally accepted in industrial control.

The popularity of PID controllers can be attributed to their robust performance in a wide range of operating conditions and their functional simplicity, which allows engineers to operate them in a simple and straightforward manner.
All PID controllers contain three unique basic coefficients, namely, proportional, integral, and derivative, which are varied to achieve optimal response (see [63] for a detailed description). A simplified form in the Laplace domain is provided by

$$G(s) = K \left( 1 + \frac{1}{sT_i} + sT_d \right), \qquad (33)$$

where K is the controller gain, $T_i$ is the adjustable integral time parameter, and $T_d$ is the rate time. For suitable analytical calculations, the following form is often chosen:

$$G(s) = k_p + \frac{k_i}{s} + k_d s, \qquad (34)$$

where the parameters are related to the standard form (33) through $k_p = K$, $k_i = K/T_i$, and $k_d = K/T_d$.

The transfer function for the DC motor considered in [62] is provided by

$$G(s) = \frac{0.9}{0.00105s^3 + 0.2104s^2 + 0.8913s} \qquad (35)$$

The optimization process involving the PSO algorithm was applied to minimize errors of the output transfer function, which can indicate the quality of regulation of the controller. Four criteria were considered:

- The integral of the time multiplied by absolute error (ITAE):

$$I_{ITAE} = \int_{0}^{T} t|e(t)|dt, \qquad (36)$$

where $e(t)$ is the control error (see [63]).

- The integral of the absolute magnitude of the error (IAE):

$$I_{IAE} = \int_{0}^{T} |e(t)| dt. \qquad (37)$$

- The integral of the square of the error (ISE):

$$I_{ISE} = \int_{0}^{T} (e(t))^2 dt. \qquad (38)$$

- The mean of the square of the error (MSE):

$$I_{MSE} = \frac{1}{n} \sum_{i=1}^{n} (e(t))^2. \qquad (39)$$

The presented data and graphical simulation results of [62] lend weight to the argument that the implementation of a chaotic Lozi map into the PSO algorithm as a random number generator can significantly improve its performance over other non-heuristic and heuristic methods when solving the problem of optimal PID controller design for DC motor systems.

In [64], the same authors addressed a fourth-order system provided by

$$G(s) = \frac{1}{s^4 + 6s^3 + 11s^2 + 6s}. \qquad (40)$$

For statistical reasons, in both articles the optimization for each criterion was repeated thirty times. The results for the fourth-order system showed the same improvement of the Lozi map-based PSO method versus other non-heuristic and heuristic methods.
All PID controllers contain three unique basic coefficients, namely, proportional, integral, and derivative, which are varied to achieve optimal response (see [63] for a detailed description). A simplified form in the Laplace domain is provided by

$$G(s) = K \left( 1 + \frac{1}{sT_i} + sT_d \right), \quad (33)$$

where K is the controller gain, $T_i$ is the adjustable integral time parameter, and $T_d$ is the rate time. For suitable analytical calculations, the following form is often chosen:

$$G(s) = k_p + \frac{k_i}{s} + k_d s, \quad (34)$$

where the parameters are related to the standard form (33) through $k_p = K$, $k_i = K/T_i$, and $k_d = K/T_d$.

The transfer function for the DC motor considered in [62] is provided by

$$G(s) = \frac{0.9}{0.00105s^3 + 0.2104s^2 + 0.8913s} \quad (35)$$

The optimization process involving the PSO algorithm was applied to minimize errors of the output transfer function, which can indicate the quality of regulation of the controller. Four criteria were considered:

- The integral of the time multiplied by absolute error (ITAE):

$$I_{ITAE} = \int_{0}^{T} t|e(t)|dt, \quad (36)$$

where $e(t)$ is the control error (see [63]).

- The integral of the absolute magnitude of the error (IAE):

$$I_{IAE} = \int_{0}^{T} |e(t)| dt. \quad (37)$$

- The integral of the square of the error (ISE):

$$I_{ISE} = \int_{0}^{T} (e(t))^2 dt. \quad (38)$$

- The mean of the square of the error (MSE):

$$I_{MSE} = \frac{1}{n} \sum_{i=1}^{n} (e(t))^2. \quad (39)$$

The presented data and graphical simulation results of [62] lend weight to the argument that the implementation of a chaotic Lozi map into the PSO algorithm as a random number generator can significantly improve its performance over other non-heuristic and heuristic methods when solving the problem of optimal PID controller design for DC motor systems.

In [64], the same authors addressed a fourth-order system provided by

$$G(s) = \frac{1}{s^4 + 6s^3 + 11s^2 + 6s}. \quad (40)$$

For statistical reasons, in both articles the optimization for each criterion was repeated thirty times. The results for the fourth-order system showed the same improvement of the Lozi map-based PSO method versus other non-heuristic and heuristic methods.
### 4.1.3. Assessment of Enhanced PSO Algorithm with Lozi Chaotic Map

In order to better evaluate the improvement conferred by the use of chaos in PSO algorithms, Pluhacek et al. [65], investigated the effect of tuning the control parameters a and b of (3) (recall that in all three above examples the parameter values were fixed to a = 1.7 and b = 0.5). Three different benchmark functions from the IEEE's CEC 2013 benchmark suite [66] were selected for the tuning experiments. The controlling parameters of the Lozi map were set as follows:

$$a: 1.3 - 1.7, \text{ step } 0.05$$

$$b: 0.1 - 0.6, \text{ step } 0.05$$

The benchmark functions were evaluated for dimension = 10, population size = 40, number of iterations = 2500, $w_{start} = 0.9$, and $w_{end} = 0.4$ according to the demands in [66]. The PSO algorithm with an adequately set Chaotic Pseudo-Random Number Generator (CPRNG) was run 100 times, and the mean results for each function and Lozi map setting were analyzed. The presented results support the claim that different settings of the chaotic map lead to significantly different CPRNG attributes, and have a great effect on the subsequent performance of the chaotic PSO algorithm. It seems that the most promising values for the parameter are $a \in [1.4-1.5]$ and $b \in [0.35-0.45]$.

In [67,68], the authors compared the classical version of PSO with inertia weight (labelled PSO weight), PSO with Lozi map (PSO Lozi) with the sole classical value of the parameters $a = 1.7$, $b = 0.5$, and Differential Evolution (DE) with the “DE/Rand/1/Bin” strategy (see the next Section), for which they used four different benchmark functions in different dimensions:

- The first De Jong function:

$$f(x) = \sum_{i=1}^{\text{dim}} x_i^2 \quad (41)$$

- The second De Jong function:

$$f(x) = \sum_{i=1}^{\text{dim}-1} 100(x_i^2 - x_{i+1})^2 + (1 - x_i)^2 \quad (42)$$

- The Rastrigin function:

$$f(x) = 10\text{dim} \sum_{i=1}^{\text{dim}} x_i^2 - 10\cos(2\pi x_i) \quad (43)$$

- The Schwefel function:

$$f(x) = \sum_{i=1}^{\text{dim}} -x_i \sin(\sqrt{|x_i|}) \quad (44)$$

The basic PSO control parameters were set based on previous experiments and the literature, as follows: population size 30, iterations/generations 10 * the dimension, and dimensions of 2, 5, 10, 20, and 40.

For statistical reasons, optimization for each dimension value was repeated thirty times.

The numerical results with the first De Jong function highlighted that implementation of the chaotic number generator improved the performance of the PSO algorithm. The results were better or comparable with those of differential evolution. In the case of higher dimensions (20 and 40), DE provided better results, and the differences between the results of PSO Weight and PSO Lozi were less significant.

- For the second De Jong function, PSO Lozi obtained the best results in almost all tests performed with this benchmark function. The performance of both PSO algorithms was better than that of DE.
### 4.1.3. Assessment of Enhanced PSO Algorithm with Lozi Chaotic Map

In order to better evaluate the improvement conferred by the use of chaos in PSO algorithms, Pluhacek et al. [65], investigated the effect of tuning the control parameters a and b of (3) (recall that in all three above examples the parameter values were fixed to a = 1.7 and b = 0.5). Three different benchmark functions from the IEEE’s CEC 2013 benchmark suite [66] were selected for the tuning experiments. The controlling parameters of the Lozi map were set as follows:

a : 1.3 – 1.7, step 0.05

b : 0.1 – 0.6, step 0.05

The benchmark functions were evaluated for dimension = 10, population size = 40, number of iterations = 2500, $w_{start} = 0.9$, and $w_{end} = 0.4$ according to the demands in [66]. The PSO algorithm with an adequately set Chaotic Pseudo-Random Number Generator (CPRNG) was run 100 times, and the mean results for each function and Lozi map setting were analyzed. The presented results support the claim that different settings of the chaotic map lead to significantly different CPRNG attributes, and have a great effect on the subsequent performance of the chaotic PSO algorithm. It seems that the most promising values for the parameter are $a \in [1.4–1.5]$ and $b \in [0.35–0.45]$.

In [67,68], the authors compared the classical version of PSO with inertia weight (labelled PSO weight), PSO with Lozi map (PSO Lozi) with the sole classical value of the parameters $a = 1.7$, $b = 0.5$, and Differential Evolution (DE) with the “DE/Rand/1/Bin” strategy (see the next Section), for which they used four different benchmark functions in different dimensions:

- The first De Jong function:

$$f(x) = \sum_{i=1}^{\text{dim}} x_i^2 \qquad (41)$$

- The second De Jong function:

$$f(x) = \sum_{i=1}^{\text{dim}-1} 100(x_i^2 - x_{i+1})^2 + (1 - x_i)^2 \qquad (42)$$

- The Rastrigin function:

$$f(x) = 10 \text{dim} \sum_{i=1}^{\text{dim}} x_i^2 - 10 \cos(2\pi x_i) \qquad (43)$$

- The Schwefel function:

$$f(x) = \sum_{i=1}^{\text{dim}} -x_i \sin(\sqrt{|x_i|}) \qquad (44)$$

The basic PSO control parameters were set based on previous experiments and the literature, as follows: population size 30, iterations/generations 10 * the dimension, and dimensions of 2, 5, 10, 20, and 40.

For statistical reasons, optimization for each dimension value was repeated thirty times.

The numerical results with the first De Jong function highlighted that implementation of the chaotic number generator improved the performance of the PSO algorithm. The results were better or comparable with those of differential evolution. In the case of higher dimensions (20 and 40), DE provided better results, and the differences between the results of PSO Weight and PSO Lozi were less significant.

- For the second De Jong function, PSO Lozi obtained the best results in almost all tests performed with this benchmark function. The performance of both PSO algorithms was better than that of DE.
- In the case of Rastrigin's benchmark function, there was no significant improvement in PSO performance; however, both PSO algorithms were significantly worse than DE for dim = 40.

- The results for Schwefel's function showed an increasing difference between the median values for PSO Weight and PSO Lozi, together with increasing dimension in favour of the PSO enhanced with the Lozi map. As in the previous case, both algorithms were surpassed by DE.

Pluhacek et al. deepened their previous analysis in [69], enlarging the tested population size from 30 to 50, 75, 100, 150, 200, 300, and 400. For smaller population sizes, they found similar results for the first and second De Jong functions as well as for the Rastrigin and Schwefel functions (for an example, see the case dim = 40 for the Second De Jong function in Table 1).

<table><caption>**Table 1.** Results for the second De Jong function for dim = 40.</caption><thead><tr><th>Result for dim = 40</th><th>PSO Weight</th><th>PSO Lozi</th><th>DE</th></tr></thead><tbody><tr><td>The worst result</td><td><b>1.03</b> × 10<sup>0</sup></td><td>1.33 × 10<sup>0</sup></td><td>1.34 × 10<sup>2</sup></td></tr><tr><td>The best result</td><td>7.93 × 10<sup>-2</sup></td><td><b>2.25</b> × 10<sup>-2</sup></td><td>5.02 × 10<sup>1</sup></td></tr><tr><td>Median</td><td>3.74 × 10<sup>-1</sup></td><td><b>2.19</b> × 10<sup>-1</sup></td><td>7.77 × 10<sup>1</sup></td></tr></tbody></table>

However, they found that increasing the size of the population led to significantly improved performance of both PSO algorithms with inertia weight (Table 2).

<table><caption>**Table 2.** Mean value for thirty runs with Schwefel's function, dim = 20, and generations = 200.</caption><thead><tr><th>Pop. number</th><th>30</th><th>50</th><th>75</th><th>100</th><th>150</th><th>200</th><th>300</th><th>400</th></tr></thead><tbody><tr><td>PSO Weight</td><td>-3697.6</td><td>-3873.6</td><td>-4141.0</td><td>-4255.8</td><td>-4329.6</td><td>-4866.4</td><td>-5316.4</td><td>-5377.7</td></tr><tr><td>PSO Lozi</td><td>-4340.1</td><td>-4560.4</td><td>-5032.8</td><td>-5241.8</td><td>-5802.0</td><td>-5998.1</td><td>-6174.6</td><td>-6225.6</td></tr><tr><td>DE</td><td><b>-6100.9</b></td><td><b>-5737.7</b></td><td><b>-5649.1</b></td><td><b>-5500.0</b></td><td>-5635.6</td><td>-5651.5</td><td>-5673.3</td><td>-5651.23</td></tr></tbody></table>

From this investigation, they concluded that the population size has very significant influence on both algorithms. With increasing population size, DE was outperformed by PSO even though both algorithms performed the same amount of CF evaluations.

Widening the scope of their investigations, in [70] Zelinka's team explored two other chaotic maps: first, the dissipative standard map [71]

$$ \begin{cases} x_{n+1} = x_{n+1} + y_{n+1} \pmod{2\pi}, \\ y_{n+1} = by_n + k\sin x_n \pmod{2\pi}, \end{cases} \qquad (45) $$

with $b = 0.6$ and $k = 8.8$, and second, the Arnold's Cat map [72]

$$ \begin{cases} x_{n+1} = x_n + y_n \pmod{1}, \\ y_{n+1} = x_n + ky_n \pmod{1}, \end{cases} \qquad (46) $$

with $k = 0.1$. They compared these to the Lozi map with $a = 1.7$ and $b = 0.5$ using the same test function (41)–(44).

After a very extensive sequence of tests, they concluded that, first, the number generator based on the Arnold's Cat chaotic map does not seem to have any significant effect on the behavior or performance of the PSO algorithm with linear decreasing inertia weight.

Second, the Lozi map used as the number generator seemed to have a significant positive effect on the speed of convergence of the algorithm. This may support the idea of using this modification in real-time optimization tasks on low-dimensional problems, especially because the risk of premature convergence and stagnation in higher dimensions seems to be a significant drawback of the PSO Lozi version. Finally, the PSO with the dissipative standard map seemed able to avoid premature convergence much better than any other
- In the case of Rastrigin's benchmark function, there was no significant improvement in PSO performance; however, both PSO algorithms were significantly worse than DE for dim = 40.

- The results for Schwefel's function showed an increasing difference between the median values for PSO Weight and PSO Lozi, together with increasing dimension in favour of the PSO enhanced with the Lozi map. As in the previous case, both algorithms were surpassed by DE.

Pluhacek et al. deepened their previous analysis in [69], enlarging the tested population size from 30 to 50, 75, 100, 150, 200, 300, and 400. For smaller population sizes, they found similar results for the first and second De Jong functions as well as for the Rastrigin and Schwefel functions (for an example, see the case dim = 40 for the Second De Jong function in Table 1).

<table><caption>**Table 1.** Results for the second De Jong function for dim = 40.</caption><thead><tr><th>Result for dim = 40</th><th>PSO Weight</th><th>PSO Lozi</th><th>DE</th></tr></thead><tbody><tr><td>The worst result</td><td><b>1.03</b> × 10<sup>0</sup></td><td>1.33 × 10<sup>0</sup></td><td>1.34 × 10<sup>2</sup></td></tr><tr><td>The best result</td><td>7.93 × 10<sup>-2</sup></td><td><b>2.25</b> × 10<sup>-2</sup></td><td>5.02 × 10<sup>1</sup></td></tr><tr><td>Median</td><td>3.74 × 10<sup>-1</sup></td><td><b>2.19</b> × 10<sup>-1</sup></td><td>7.77 × 10<sup>1</sup></td></tr></tbody></table>

However, they found that increasing the size of the population led to significantly improved performance of both PSO algorithms with inertia weight (Table 2).

<table><caption>**Table 2.** Mean value for thirty runs with Schwefel's function, dim = 20, and generations = 200.</caption><thead><tr><th>Pop. number</th><th>30</th><th>50</th><th>75</th><th>100</th><th>150</th><th>200</th><th>300</th><th>400</th></tr></thead><tbody><tr><td>PSO Weight</td><td>-3697.6</td><td>-3873.6</td><td>-4141.0</td><td>-4255.8</td><td>-4329.6</td><td>-4866.4</td><td>-5316.4</td><td>-5377.7</td></tr><tr><td>PSO Lozi</td><td>-4340.1</td><td>-4560.4</td><td>-5032.8</td><td>-5241.8</td><td>-5802.0</td><td>-5998.1</td><td>-6174.6</td><td>-6225.6</td></tr><tr><td>DE</td><td><b>-6100.9</b></td><td><b>-5737.7</b></td><td><b>-5649.1</b></td><td><b>-5500.0</b></td><td>-5635.6</td><td>-5651.5</td><td>-5673.3</td><td>-5651.23</td></tr></tbody></table>

From this investigation, they concluded that the population size has very significant influence on both algorithms. With increasing population size, DE was outperformed by PSO even though both algorithms performed the same amount of CF evaluations.

Widening the scope of their investigations, in [70] Zelinka's team explored two other chaotic maps: first, the dissipative standard map [71]

$$ \begin{cases} x_{n+1} = x_{n+1} + y_{n+1} \pmod{2\pi}, \\ y_{n+1} = by_n + k\sin x_n \pmod{2\pi}, \end{cases} \qquad (45) $$

with $b = 0.6$ and $k = 8.8$, and second, the Arnold's Cat map [72]

$$ \begin{cases} x_{n+1} = x_n + y_n \pmod{1}, \\ y_{n+1} = x_n + ky_n \pmod{1}, \end{cases} \qquad (46) $$

with $k = 0.1$. They compared these to the Lozi map with $a = 1.7$ and $b = 0.5$ using the same test function (41)–(44).

After a very extensive sequence of tests, they concluded that, first, the number generator based on the Arnold's Cat chaotic map does not seem to have any significant effect on the behavior or performance of the PSO algorithm with linear decreasing inertia weight.

Second, the Lozi map used as the number generator seemed to have a significant positive effect on the speed of convergence of the algorithm. This may support the idea of using this modification in real-time optimization tasks on low-dimensional problems, especially because the risk of premature convergence and stagnation in higher dimensions seems to be a significant drawback of the PSO Lozi version. Finally, the PSO with the dissipative standard map seemed able to avoid premature convergence much better than any other
version in this study. However, a drawback was its very slow speed of convergence; in other terms, a higher number of generations were needed to find the optimal value, or at least a satisfactory solution.

Finally, based on these conclusions, in [73] they found a new way to improve PSO based on the dissipative standard map and Lozi map by using them sequentially.

## 4.2. Differential Evolution (DE)

Meta-heuristic optimization methods have seen incredible advancements over the two last decades. More efficient computational algorithms have enabled solving of complex problems. One of these modern meta-heuristics is Differential Evolution (DE), an evolutionary computation method which has become widely used in engineering optimization due to its simple structure, ease of use, fast convergence speed, versatility, and robustness.

The main difference between genetic algorithms and DE is that in genetic algorithms mutation is the result of small perturbations to the genes of an individual (potential solution) while in DE mutation is the result of arithmetical combinations of individuals.

### 4.2.1. Overspeed Protection System of a Gas Turbine

In [58], dos Santos Coehlo considered the reliability-redundancy optimization problem of an overspeed protection system for a gas turbine. The fundamental idea behind DE is a scheme whereby it generates the trial parameter vectors. A DE population is subject to mutation, crossover, and selection operators. At each time step, DE mutates vectors by adding weighted random vector differentials to these operators. If the cost of the trial vector is better than that of the target, the target vector is replaced by trial vector in the next generation.

Among the many different strategies possible for DE, dos Santos Coehlo implemented the “DE/rand/1/bin” strategy, meaning that the target vector was randomly selected and only one difference vector was used. The bin acronym indicates that recombination was controlled by a binomial decision rule. Their strategy consisted of seven steps [58]

* Step 1: Choice of the control parameters

* Step 2: Initialization of population with M individuals

* Step 3: Evaluation of individuals

* Step 4: Mutation operation (or differential operation)

* Step 5: Recombination operation

* Step 6: Selection operation

* Step 7: Verification of the stopping criterion

The mutation operation in Step 4 is an operation that adds a vector differential to a population vector of individuals according to

$$z_i(t+1) = x_{i,r_i}(t) + f_m(t)[x_{i,r_2}(t) - x_{i,r_3}(t)], \quad (47)$$

where $i = 1, 2, \dots, M$ is the individual's index of population; $t$ is the generation (time), $x_i(t) = [x_{i1}(t), x_{i2}(t), \dots, x_{in}(t)]^T$ stands for the $i$-th individual of a population of real-valued $N$-dimensional vectors, $z_i(t) = [z_{i1}(t), z_{i2}(t), \dots, z_{in}(t)]^T$ stands for the $i$-th individual of a mutant vector, and $r_1, r_2, \dots, r_3$ are mutually different integers that are different from the running index $i$, which is randomly selected with a uniform distribution from the set $\{1, 2, \dots, i-1, i+1, \dots, N\}$. The mutation factor $f_m(t) > 0$ is a real parameter which controls the amplification of the difference between two individuals with indexes $r_2$ and $r_3$.

It is well known that DE is particularly sensitive to its control parameters, particularly the mutation factor $f_m(t)$. In this context, due to the ergodic and dynamic properties of the Lozi map variables, dos Santos Coehlo introduced Lozi map-based DE (LDE), which can be useful in mutation factor design for escaping from local optima in optimization problems. The LDE approach was employed to increase the diversity of individuals' population
version in this study. However, a drawback was its very slow speed of convergence; in other terms, a higher number of generations were needed to find the optimal value, or at least a satisfactory solution.

Finally, based on these conclusions, in [73] they found a new way to improve PSO based on the dissipative standard map and Lozi map by using them sequentially.

## 4.2. Differential Evolution (DE)

Meta-heuristic optimization methods have seen incredible advancements over the two last decades. More efficient computational algorithms have enabled solving of complex problems. One of these modern meta-heuristics is Differential Evolution (DE), an evolutionary computation method which has become widely used in engineering optimization due to its simple structure, ease of use, fast convergence speed, versatility, and robustness.

The main difference between genetic algorithms and DE is that in genetic algorithms mutation is the result of small perturbations to the genes of an individual (potential solution) while in DE mutation is the result of arithmetical combinations of individuals.

### 4.2.1. Overspeed Protection System of a Gas Turbine

In [58], dos Santos Coehlo considered the reliability-redundancy optimization problem of an overspeed protection system for a gas turbine. The fundamental idea behind DE is a scheme whereby it generates the trial parameter vectors. A DE population is subject to mutation, crossover, and selection operators. At each time step, DE mutates vectors by adding weighted random vector differentials to these operators. If the cost of the trial vector is better than that of the target, the target vector is replaced by trial vector in the next generation.

Among the many different strategies possible for DE, dos Santos Coehlo implemented the “DE/rand/1/bin” strategy, meaning that the target vector was randomly selected and only one difference vector was used. The bin acronym indicates that recombination was controlled by a binomial decision rule. Their strategy consisted of seven steps [58]

* Step 1: Choice of the control parameters

* Step 2: Initialization of population with M individuals

* Step 3: Evaluation of individuals

* Step 4: Mutation operation (or differential operation)

* Step 5: Recombination operation

* Step 6: Selection operation

* Step 7: Verification of the stopping criterion

The mutation operation in Step 4 is an operation that adds a vector differential to a population vector of individuals according to

$$z_i(t+1) = x_{i,r_i}(t) + f_m(t)[x_{i,r_2}(t) - x_{i,r_3}(t)], \quad (47)$$

where $i = 1,2,...,M$ is the individual's index of population; $t$ is the generation (time), $x_i(t) = [x_{i1}(t), x_{i2}(t),...,x_{in}(t)]^T$ stands for the $i$-th individual of a population of real-valued N-dimensional vectors, $z_i(t) = [z_{i1}(t), z_{i2}(t),...,z_{in}(t)]^T$ stands for the $i$-th individual of a mutant vector, and $r_1, r_2$, and $r_3$ are mutually different integers that are different from the running index $i$, which is randomly selected with a uniform distribution from the set $\{1,2,...,i-1,i+1,...,N\}$. The mutation factor $f_m(t) > 0$ is a real parameter which controls the amplification of the difference between two individuals with indexes $r_2$ and $r_3$.

It is well known that DE is particularly sensitive to its control parameters, particularly the mutation factor $f_m(t)$. In this context, due to the ergodic and dynamic properties of the Lozi map variables, dos Santos Coehlo introduced Lozi map-based DE (LDE), which can be useful in mutation factor design for escaping from local optima in optimization problems. The LDE approach was employed to increase the diversity of individuals' population
and prevent the premature convergence. The LDE approach is a modification of (47) provided by

$$z_i(t+1) = x_{i,r_i}(t) + L(t)[x_{i,r_2}(t) - x_{i,r_3}(t)], \quad (48)$$

where $L(t)$ is the Lozi map (3) normalized following (30) in the range [0.1, 0.9].

In each case study, fifty independent runs were carried out for each of the optimization methods involving fifty different initial trial solutions for each optimization method. Moreover, the DE and LDE approaches were adopted using 2250 cost function evaluations in each run. Both the DE and the LDE optimization methods were successfully applied to estimate the parameters of the overspeed protection system of a gas turbine in a reliability-redundancy optimization problem including discrete and continuous variables. Among seven variants of each of these methods, three DE variants and LDE five variants succeeded in finding the best solution for the tested methods. The author concluded that the LDE strategies have potential to be further applied in constrained optimization problems.

### 4.2.2. Chemical Reactor Optimization

Senkerik et al. used Differential Evolution in the optimization task of the batch reactor geometry [74]. The novelty of this approach is that Lozi map was not only used as a chaotic pseudo-random number generator to drive the mutation process as in [58], it was used for the crossover process in DE. The obtained results were compared with original reactor geometry and process parameters adjustment. The statistical analysis of the results provided by chaos-driven DE was compared with canonical DE strategy.

The authors used a simplified mathematical model of the reactor consisting of two physical inputs: one for chemical substances **Chemical FK (Filter Cake)**, with parameters temperature $T_{FK}$, mass flow rate $\dot{m}_{FK}$, and specific heat $c_{FK}$, and one for the cooling medium of the temperature $T_{VP}$, mass flow rate $\dot{m}_{VP}$, and specific heat $c_{VP}$. The reactor had one output, with the cooling medium flowing through the jacket inner space of the reactor (with the volume related to the mass $m_{VR}$) and flowing out through the second output, with the following parameters: mass flow rate $m_V$, temperature $T_V$, and specific heat $c_V$.

There was an initial batch inside the reactor, with the mass parameter $m_P$ introduced at the beginning of the process. The chemical FK was then added to this initial batch, resulting in a reaction mixture inside the reactor with the following parameters: total mass $m$, temperature $T$, and specific heat $c_R$. In addition, the partially unreacted portions of the chemical FK were described by the concentration parameter $a_{FK}$.

This “batch” technique allowed for partial control of the temperature of the reaction mixture through controlled feeding of the input chemical FK. In general, this reaction is highly exothermic; thus, the most important parameter is the temperature of the reaction mixture. This temperature must not exceed 100 °C because of safety and product quality requirements. The original design of the reactor was based on standard chemical-technological methods as part of a proposal involving the physical dimensions of the reactor and the parameters of the chemical substances. In the paper, these values were called the **original parameters**. The main objective of the optimization was to achieve the processing of large amounts of the chemical FK in a very short time. A system of four balance equations (49) was used to describe the reactor process, while another Equation (50) defined the term $k$:

$$\left\{
\begin{array}{l}
\dot{m}_{FK} = m'[t], \\
\dot{m}_{FK} = m[t]a'_{FK}[t] + km[t]a_{FK}[t], \\
\dot{m}_{FKc_{FK}T_{FK}} + \Delta H_r km[t]a_{FK}[t] = KS(T[t] - T_V[t]) + m[t]c_R T'[t], \\
\dot{m}_{VCV}T_{VP} + KS(T[t] - T_V[t]) = \dot{m}_{VCV}T_V[t] + m_{VRCV}[t]T'_V[t],
\end{array}
\right.
\quad (49)$$

$$k = Ae^{-\frac{E}{RT}[t]}. \quad (50)$$
and prevent the premature convergence. The LDE approach is a modification of (47) provided by

$$z_i(t+1) = x_{i,r_i}(t) + L(t)[x_{i,r_2}(t) - x_{i,r_3}(t)], \quad (48)$$

where $L(t)$ is the Lozi map (3) normalized following (30) in the range [0.1, 0.9].

In each case study, fifty independent runs were carried out for each of the optimization methods involving fifty different initial trial solutions for each optimization method. Moreover, the DE and LDE approaches were adopted using 2250 cost function evaluations in each run. Both the DE and the LDE optimization methods were successfully applied to estimate the parameters of the overspeed protection system of a gas turbine in a reliability-redundancy optimization problem including discrete and continuous variables. Among seven variants of each of these methods, three DE variants and LDE five variants succeeded in finding the best solution for the tested methods. The author concluded that the LDE strategies have potential to be further applied in constrained optimization problems.

### 4.2.2. Chemical Reactor Optimization

Senkerik et al. used Differential Evolution in the optimization task of the batch reactor geometry [74]. The novelty of this approach is that Lozi map was not only used as a chaotic pseudo-random number generator to drive the mutation process as in [58], it was used for the crossover process in DE. The obtained results were compared with original reactor geometry and process parameters adjustment. The statistical analysis of the results provided by chaos-driven DE was compared with canonical DE strategy.

The authors used a simplified mathematical model of the reactor consisting of two physical inputs: one for chemical substances **Chemical FK (Filter Cake)**, with parameters temperature $T_{FK}$, mass flow rate $\dot{m}_{FK}$, and specific heat $c_{FK}$, and one for the cooling medium of the temperature $T_{VP}$, mass flow rate $\dot{m}_{VP}$, and specific heat $c_{VP}$. The reactor had one output, with the cooling medium flowing through the jacket inner space of the reactor (with the volume related to the mass $m_{VR}$) and flowing out through the second output, with the following parameters: mass flow rate $m_V$, temperature $T_V$, and specific heat $c_V$.

There was an initial batch inside the reactor, with the mass parameter $m_P$ introduced at the beginning of the process. The chemical FK was then added to this initial batch, resulting in a reaction mixture inside the reactor with the following parameters: total mass $m$, temperature $T$, and specific heat $c_R$. In addition, the partially unreacted portions of the chemical FK were described by the concentration parameter $a_{FK}$.

This “batch” technique allowed for partial control of the temperature of the reaction mixture through controlled feeding of the input chemical FK. In general, this reaction is highly exothermic; thus, the most important parameter is the temperature of the reaction mixture. This temperature must not exceed 100 °C because of safety and product quality requirements. The original design of the reactor was based on standard chemical-technological methods as part of a proposal involving the physical dimensions of the reactor and the parameters of the chemical substances. In the paper, these values were called the **original parameters**. The main objective of the optimization was to achieve the processing of large amounts of the chemical FK in a very short time. A system of four balance equations (49) was used to describe the reactor process, while another Equation (50) defined the term $k$:

$$\left\{
\begin{array}{l}
\dot{m}_{FK} = m'[t], \\
\dot{m}_{FK} = m[t]a'_{FK}[t] + km[t]a_{FK}[t], \\
\dot{m}_{FKc_{FK}T_{FK}} + \Delta H_r km[t]a_{FK}[t] = KS(T[t] - T_V[t]) + m[t]c_R T'[t], \\
\dot{m}_{VCV}T_{VP} + KS(T[t] - T_V[t]) = \dot{m}_{VCV}T_V[t] + m_{VRCV}[t]T'_V[t],
\end{array}
\right.
\quad (49)$$

$$k = Ae^{-\frac{E}{RT}[t]}. \quad (50)$$
Similar to dos Santos Coehlo [58], Senkerik et al. implemented the “DE/rand/1/bin” strategy; for the mutation operation, they used

$$u_{j,i,G+1} = x_{j,r_1,G} + F \cdot (x_{j,r_2,G} - x_{j,r_3,G}) \quad (51)$$

to minimize the cost function (CF)

$$f_{cost} = \sum_{t=0}^{t_1} |w - T[t]| + \sum_{t=0}^{t_1} a_{FK}[t] + pen.1 + pen.2 \quad (52)$$

with two penalizations:

$$pen.1 = \begin{cases} 0, & \text{Max}(T[\tau]) \le 323.15, \\ 50,000, & \text{else,} \end{cases} \quad (53)$$

for $\tau \in (t_2, t_3)$ and

$$pen.2 = \begin{cases} 0, & \text{Max}(T[\tau]) \le 373.15, \\ 50,000, & \text{else,} \end{cases} \quad (54)$$

for $\tau \in (0, t_3)$. The time intervals were set as follows: $t_1 = 15,000$ s, $t_1 = 20,000$ s, $t_1 = 25,000$ s.

The first penalization helps to find solutions in which the temperature of reaction mixture quickly cools down to its initial state, which shortens the process duration.

The second corresponds to the critical temperature. The best optimization results are shown in Tables 3 and 4. Their analysis showed that the temperature of the reaction mixture did not exceed the critical value. The maximum temperature was 373.11 K (99.95 °C). The required temperature $w$ used in the cost function was 370.0 K (96.85 °C).

Senkerik et al. concluded that “based on statistical analysis, it is possible to claim that embedding the chaotic dynamics in the form of a chaotic pseudo-random number generator into differential evolution algorithm significantly improves the performance of DE.”

<table><caption>**Table 3.** Statistical results of all 100 runs of Chaos DE and Canonical DE.</caption><thead><tr><th>DE Version</th><th>Avg. CF</th><th>Median CF</th><th>Std. Dev.</th><th>Max CF</th><th>Min CF</th></tr></thead><tbody><tr><td>Canonical DE</td><td>21,259.0</td><td>21,234.1</td><td>94.1</td><td>21,588.2</td><td>21,089.2</td></tr><tr><td>Chaos DE Lozi</td><td>21,107.6</td><td>21,107.4</td><td>25.4</td><td>21,160.7</td><td>21,059.0</td></tr></tbody></table>

<table><caption>**Table 4.** Average CF values for generation Nos. 50, 100, 150 and 200 from all 100 runs of Chaos DE and Canonical DE.</caption><thead><tr><th>Generation No.:</th><th>50</th><th>100</th><th>150</th><th>200</th></tr></thead><tbody><tr><td>Avg. CF for Canonical DE</td><td>25,320.1</td><td>22,719.9</td><td>21,619.8</td><td>21,259.0</td></tr><tr><td>Avg. CF for Chaos DE Lozi</td><td>24,707.1</td><td>21,992.1</td><td>21,265.2</td><td>21,107.6</td></tr></tbody></table>

### 4.2.3. Evolutionary Chaos Control Optimization

An original study introduced by Zelinka's team sought to utilize hidden chaotic dynamics in pseudo-random sequences generated via chaotic map to help the differential evolution algorithm search for the best controller settings for the same chaotic system. In [75–77], they considered (55) with the standard parameter values of $a = 1.7$ and $b = 0.5$, and tried to stabilize desired Unstable Periodic Orbits (UPO) using the chaos-driven DE for tuning the parameters of the Extended Delay Feedback (ETDAS) control method proposed by Pyragas [78].

$$\mathcal{L}_{a,b} \begin{pmatrix} X \\ Y \end{pmatrix} = \begin{pmatrix} 1 - a|X| + Y, \\ bX. \end{pmatrix} \quad (55)$$
Similar to dos Santos Coehlo [58], Senkerik et al. implemented the “DE/rand/1/bin” strategy; for the mutation operation, they used

$$u_{j,i,G+1} = x_{j,r_1,G} + F \cdot (x_{j,r_2,G} - x_{j,r_3,G}) \quad (51)$$

to minimize the cost function (CF)

$$f_{cost} = \sum_{t=0}^{t_1} |w - T[t]| + \sum_{t=0}^{t_1} a_{FK}[t] + pen.1 + pen.2 \quad (52)$$

with two penalizations:

$$pen.1 = \begin{cases} 0, & \text{Max}(T[\tau]) \le 323.15, \\ 50,000, & \text{else,} \end{cases} \quad (53)$$

for $\tau \in (t_2, t_3)$ and

$$pen.2 = \begin{cases} 0, & \text{Max}(T[\tau]) \le 373.15, \\ 50,000, & \text{else,} \end{cases} \quad (54)$$

for $\tau \in (0, t_3)$. The time intervals were set as follows: $t_1 = 15,000$ s, $t_1 = 20,000$ s, $t_1 = 25,000$ s.

The first penalization helps to find solutions in which the temperature of reaction mixture quickly cools down to its initial state, which shortens the process duration.

The second corresponds to the critical temperature. The best optimization results are shown in Tables 3 and 4. Their analysis showed that the temperature of the reaction mixture did not exceed the critical value. The maximum temperature was 373.11 K (99.95 °C). The required temperature $w$ used in the cost function was 370.0 K (96.85 °C).

Senkerik et al. concluded that “based on statistical analysis, it is possible to claim that embedding the chaotic dynamics in the form of a chaotic pseudo-random number generator into differential evolution algorithm significantly improves the performance of DE.”

<table><caption>**Table 3.** Statistical results of all 100 runs of Chaos DE and Canonical DE.</caption><thead><tr><th>DE Version</th><th>Avg. CF</th><th>Median CF</th><th>Std. Dev.</th><th>Max CF</th><th>Min CF</th></tr></thead><tbody><tr><td>Canonical DE</td><td>21,259.0</td><td>21,234.1</td><td>94.1</td><td>21,588.2</td><td>21,089.2</td></tr><tr><td>Chaos DE Lozi</td><td>21,107.6</td><td>21,107.4</td><td>25.4</td><td>21,160.7</td><td>21,059.0</td></tr></tbody></table>

<table><caption>**Table 4.** Average CF values for generation Nos. 50, 100, 150 and 200 from all 100 runs of Chaos DE and Canonical DE.</caption><thead><tr><th>Generation No.:</th><th>50</th><th>100</th><th>150</th><th>200</th></tr></thead><tbody><tr><td>Avg. CF for Canonical DE</td><td>25,320.1</td><td>22,719.9</td><td>21,619.8</td><td>21,259.0</td></tr><tr><td>Avg. CF for Chaos DE Lozi</td><td>24,707.1</td><td>21,992.1</td><td>21,265.2</td><td>21,107.6</td></tr></tbody></table>

### 4.2.3. Evolutionary Chaos Control Optimization

An original study introduced by Zelinka's team sought to utilize hidden chaotic dynamics in pseudo-random sequences generated via chaotic map to help the differential evolution algorithm search for the best controller settings for the same chaotic system. In [75–77], they considered (55) with the standard parameter values of $a = 1.7$ and $b = 0.5$, and tried to stabilize desired Unstable Periodic Orbits (UPO) using the chaos-driven DE for tuning the parameters of the Extended Delay Feedback (ETDAS) control method proposed by Pyragas [78].

$$\mathcal{L}_{a,b} \begin{pmatrix} X \\ Y \end{pmatrix} = \begin{pmatrix} 1 - a|X| + Y, \\ bX. \end{pmatrix} \quad (55)$$
This control method takes the form

$$ egin{cases} x_{n+1} = 1 - a|X_n| + bY_n + F_n, \\ F_n = K[(1 - R)S_{n-m} - x_n], \\ S_n = x_n + RS_{n-m}, \end{cases} \tag{56} $$

where K and R are adjustable constants which have to be evolutionary tuned, F is the perturbation, S is provided by a delay equation utilizing the previous states of the system, and m is the period of the m-periodic orbit to be stabilized. The perturbation F_n in Equation (56) may have an arbitrarily large value, which can cause divergence of the system outside the output interval of the Lozi map system [-1.4, 1.4]. Therefore, F_n should have a value between [-F_max, F_max].

The suitable F_max value was obtained from the evolutionary optimization process.

In these articles, two cost function designs, CF SIMPLE and CF UNI, were utilized and compared. The goal of the first basic CF SIMPLE, which can be used only for the stabilization of p-1 orbit, was to minimize the area created by the difference between the required state and the real system output on the whole simulation interval [0, τi]:

$$ CF_{SIMPLE} = \sum_{t=0}^{\tau_i} |TS_t - AS_t|, \tag{57} $$

where TS_t is the target state and AS_t the actual state.

Nevertheless, this simple approach has one prominent disadvantage, which is the inclusion of the initial chaotic transient behavior of the not-yet-stabilized system in the cost function value. To avoid this disadvantage, a universal cost function suppressing the initial chaotic transient behavior was introduced to ensure the stabilization of either the p-1 orbit (stable state) or the higher periodic orbit while retaining the possibility of adding penalization rules:

$$ CF_{UNIVERSAL} = pen_1 + \sum_{t=\tau_1}^{\tau_2} |TS_t - AS_t|, \tag{58} $$

where τ1 is the first minimal value of the difference between TS and AS and τ2 indicates the end of the optimization interval (τ1 + τs), with pen1 = 0 if τi - τ2 ≥ τs and pen1 = 10 * (τi - τ2) if τi - τ2 < τs (i.e., late stabilization).

After hundreds of numerical tests, the researchers on Zelinka's team concluded that embedding the chaotic dynamics into the differential evolution algorithm in the form of a chaotic pseudo-random number generator may help to improve their performance and robustness. In this way, optimal solutions were obtained securing very fast and precise stabilization for a convenient cost function surface in the CF SIMPLE case and a very chaotic and nonlinear cost function surface in the CF UNIVERSAL case.

### 4.2.4. Comparison of PSO with DE

Having presented the PSO and DE optimization methods, it is natural to compare them.

In [79], Senkerik et al. compared the swarm-based PSO technique to Differential Evolution on the task described above involving optimal evolutionary tuning of controller parameters for the stabilization of a Lozi discrete chaotic system. The novelty of this approach was that the most utilized examples of evolutionary/swarm based algorithms were compared directly on the highly nonlinear and complex multimodal optimization and simulation task. The simulations and comparisons were performed for three different required final behaviors of the chaotic system which were simulated in the following form:

* Case study 1: p-1 UPO, Lozi map as controlled system with CF SIMPLE (57)

* Case study 2: p-2 UPO with the same control

* Case study 3: higher order p-4 UPO with the same control
This control method takes the form

$$ egin{cases} x_{n+1} = 1 - a|X_n| + bY_n + F_n, \\ F_n = K[(1 - R)S_{n-m} - x_n], \\ S_n = x_n + RS_{n-m}, \end{cases} \quad (56) $$

where K and R are adjustable constants which have to be evolutionary tuned, F is the perturbation, S is provided by a delay equation utilizing the previous states of the system, and m is the period of the m-periodic orbit to be stabilized. The perturbation F_n in Equation (56) may have an arbitrarily large value, which can cause divergence of the system outside the output interval of the Lozi map system [-1.4, 1.4]. Therefore, F_n should have a value between [-F_max, F_max].

The suitable F_max value was obtained from the evolutionary optimization process.

In these articles, two cost function designs, CF SIMPLE and CF UNI, were utilized and compared. The goal of the first basic CF SIMPLE, which can be used only for the stabilization of p-1 orbit, was to minimize the area created by the difference between the required state and the real system output on the whole simulation interval [0, τi]:

$$ CF_{SIMPLE} = \sum_{t=0}^{\tau_i} |TS_t - AS_t|, \quad (57) $$

where TS_t is the target state and AS_t the actual state.

Nevertheless, this simple approach has one prominent disadvantage, which is the inclusion of the initial chaotic transient behavior of the not-yet-stabilized system in the cost function value. To avoid this disadvantage, a universal cost function suppressing the initial chaotic transient behavior was introduced to ensure the stabilization of either the p-1 orbit (stable state) or the higher periodic orbit while retaining the possibility of adding penalization rules:

$$ CF_{UNIVERSAL} = pen_1 + \sum_{t=\tau_1}^{\tau_2} |TS_t - AS_t|, \quad (58) $$

where τ1 is the first minimal value of the difference between TS and AS and τ2 indicates the end of the optimization interval (τ1 + τs), with pen1 = 0 if τi - τ2 ≥ τs and pen1 = 10 * (τi - τ2) if τi - τ2 < τs (i.e., late stabilization).

After hundreds of numerical tests, the researchers on Zelinka's team concluded that embedding the chaotic dynamics into the differential evolution algorithm in the form of a chaotic pseudo-random number generator may help to improve their performance and robustness. In this way, optimal solutions were obtained securing very fast and precise stabilization for a convenient cost function surface in the CF SIMPLE case and a very chaotic and nonlinear cost function surface in the CF UNIVERSAL case.

### 4.2.4. Comparison of PSO with DE

Having presented the PSO and DE optimization methods, it is natural to compare them.

In [79], Senkerik et al. compared the swarm-based PSO technique to Differential Evolution on the task described above involving optimal evolutionary tuning of controller parameters for the stabilization of a Lozi discrete chaotic system. The novelty of this approach was that the most utilized examples of evolutionary/swarm based algorithms were compared directly on the highly nonlinear and complex multimodal optimization and simulation task. The simulations and comparisons were performed for three different required final behaviors of the chaotic system which were simulated in the following form:

* Case study 1: p-1 UPO, Lozi map as controlled system with CF SIMPLE (57)

* Case study 2: p-2 UPO with the same control

* Case study 3: higher order p-4 UPO with the same control
Within their research, a total of fifty simulations for each experiments was performed in the *Wolfram Mathematica* environment. All experiments used different initializations, i.e., a different initial population was generated in each run of DE/PSO.

The parameter settings for DE were population size = 25, *F* = 0.5, *Cr* = 0.9, number of generations = 300, and CF Evaluations (CFE) = 7500.

For PSO, they were population size = 25, *c*₁ = *c*₂ = 2.0, number of iterations = 300, and CF Evaluations (CFE) = 7500.

All simulations were successful and provided new optimal settings for the ETDAS control method, securing the fast stabilization of the chaotic system at the required behaviors, which were p-1 UPO (stable state), p-2 UPO (oscillation between two values), and p-4 UPO.

These results allowed the authors to compare DE and PSO in each case.

* Case Study 1: from the statistical point of view, the performance of both heuristics was similar. Nevertheless, DE converged towards the optimal solution in only twenty generations, and all fifty runs provided identical results. The convergence for PSO was much slower. Therefore, DE is more suitable for online control of nonlinear chaotic dynamics.

* Case Study 2: as in Case Study 1, the structure and simulation outputs showed similar features, with an even larger difference in favor of the DE algorithm.

* Case Study 3: dealing with the most complex and highly nonlinear dynamics, this case showed several interesting features. The classical geometrical (vector crossover)-based evolutionary algorithm was stacked in many suboptimal solutions, with only a small chance of leaving this area of the solution space. The PSO (swarm-based) algorithm was statistically better in the searching process, with a lower standard deviation and range of CF. Even when DE found a lower final CF value, PSO seemed to be a better and more robust choice for optimization in extremely nonlinear solution spaces due to its better natural exploration ability.

Senkerik et al. in [76] continued this comparison, using both *CF<sub>SIMPLE</sub>* and *CF<sub>UNIVERSAL</sub>*:

* Case Study 4: p-1 UPO, Lozi map as controlled system with *CF<sub>UNIVERSAL</sub>* (58)

* Case Study 5: p-2 UPO with the same control

* Case Study 6: higher order p-4 UPO with the same control

The authors found that *CF<sub>UNIVERSAL</sub>* led to faster and more precise stabilization of the chaotic system.

### 4.2.5. Optimal Tuning of Speed Control for a Brushless DC Motor

As a last example involving the DE algorithm, we consider the work of Rodríguez-Molina et al. [80], who assessed the efficiency of the controller performance in a Brushless Direct current (BLDC) motor in an uncertain environment. The performance of the BLDC depends highly on the adaptability of the controller gains. The authors proposed a chaotic adaptive tuning strategy for controller gains (CATSCG) for speed regulation of BLDC motors. The CATSCG includes two sequential dynamic optimization stages based on identification and predictive processes, with a novel chaotic online differential evolution (CODE) used to provide the controller gain at each predefined time interval.

The CATSCG consisted of a two-stage tuning strategy; the dynamics associated with the future behavior of the BLDC motor are estimated in the first stage using the experience gained from the past optimization process. These dynamics are used in the second stage to find the controller gains that directly impact the closed-loop system responses through the novel Chaotic Online Differential Evolution (CODE) approach. Furthermore, CODE includes the Lozi chaotic initial populations in the differential evolution algorithm to enhance the diversity of candidate solutions in the dynamic environment, which further improves the quality of the controller gains under the effects of uncertainties.

The proposed chaotic online differential evolution (CODE) is an optimizer based on the DE/rand/1/bin variant of differential evolution (see Section 4.2.1). It incorporates
Within their research, a total of fifty simulations for each experiments was performed in the *Wolfram Mathematica* environment. All experiments used different initializations, i.e., a different initial population was generated in each run of DE/PSO.

The parameter settings for DE were population size = 25, *F* = 0.5, *Cr* = 0.9, number of generations = 300, and CF Evaluations (CFE) = 7500.

For PSO, they were population size = 25, $c_1 = c_2 = 2.0$, number of iterations = 300, and CF Evaluations (CFE) = 7500.

All simulations were successful and provided new optimal settings for the ETDAS control method, securing the fast stabilization of the chaotic system at the required behaviors, which were p-1 UPO (stable state), p-2 UPO (oscillation between two values), and p-4 UPO.

These results allowed the authors to compare DE and PSO in each case.

* Case Study 1: from the statistical point of view, the performance of both heuristics was similar. Nevertheless, DE converged towards the optimal solution in only twenty generations, and all fifty runs provided identical results. The convergence for PSO was much slower. Therefore, DE is more suitable for online control of nonlinear chaotic dynamics.

* Case Study 2: as in Case Study 1, the structure and simulation outputs showed similar features, with an even larger difference in favor of the DE algorithm.

* Case Study 3: dealing with the most complex and highly nonlinear dynamics, this case showed several interesting features. The classical geometrical (vector crossover)-based evolutionary algorithm was stacked in many suboptimal solutions, with only a small chance of leaving this area of the solution space. The PSO (swarm-based) algorithm was statistically better in the searching process, with a lower standard deviation and range of CF. Even when DE found a lower final CF value, PSO seemed to be a better and more robust choice for optimization in extremely nonlinear solution spaces due to its better natural exploration ability.

Senkerik et al. in [76] continued this comparison, using both $CF_{SIMPLE}$ and $CF_{UNIVERSAL}$:

* Case Study 4: p-1 UPO, Lozi map as controlled system with $CF_{UNIVERSAL}$ (58)

* Case Study 5: p-2 UPO with the same control

* Case Study 6: higher order p-4 UPO with the same control

The authors found that $CF_{UNIVERSAL}$ led to faster and more precise stabilization of the chaotic system.

### 4.2.5. Optimal Tuning of Speed Control for a Brushless DC Motor

As a last example involving the DE algorithm, we consider the work of Rodríguez-Molina et al. [80], who assessed the efficiency of the controller performance in a Brushless Direct current (BLDC) motor in an uncertain environment. The performance of the BLDC depends highly on the adaptability of the controller gains. The authors proposed a chaotic adaptive tuning strategy for controller gains (CATSCG) for speed regulation of BLDC motors. The CATSCG includes two sequential dynamic optimization stages based on identification and predictive processes, with a novel chaotic online differential evolution (CODE) used to provide the controller gain at each predefined time interval.

The CATSCG consisted of a two-stage tuning strategy; the dynamics associated with the future behavior of the BLDC motor are estimated in the first stage using the experience gained from the past optimization process. These dynamics are used in the second stage to find the controller gains that directly impact the closed-loop system responses through the novel Chaotic Online Differential Evolution (CODE) approach. Furthermore, CODE includes the Lozi chaotic initial populations in the differential evolution algorithm to enhance the diversity of candidate solutions in the dynamic environment, which further improves the quality of the controller gains under the effects of uncertainties.

The proposed chaotic online differential evolution (CODE) is an optimizer based on the DE/rand/1/bin variant of differential evolution (see Section 4.2.1). It incorporates
a chaotic initialization and an elitist adaptation mechanism to improve the exploitative capacity and speed convergence of the original algorithm.

In the case of CODE, the random numbers utilized in the population initialization are generated by a Lozi chaotic map with $a = 1.7$ and $b = 0.5$. This is intended to increase the initial diversity of solutions at the beginning of each CODE run. Concerning the initial condition of the Lozi discrete dynamics adopted in the adaptive tuning proposal, it is established randomly at the very first instant of the control strategy execution (i.e., when $t = 0$). After that, the Lozi map dynamics evolve one step forward each time a new chaotic random number is required (i.e., when it is required to generate a new random variable while generating each individual from the initial population of CODE).

Rodríguez-Molina et al. highlighted that among the tested optimizers (online differential evolution (ODE), online genetic algorithm (OGA), and online particle swarm optimization (OPSO)) in the adaptive controller tuning, the statistical results evidence that the use of CODE suitably balances the search capacities of exploration and exploitation of the algorithm such that the proposed CATSCG maintains the minimum possible motor velocity error under the effects of disturbances, uncertainties, noise, and reference velocity changes. They attributed the outstanding performance of CATSCG to the use of the chaotic initialization based on the Lozi map.

### 4.3. Self-Organizing Migrating Algorithm (SOMA)

In 2004, Zelinka introduced SOMA (Self-Organizing Migrating Algorithm) [59], a new class of stochastic optimization algorithms. Evolutionary algorithms work on populations of candidate solutions that are evolved in generations (i.e., two parents create one new individual, called the offspring) in which only the best-suited (or fittest) individuals are likely to survive. Instead, SOMA which can work on a population of individuals, is based on the self-organizing behavior of groups of individuals in a "social environment", e.g., a herd of animals looking for food.

#### 4.3.1. SOMA Philosophy

A group of animals such as wolves or other predators makes for a good example. If they are looking for food, they usually cooperate and compete such that if one member of the group is successful (has found food or shelter) then the other animals of the group change their trajectories towards the most successful member. If a member of this group is more successful than the previous best one (it has found more food, etc.), then again all members change their trajectories towards the new successful member. This is repeated until all members meet around one food source. This principle from the real world is of course strongly simplified; nonetheless, it can be said that it is the competitive-cooperative behavior of intelligent agents that allows SOMA to carry out very successful searches.

SOMA can be classified as an evolutionary algorithm, despite the fact that no new generations of individuals are created during the search based on philosophy of this algorithm. Only the positions of the individuals in the search space are changed during a generation, which is called a "migration loop". Individuals are generated by random according to what is called the "specimen of the individual" principle. However, it is possible to classify SOMA as an evolutionary algorithm, as the final result after one migration loop is equivalent to the result from one generation derived by the classic EA algorithms.

Recently, Zelinka et al. used SOMA [81] for the design of quantum computing circuits for future quantum computers.

One key point in SOMA is the possibility of avoiding premature convergence by means of a so-called *PRTVector* (or Perturbation Vector), which limits the movement of particles to certain dimensions. Pluhacek et al. [82], considering that there is no conceptual barrier between DE, PSO, and SOMA, found that it is possible to noticeably improve the performance of a PSO algorithm driven by a Lozi chaotic map by implementing the *PRTVector* mechanism from the SOMA algorithm.
a chaotic initialization and an elitist adaptation mechanism to improve the exploitative capacity and speed convergence of the original algorithm.

In the case of CODE, the random numbers utilized in the population initialization are generated by a Lozi chaotic map with $a = 1.7$ and $b = 0.5$. This is intended to increase the initial diversity of solutions at the beginning of each CODE run. Concerning the initial condition of the Lozi discrete dynamics adopted in the adaptive tuning proposal, it is established randomly at the very first instant of the control strategy execution (i.e., when $t = 0$). After that, the Lozi map dynamics evolve one step forward each time a new chaotic random number is required (i.e., when it is required to generate a new random variable while generating each individual from the initial population of CODE).

Rodríguez-Molina et al. highlighted that among the tested optimizers (online differential evolution (ODE), online genetic algorithm (OGA), and online particle swarm optimization (OPSO)) in the adaptive controller tuning, the statistical results evidence that the use of CODE suitably balances the search capacities of exploration and exploitation of the algorithm such that the proposed CATSCG maintains the minimum possible motor velocity error under the effects of disturbances, uncertainties, noise, and reference velocity changes. They attributed the outstanding performance of CATSCG to the use of the chaotic initialization based on the Lozi map.

### 4.3. Self-Organizing Migrating Algorithm (SOMA)

In 2004, Zelinka introduced SOMA (Self-Organizing Migrating Algorithm) [59], a new class of stochastic optimization algorithms. Evolutionary algorithms work on populations of candidate solutions that are evolved in generations (i.e., two parents create one new individual, called the offspring) in which only the best-suited (or fittest) individuals are likely to survive. Instead, SOMA which can work on a population of individuals, is based on the self-organizing behavior of groups of individuals in a "social environment", e.g., a herd of animals looking for food.

#### 4.3.1. SOMA Philosophy

A group of animals such as wolves or other predators makes for a good example. If they are looking for food, they usually cooperate and compete such that if one member of the group is successful (has found food or shelter) then the other animals of the group change their trajectories towards the most successful member. If a member of this group is more successful than the previous best one (it has found more food, etc.), then again all members change their trajectories towards the new successful member. This is repeated until all members meet around one food source. This principle from the real world is of course strongly simplified; nonetheless, it can be said that it is the competitive-cooperative behavior of intelligent agents that allows SOMA to carry out very successful searches.

SOMA can be classified as an evolutionary algorithm, despite the fact that no new generations of individuals are created during the search based on philosophy of this algorithm. Only the positions of the individuals in the search space are changed during a generation, which is called a "migration loop". Individuals are generated by random according to what is called the "specimen of the individual" principle. However, it is possible to classify SOMA as an evolutionary algorithm, as the final result after one migration loop is equivalent to the result from one generation derived by the classic EA algorithms.

Recently, Zelinka et al. used SOMA [81] for the design of quantum computing circuits for future quantum computers.

One key point in SOMA is the possibility of avoiding premature convergence by means of a so-called *PRTVector* (or Perturbation Vector), which limits the movement of particles to certain dimensions. Pluhacek et al. [82], considering that there is no conceptual barrier between DE, PSO, and SOMA, found that it is possible to noticeably improve the performance of a PSO algorithm driven by a Lozi chaotic map by implementing the *PRTVector* mechanism from the SOMA algorithm.
Through several examples of objective function, such as the Sphere function, Rosen-brock’s function

$$f(x) = \sum_{i=1}^{D-1} \left| 100(x_i^2 - x_{i+1}^2)^2 - (1-x_i)^2 \right|, \qquad (59)$$

noise function, Rastrigin’s function (43), Schwefel’s function (44), Ackley’s function,

$$f(x) = -20 \exp \left( -0.2 \sqrt{\frac{1}{D} \sum_{i=1}^{D} x_i^2} \right) - \exp \left( \frac{1}{D} \sum_{i=1}^{D} x_i^2 \cos 2\pi x_1 \right) + 20 + e, \qquad (60)$$

etc., they highlighted that it is possible to find better solutions even faster than with the original design of PSO driven by Lozi map.

### 4.3.2. Performance Comparison of PID Controller Using DE and SOMA Versus PSO

The comparison of PSO performance with DE (see Section 4.2.4) was extended to SOMA by Davendra et al. [63] for the problem of a PID controller (see Section 4.1.2) using three transfer functions:

* A third-order system

$$G(s) = \frac{0.1}{s(3s + 1)(0.8s + 1)}, \qquad (61)$$

* A fourth-order system comparable with the Ball and Hoop system (40),

* A third-order system for an electric DC Motor (35).

For all these problems, four unique optimization specifications (IAE (37), ISE (38), ITAE (36), and MSE (39)) were utilized. The obtained values were then used to calculate the following three performance indices:

* Rise time: the time it takes for the process output to achieve the new desired value. One-third of the dominant process time constant is typical.

* Settling time: the time it takes for the process output to die to between, say, ±5% of the setpoint.

* Peak overshoot: the magnitude by which the controlled “variable swings” past the setpoint ±5%–10% overshoot is normally acceptable for most loops.

The Lozi map was selected as the chaotic map of choice due to its superior performance in previous research [83], in which additional comparisons for PID controllers using DE and PSO with the Lozi map can be found. The results obtained by chaos-driven DE and SOMA were compared with the values obtained using the Ziegler–Nichols method [84] and other published heuristics. For all the attempted problems, DE and SOMA performed better than the Ziegler–Nichols method. In the ball and hoop system (40), DE and SOMA performed better than the Genetic Algorithm (GA), while in the electric DC motor problem (35) they obtained better overshoot and settling times compared to other heuristics (GA, PSO, Continuous Cycling [85], and Evolutionary Programming [86]). The authors found that DE and SOMA both achieved consistently better results.

### 4.4. Other Genetic and Evolutionary Algorithms

In addition to PSO, DE, and SOMA there are a large number of other genetic and evolutionary optimization algorithms [57]. Among these, many use the Lozi map as a chaotic pseudo-random number generator. In this section, we describe several of these (for the tribes optimization approach, see Section 7.1.5).

#### 4.4.1. Ant Lion Approach

The Ant Lion Optimizer (ALO) is a nature-inspired stochastic metaheuristic algorithm that mimics the hunting behavior of ant lions in nature. It uses the steps of hunting prey (such as a random walk of ants), building traps, entrapment of ants, catching prey, and rebuilding traps. This algorithm was proposed by Mirjalili [87] and adapted by Coelho
Through several examples of objective function, such as the Sphere function, Rosen-brock’s function

$$f(x) = \sum_{i=1}^{D-1} \left| 100(x_i^2 - x_{i+1}^2)^2 - (1-x_i)^2 \right|, \qquad (59)$$

noise function, Rastrigin’s function (43), Schwefel’s function (44), Ackley’s function,

$$f(x) = -20 \exp \left( -0.2 \sqrt{\frac{1}{D} \sum_{i=1}^{D} x_i^2} \right) - \exp \left( \frac{1}{D} \sum_{i=1}^{D} x_i^2 \cos 2\pi x_1 \right) + 20 + e, \qquad (60)$$

etc., they highlighted that it is possible to find better solutions even faster than with the original design of PSO driven by Lozi map.

### 4.3.2. Performance Comparison of PID Controller Using DE and SOMA Versus PSO

The comparison of PSO performance with DE (see Section 4.2.4) was extended to SOMA by Davendra et al. [63] for the problem of a PID controller (see Section 4.1.2) using three transfer functions:

* A third-order system

$$G(s) = \frac{0.1}{s(3s + 1)(0.8s + 1)}, \qquad (61)$$

* A fourth-order system comparable with the Ball and Hoop system (40),

* A third-order system for an electric DC Motor (35).

For all these problems, four unique optimization specifications (IAE (37), ISE (38), ITAE (36), and MSE (39)) were utilized. The obtained values were then used to calculate the following three performance indices:

* Rise time: the time it takes for the process output to achieve the new desired value. One-third of the dominant process time constant is typical.

* Settling time: the time it takes for the process output to die to between, say, ±5% of the setpoint.

* Peak overshoot: the magnitude by which the controlled “variable swings” past the setpoint ±5%–10% overshoot is normally acceptable for most loops.

The Lozi map was selected as the chaotic map of choice due to its superior performance in previous research [83], in which additional comparisons for PID controllers using DE and PSO with the Lozi map can be found. The results obtained by chaos-driven DE and SOMA were compared with the values obtained using the Ziegler–Nichols method [84] and other published heuristics. For all the attempted problems, DE and SOMA performed better than the Ziegler–Nichols method. In the ball and hoop system (40), DE and SOMA performed better than the Genetic Algorithm (GA), while in the electric DC motor problem (35) they obtained better overshoot and settling times compared to other heuristics (GA, PSO, Continuous Cycling [85], and Evolutionary Programming [86]). The authors found that DE and SOMA both achieved consistently better results.

### 4.4. Other Genetic and Evolutionary Algorithms

In addition to PSO, DE, and SOMA there are a large number of other genetic and evolutionary optimization algorithms [57]. Among these, many use the Lozi map as a chaotic pseudo-random number generator. In this section, we describe several of these (for the tribes optimization approach, see Section 7.1.5).

#### 4.4.1. Ant Lion Approach

The Ant Lion Optimizer (ALO) is a nature-inspired stochastic metaheuristic algorithm that mimics the hunting behavior of ant lions in nature. It uses the steps of hunting prey (such as a random walk of ants), building traps, entrapment of ants, catching prey, and rebuilding traps. This algorithm was proposed by Mirjalili [87] and adapted by Coelho
et al. [88], who based it on the Lozi Map within the scope of Multiobjective Transformer Design Optimization.

Ant lions, or doodlebugs, belong to the Myrmeleontidae family and Neuroptera order (net-winged insects). The lifecycle of antlions includes two main phases: larvae and adult. A natural total lifespan can be up to three years, which mostly occurs in larvae (only 3–5 weeks for adulthood). Antlions undergo metamorphosis in a cocoon to become adults. They mostly hunt as larvae, with the adulthood period used for reproduction [87]. Their name originates from their unique hunting behaviour and favourite prey. An antlion larva digs a cone-shaped pit in sand by moving along a circular path and throwing out sand with its massive jaws. After digging the trap, the larva hides underneath the bottom of the cone and waits for insects (preferably ants) to become trapped in the pit. When the ant lion realizes that prey is in the trap, it tries to catch it. However, insects are usually not caught immediately, and try to escape the trap. In this case, the ant lion intelligently throws sand towards to edge of the pit in order to slide the prey to the bottom of the pit. When prey is caught in the ant lion's jaws, the prey is pulled under the soil and consumed. After consuming the prey, the ant lions throws the remains outside the pit and repairs the pit for the next hunt.

The ant lion algorithm mimics the interaction between ant lions and ants in the trap. To model this interactions, ants are required to move over the search space and ant lions are expected to hunt them and become fitter using their traps. Because ants move stochastically in nature when searching for food, in the original ALO algorithm a random walk is chosen to model the ants' movement. The steps of the ALO for single-objective optimization problems are briefly described in the following pseudo-code [88]:

* 1: Define the objective function and population size (NP)

* 2: Generate the populations of ant lions and ants

* 3: Evaluate the fitness of the ant lions

* 4: Initialize the generation counter $t = 1$

* 5: For $t < \text{maximum of iterations}$:

* 6: For each ant from 1 to NP:

* 7: Perform a random walk around a random ant lion,

* 8: Perform a random walk around the elite individual (best ant lion),

* 9: Update the position and check the search boundaries.

* 10: Evaluate the fitness of the ants

* 11: End for,

* 12: Update ant lion positions based on the ants

* 13: Update the elite individual and keep it in the population

* 14: Update the generation counter $t = t + 1$

* 15: End while,

* 16: Return the fittest (elite) ant lion.

On the other hand, multi-objective optimization refers to the optimization of more than one objective; these are generally in conflict with each other, resulting in no single solution to the problem. For this kind of optimization problems, the aim becomes to find good “trade-off” solutions that represent the best possible compromises among the objectives. The tradeoff between obtaining a well-converged and well-distributed set of Pareto-optimal solutions is an important issue in multi-objective optimization.

The MOALO is a classical multi-objective variant that uses the crowding distance [89] and the dominance concept for selecting the elite (best ant lion) and a tournament mechanism to select the ant lions to perform the random walk. One of the main pillars of optimization meta-heuristics, such as ALO, is their reliance on randomness or stochasticity, which is used to spark a path towards a desired goal. The current norm is the use of pseudo-random number generators. Unlike the MOALO, the proposed L-MOALO employs a mechanism with a chaotic dynamical behavior based on the Lozi map (similar to (30)) instead of pseudo-random number generators to enhance exploration during the ants' movement.
et al. [88], who based it on the Lozi Map within the scope of Multiobjective Transformer Design Optimization.

Ant lions, or doodlebugs, belong to the Myrmeleontidae family and Neuroptera order (net-winged insects). The lifecycle of antlions includes two main phases: larvae and adult. A natural total lifespan can be up to three years, which mostly occurs in larvae (only 3–5 weeks for adulthood). Antlions undergo metamorphosis in a cocoon to become adults. They mostly hunt as larvae, with the adulthood period used for reproduction [87]. Their name originates from their unique hunting behaviour and favourite prey. An antlion larva digs a cone-shaped pit in sand by moving along a circular path and throwing out sand with its massive jaws. After digging the trap, the larva hides underneath the bottom of the cone and waits for insects (preferably ants) to become trapped in the pit. When the ant lion realizes that prey is in the trap, it tries to catch it. However, insects are usually not caught immediately, and try to escape the trap. In this case, the ant lion intelligently throws sand towards to edge of the pit in order to slide the prey to the bottom of the pit. When prey is caught in the antlion's jaws, the prey is pulled under the soil and consumed. After consuming the prey, the ant lions throws the remains outside the pit and repairs the pit for the next hunt.

The ant lion algorithm mimics the interaction between ant lions and ants in the trap. To model this interactions, ants are required to move over the search space and ant lions are expected to hunt them and become fitter using their traps. Because ants move stochastically in nature when searching for food, in the original ALO algorithm a random walk is chosen to model the ants' movement. The steps of the ALO for single-objective optimization problems are briefly described in the following pseudo-code [88]:

1. *Define the objective function and population size (NP)*

2. *Generate the populations of ant lions and ants*

3. *Evaluate the fitness of the ant lions*

4. *Initialize the generation counter t = 1*

5. *For t < maximum of iterations:*

6. *For each ant from 1 to NP:*

7. *Perform a random walk around a random ant lion,*

8. *Perform a random walk around the elite individual (best ant lion),*

9. *Update the position and check the search boundaries.*

10. *Evaluate the fitness of the ants*

11. *End for,*

12. *Update ant lion positions based on the ants*

13. *Update the elite individual and keep it in the population*

14. *Update the generation counter t = t + 1*

15. *End while,*

16. *Return the fittest (elite) ant lion.*

On the other hand, multi-objective optimization refers to the optimization of more than one objective; these are generally in conflict with each other, resulting in no single solution to the problem. For this kind of optimization problems, the aim becomes to find good “trade-off” solutions that represent the best possible compromises among the objectives. The tradeoff between obtaining a well-converged and well-distributed set of Pareto-optimal solutions is an important issue in multi-objective optimization.

The MOALO is a classical multi-objective variant that uses the crowding distance [89] and the dominance concept for selecting the elite (best ant lion) and a tournament mechanism to select the ant lions to perform the random walk. One of the main pillars of optimization meta-heuristics, such as ALO, is their reliance on randomness or stochasticity, which is used to spark a path towards a desired goal. The current norm is the use of pseudo-random number generators. Unlike the MOALO, the proposed L-MOALO employs a mechanism with a chaotic dynamical behavior based on the Lozi map (similar to (30)) instead of pseudo-random number generators to enhance exploration during the ants' movement.
The results of [88] obtained from simulations have shown that the MOALO and L-MOALO algorithms can be successfully applied to a Transformer Design Optimization (TDO) problem with two competing objectives, both providing a set of different solutions that present a trade-off between the objective costs.

### 4.4.2. Computational Intelligence Algorithm

Martinez-Vargas et al. [90] proposed a leader-follower optimization algorithm based on the combination of five different Population-based Algorithms (PAs) to tackle the high-dimensional optimization problem known as the Base Station (BS) deactivation problem in telephony. The general idea is to find the minimal set of active BSs in a 6G cellular network to maintain service for at least 99% of mobile users (MUs) who are reassigned due to BSs being shut down. Computational Intelligence (CI) is the key enabler for future 6G systems to autonomously manage their elements and build an intelligent wireless environment. The five follower PAs are: binary particle swarm optimization (BPSO), chaotic BPSO (BPSOC), a Genetic Algorithm (GA) with a steady-state population model (AGE), a GA with a generational model (AGG), and a binary artificial bee colony.

Each follower algorithm searches for its best solution (that is, the best local solution) in its set of candidate solutions during some number of iterations. Meanwhile, the leader algorithm checks whether the followers' status is waiting or working. If the follower algorithms have the waiting status, the leader algorithm asks for the best local solution they have found thus far. Then, the leader gathers those best local solutions and chooses the best one among them as the best global solution. Consequently, the leader sets its status to working and sends the best global solution to the followers. When the followers receive the best global solution, they search for the worst candidate solution in their group of candidate solutions. The best global solution takes the place of the worst candidate solution (elitism). After the number of iterations is met, the leader resolves the problem, that is, it finds the lowest number of active BSs and reassigns those MUs whose BSs have been deactivated.

Among the follower algorithms, BPSOC has a unique mechanism to move through the search space. This mechanism uses chaos sequences to improve diversity and prevent premature convergence to a local optimum. To this end, BPSOC uses a Lozi map embedded in

$$v_{id} = \omega * v_{id} + c_1 * v * \textbf{chaos}() * (g_{best} - x_{id}) + c_2 * \textbf{chaos}() * (p_{best} - x_{id}), \quad (62)$$

on the `chaos()` function.

The BPSO and BPSOC algorithms have exploitation bias. The diversity trend throughout iterations in these two followers is acute. This means that the convergence velocity towards $p_{best}$ and $g_{best}$ is fast. Furthermore, diversity is slightly greater in BPSOC because of the Lozi map effect (this is a unique mechanism in BPSOC).

In conclusion, Martinez-Vargas et al. found that the leader-follower optimization algorithm can find solutions even in challenging scenarios, and that BPSO and BPSOC find better solutions by performing deeper exploitation.

### 4.4.3. Mixed Fruit Fly Optimization

The Fruit Fly Optimization Algorithm (FOA) is a meta-heuristic technique proposed by Pan [91] that is inspired by the behavior of fruit flies. It simulates the foraging behavior of fruit flies. In comparison with other swarm intelligence-based algorithms, FOA has a sample framework with few parameters, and is easily understood and implemented. It has been improved as the Mixed Fruit Fly Optimization Algorithm LGM-FOA (Logistic Mapping-FOA) on the basis of the logistic map of Mitic et al. [92]; however, the algorithm shows an ideal state with respect to convergence precision and stability in the optimization process, as there are three discontinuous points from the logistic map. To solve this problem, Luo et al. [93] proposed a new mixed fruit fly algorithm that uses the Lozi map instead of the logistic map to conduct a global search for the optimal parameter values. It uses the value as the center to perform tiny fluctuations in order to obtain a final optimal value of quadratic optimization, thereby improving the initial value selection method of LGM-
The results of [88] obtained from simulations have shown that the MOALO and L-MOALO algorithms can be successfully applied to a Transformer Design Optimization (TDO) problem with two competing objectives, both providing a set of different solutions that present a trade-off between the objective costs.

### 4.4.2. Computational Intelligence Algorithm

Martinez-Vargas et al. [90] proposed a leader-follower optimization algorithm based on the combination of five different Population-based Algorithms (PAs) to tackle the high-dimensional optimization problem known as the Base Station (BS) deactivation problem in telephony. The general idea is to find the minimal set of active BSs in a 6G cellular network to maintain service for at least 99% of mobile users (MUs) who are reassigned due to BSs being shut down. Computational Intelligence (CI) is the key enabler for future 6G systems to autonomously manage their elements and build an intelligent wireless environment. The five follower PAs are: binary particle swarm optimization (BPSO), chaotic BPSO (BPSOC), a Genetic Algorithm (GA) with a steady-state population model (AGE), a GA with a generational model (AGG), and a binary artificial bee colony.

Each follower algorithm searches for its best solution (that is, the best local solution) in its set of candidate solutions during some number of iterations. Meanwhile, the leader algorithm checks whether the followers' status is waiting or working. If the follower algorithms have the waiting status, the leader algorithm asks for the best local solution they have found thus far. Then, the leader gathers those best local solutions and chooses the best one among them as the best global solution. Consequently, the leader sets its status to working and sends the best global solution to the followers. When the followers receive the best global solution, they search for the worst candidate solution in their group of candidate solutions. The best global solution takes the place of the worst candidate solution (elitism). After the number of iterations is met, the leader resolves the problem, that is, it finds the lowest number of active BSs and reassigns those MUs whose BSs have been deactivated.

Among the follower algorithms, BPSOC has a unique mechanism to move through the search space. This mechanism uses chaos sequences to improve diversity and prevent premature convergence to a local optimum. To this end, BPSOC uses a Lozi map embedded in

$$v_{id} = \omega * v_{id} + c_1 * v * \textbf{chaos}() * (g_{best} - x_{id}) + c_2 * \textbf{chaos}() * (p_{best} - x_{id}), \quad (62)$$

on the `chaos()` function.

The BPSO and BPSOC algorithms have exploitation bias. The diversity trend throughout iterations in these two followers is acute. This means that the convergence velocity towards $p_{best}$ and $g_{best}$ is fast. Furthermore, diversity is slightly greater in BPSOC because of the Lozi map effect (this is a unique mechanism in BPSOC).

In conclusion, Martinez-Vargas et al. found that the leader-follower optimization algorithm can find solutions even in challenging scenarios, and that BPSO and BPSOC find better solutions by performing deeper exploitation.

### 4.4.3. Mixed Fruit Fly Optimization

The Fruit Fly Optimization Algorithm (FOA) is a meta-heuristic technique proposed by Pan [91] that is inspired by the behavior of fruit flies. It simulates the foraging behavior of fruit flies. In comparison with other swarm intelligence-based algorithms, FOA has a sample framework with few parameters, and is easily understood and implemented. It has been improved as the Mixed Fruit Fly Optimization Algorithm LGM-FOA (Logistic Mapping-FOA) on the basis of the logistic map of Mitic et al. [92]; however, the algorithm shows an ideal state with respect to convergence precision and stability in the optimization process, as there are three discontinuous points from the logistic map. To solve this problem, Luo et al. [93] proposed a new mixed fruit fly algorithm that uses the Lozi map instead of the logistic map to conduct a global search for the optimal parameter values. It uses the value as the center to perform tiny fluctuations in order to obtain a final optimal value of quadratic optimization, thereby improving the initial value selection method of LGM-