FOA. In support of the simulation between vector machine regression forecasting and the original Fruit Fly Algorithm, Particle Swarm Optimization (PSO), and LGM-FOA, this result testifies that the convergence accuracy of this new mixed fruit fly algorithm has obvious advantages.

### 4.4.4. Hybrid Chaos Immune Algorithm

Liu et al. [94], seeking to improve the performance of the immune algorithm, integrated chaos optimization into the immune clone selection algorithm. After decoding, a portion of the antibodies are mapped onto the Lozi system, then every optical value is collected at each iteration. Those values are added into the new antibody group to improve the antibody-antigen fitness value. Comparative results indicate that the optimization method of the hybrid chaos immune algorithm with self-adaptive parameter adjustment has many advantages, such as better self-adaptive capacity, higher computation efficiency, and better design accuracy.

## 4.5. Non-Evolutionary Chaotic Optimization Algorithms

In addition to evolutionary chaotic optimization algorithms, the Lozi map has been used as the core of more classical optimization algorithms.

### 4.5.1. Chaotic Optimization Algorithms

The tuning of PID controller (see Sections 4.1.2 and 4.3.2) was addressed by Coelho [95] using a non-evolutionary chaotic optimization approach based on the Lozi map. He considered an automatic regulator voltage (AVR) system.

The problem of the dynamic stability of power systems has challenged power system engineers for decades. In a synchronous generator, the electromechanical coupling between the rotor and the rest of the system causes it to behave in a manner similar to a spring mass damper system, which exhibits oscillatory behavior around the equilibrium state following any disturbance, such as a sudden change in load, change in transmission line parameters, or fluctuations in the turbine output. Synchronous generator excitation control is one of the most important measures to enhance power system stability and to guarantee the quality of electrical power.

The transfer function of such a system is provided by

$$G(s) = \frac{0.07s + 7}{0.0004s^4 + 0.0454s^3 + 0.555sr + 1.51s + 8} \quad (63)$$

The proposed chaotic optimization algorithm (CAO) (called the Chaotic Optimization Method based on Lozi map (COLM) by Coelho) is described as follows.

The Lozi map for this CAO is defined by

$$\left\{ \begin{aligned} y_1(k) &= 1 - a|y_1(k-1)| + y(k-1), \\ y(k) &= by_1(k-1), \\ z(k) &= \frac{y(k)-a}{\beta-a}, \end{aligned} \right. \quad (64)$$

where k is the iteration number. Similar to (30), the values of y are normalized in the range [0, 1] to each decision variable in the n-dimensional space of the optimization problem. Thus, we have $y_1 \in [-0.6418, 0.6716]$ and $[\alpha, \beta] = [-0.6418, 0.6716]$ for $a = 1.7$ and $b = 0.5$. Many unconstrained optimization problems with continuous variables can be formulated as the following functional optimization problem.

Find X to minimize $f(X)$, $X = [x_1, x_2, \dots, x_n]$, where f is the objective function and X is the decision solution vector consisting of n variables. Here, $x_i$ bounded by lower limits ($L_i$) and upper limits ($U_i$). The COLM can be illustrated as follows.

**Step 1:** Initialization of variables and initial conditions. Set $k = 1, y_1(0), y(0)$ and set the

initial best objective function $\bar{f}$.
FOA. In support of the simulation between vector machine regression forecasting and the original Fruit Fly Algorithm, Particle Swarm Optimization (PSO), and LGM-FOA, this result testifies that the convergence accuracy of this new mixed fruit fly algorithm has obvious advantages.

### 4.4.4. Hybrid Chaos Immune Algorithm

Liu et al. [94], seeking to improve the performance of the immune algorithm, integrated chaos optimization into the immune clone selection algorithm. After decoding, a portion of the antibodies are mapped onto the Lozi system, then every optical value is collected at each iteration. Those values are added into the new antibody group to improve the antibody-antigen fitness value. Comparative results indicate that the optimization method of the hybrid chaos immune algorithm with self-adaptive parameter adjustment has many advantages, such as better self-adaptive capacity, higher computation efficiency, and better design accuracy.

## 4.5. Non-Evolutionary Chaotic Optimization Algorithms

In addition to evolutionary chaotic optimization algorithms, the Lozi map has been used as the core of more classical optimization algorithms.

### 4.5.1. Chaotic Optimization Algorithms

The tuning of PID controller (see Sections 4.1.2 and 4.3.2) was addressed by Coelho [95] using a non-evolutionary chaotic optimization approach based on the Lozi map. He considered an automatic regulator voltage (AVR) system.

The problem of the dynamic stability of power systems has challenged power system engineers for decades. In a synchronous generator, the electromechanical coupling between the rotor and the rest of the system causes it to behave in a manner similar to a spring mass damper system, which exhibits oscillatory behavior around the equilibrium state following any disturbance, such as a sudden change in load, change in transmission line parameters, or fluctuations in the turbine output. Synchronous generator excitation control is one of the most important measures to enhance power system stability and to guarantee the quality of electrical power.

The transfer function of such a system is provided by

$$G(s) = \frac{0.07s + 7}{0.0004s^4 + 0.0454s^3 + 0.555sr + 1.51s + 8} \quad (63)$$

The proposed chaotic optimization algorithm (CAO) (called the Chaotic Optimization Method based on Lozi map (COLM) by Coelho) is described as follows.

The Lozi map for this CAO is defined by

$$\left\{ \begin{array}{l} y_1(k) = 1 - a|y_1(k-1)| + y(k-1), \\ y(k) = by_1(k-1), \\ z(k) = \frac{y(k)-a}{\beta-a}, \end{array} \right. \quad (64)$$

where k is the iteration number. Similar to (30), the values of y are normalized in the range [0, 1] to each decision variable in the n-dimensional space of the optimization problem. Thus, we have $y_1 \in [-0.6418, 0.6716]$ and $[\alpha, \beta] = [-0.6418, 0.6716]$ for $a = 1.7$ and $b = 0.5$. Many unconstrained optimization problems with continuous variables can be formulated as the following functional optimization problem.

Find X to minimize $f(X)$, $X = [x_1, x_2, \dots, x_n]$, where f is the objective function and X is the decision solution vector consisting of n variables. Here, $x_i$ bounded by lower limits ($L_i$) and upper limits ($U_i$). The COLM can be illustrated as follows.

**Step 1:** Initialization of variables and initial conditions. Set $k = 1, y_1(0), y(0)$ and set the

initial best objective function $\bar{f}$.
**Step 2:** Algorithm for chaotic global search.

Begin

While $k \le M_g$ do

$$x_i(k) = L_i + z_i(k) \times (U_i - L_i)$$

If $f(X(k)) < \bar{f}$ Then

$$\bar{X} = X(k)$$

$$\bar{f} = f(X(k))$$

End if

$$k = k + 1$$

End while

End

**Step 3:** Algorithm for chaotic local search.

Begin

While $k \le (M_g + M_L)$ do

for $i = 1$ to $n$

If $r < 0.5$ then (where $r$ is a uniformly distributed random variable with range $[0, 1]$)

$$x_i(k) = \bar{x}_i + \lambda z_i(k) \times |U_i - \bar{X}_i|$$

Else

$$x_i(k) = \bar{x}_i - \lambda z_i(k) \times |\bar{X}_i - L_1|$$

End if

End for

If $f(X(k)) < \bar{f}$ then

$$\bar{X} = X(k)$$

$$\bar{f} = f(X(k))$$

End if

$$k = k + 1$$

End while

End

Here, $M_g$ and $M_L$ are the maximum number of iterations of the chaotic global and local searches, respectively, while $M_g + M_L$ is the stopping criterion in iterations. The step size in chaotic local search is $\lambda$. The best objective function is $\bar{f}$, and $\bar{X}$ is the best solution from the current run of the chaotic search.

A suitable value for the step size usually provides a balance between global and local exploration abilities, and consequently a reduction in the number of iterations required to locate the optimum solution. In [95], the tested values of step size in COLM were described as follows:

* COLM(1): the step size $\lambda$ equals 0.01

* COLM(2): the step size $\lambda$ equals 0.05

* COLM(3): the step size $\lambda$ equals 0.10

* COLM(4): the step size $\lambda$ linearly decreases from 0.10 to 0.01

* COLM(5): the step size $\lambda$ linearly increases from 0.01 to 0.10

In each case study, fifty independent runs were carried out for each of the COLM methods involving fifty different initial trial conditions $y1(0), y(0)$.

From the case studies and comparison of the results through the five tested COLM approaches, it has been shown that the step size parameter $\lambda$ is essential to obtaining a good convergence profile. In this context, the $\lambda$ parameter regulates the trade-off between the global and local exploration abilities of the chaotic local search.

Shayeghi et al. [96,97], using the COA introduced by Coelho, proposed a multi-objective design for multi-machine Power System Stabilizers (PSSs). The PSS parameter tuning problem was converted to an optimization problem which was solved by COLM using the COLM(4) strategy. The authors suggested that COLM, which has the features of easy implementation, short execution time, and robust mechanisms of escaping from
**Step 2:** Algorithm for chaotic global search.

Begin

While $k \le M_g$ do

$$x_i(k) = L_i + z_i(k) \times (U_i - L_i)$$

If $f(X(k)) < \bar{f}$ Then

$$\bar{X} = X(k)$$

$$\bar{f} = f(X(k))$$

End if

$$k = k + 1$$

End while

End

**Step 3:** Algorithm for chaotic local search.

Begin

While $k \le (M_g + M_L)$ do

for $i = 1$ to $n$

If $r < 0.5$ then (where $r$ is a uniformly distributed random variable with range $[0, 1]$)

$$x_i(k) = \bar{x}_i + \lambda z_i(k) \times |U_i - \bar{X}_i|$$

Else

$$x_i(k) = \bar{x}_i - \lambda z_i(k) \times |\bar{X}_i - L_1|$$

End if

End for

If $f(X(k)) < \bar{f}$ then

$$\bar{X} = X(k)$$

$$\bar{f} = f(X(k))$$

End if

$$k = k + 1$$

End while

End

Here, $M_g$ and $M_L$ are the maximum number of iterations of the chaotic global and local searches, respectively, while $M_g + M_L$ is the stopping criterion in iterations. The step size in chaotic local search is $\lambda$. The best objective function is $\bar{f}$, and $\bar{X}$ is the best solution from the current run of the chaotic search.

A suitable value for the step size usually provides a balance between global and local exploration abilities, and consequently a reduction in the number of iterations required to locate the optimum solution. In [95], the tested values of step size in COLM were described as follows:

* COLM(1): the step size $\lambda$ equals 0.01

* COLM(2): the step size $\lambda$ equals 0.05

* COLM(3): the step size $\lambda$ equals 0.10

* COLM(4): the step size $\lambda$ linearly decreases from 0.10 to 0.01

* COLM(5): the step size $\lambda$ linearly increases from 0.01 to 0.10

In each case study, fifty independent runs were carried out for each of the COLM methods involving fifty different initial trial conditions $y1(0), y(0)$.

From the case studies and comparison of the results through the five tested COLM approaches, it has been shown that the step size parameter $\lambda$ is essential to obtaining a good convergence profile. In this context, the $\lambda$ parameter regulates the trade-off between the global and local exploration abilities of the chaotic local search.

Shayeghi et al. [96,97], using the COA introduced by Coelho, proposed a multi-objective design for multi-machine Power System Stabilizers (PSSs). The PSS parameter tuning problem was converted to an optimization problem which was solved by COLM using the COLM(4) strategy. The authors suggested that COLM, which has the features of easy implementation, short execution time, and robust mechanisms of escaping from
the local optimum, is a promising tool for engineering applications. The chaotic sequences increase its convergence rate and improve the precision of the results.

The effectiveness of the proposed method was tested on a multi-machine power system for a wide range of loading conditions and disturbances. Eigenvalues analysis provided the satisfactory damping level for the system modes, especially the low frequency modes, for systems with the proposed COA-based tuned PSSs, especially according to the time domain-based multi-objective function. Time domain simulations showed that the oscillations of the synchronous machines can be quickly and effectively damped for power systems over a wide range of loading conditions using the proposed PSSs.

The same authors used this algorithm for an output feedback damping controller design of a static synchronous compensator (STATCOM) [98].

The same algorithm was used by Jalilvand et al. [99] to design a Unified Power Flow Controller (UPFC) based on damping controllers using COA. Their eigenvalues analysis and simulation results showed that it has good performance for damping low-frequency oscillations and can improve transient stability under different operating conditions.

The COLM algorithm was later improved by Hamaizia and Lozi [100] by locally averaging the global search, then performing a few steps of chaotic local search around every point obtained by the chaotic series. In another study, Bououden and Abdelouahab [101] used transformations to modify the density of the Lozi map and combined Lozi and logistic maps during global and local search [102].

### 4.5.2. Improved Lozi Map-Based Chaotic Optimization Algorithm (ILCOA)

The COLM algorithm (called the Lozi Chaotic Optimization Algorithm (LCOA)) was improved by Alizadeh et al. in [103]. ILCOA was employed to solve a single-machine Power System Stabilizer (PSS) design problem, and its promising features in comparison with LCOA were highlighted.

In ILCOA, the chaotic variable interval is partitioned into three areas, namely, $A_1 = [0, C_1]$, $A_2 = [C_1, C_2]$ and $A_3 = [C_2, 1]$, in which $0 < C_1 < 0.5$ and $0.5 < C_2 < 1$ are adjustable parameters.

During the chaotic global search, each time a value of the chaotic variable $z_i$ is generated from the Lozi map and its validity is evaluated. If the chaotic variable $z_i$ belongs to $A_1$ or $A_3$, it must be regenerated; otherwise, it is reserved and used for calculating the objective function and checking its optimum, as it certainly belongs to the area $A_2$.

The lower ($L_i$) and upper ($U_i$) limits of the solution variables have been reformed as well. Thus, in the chaotic global search subroutine of ILCOA, $U_i$ and $L_i$ are replaced by $U_i^{new}$ and $L_i^{new}$, respectively, where

$$U_i^{new} = \frac{1 - C_1}{C_2 - C_1} U_i + \frac{1 - C_2}{C_2 - C_1} L_i, \qquad (65)$$

$$L_i^{new} = \frac{-C_1}{C_2 - C_1} U_i + \frac{C_2}{C_2 - C_1} L_i, \qquad (66)$$

$$z_i^{new} = \frac{1}{C_2 - C_1} \times (z_i - C_1). \qquad (67)$$

and where, similar to the chaotic global search subroutine, $C_1 \le z_i \le C_2$; therefore, $0 \le z_i^{new} \le 1$.

There is another parameter in the chaotic local search subroutine, namely, the step size $\lambda$, which is an important parameter in the convergence behavior of optimization methods. It adjusts small ergodic ranges around $\bar{X}$. The selection of the step size $\lambda$ has a direct effect on the process of generating a new trial solution. Balance between the global and local exploration abilities is provided by an appropriate value of $\lambda$. In [103], the authors introduced a new selection strategy for step size $\lambda$, namely, the semi-exponential step size
the local optimum, is a promising tool for engineering applications. The chaotic sequences increase its convergence rate and improve the precision of the results.

The effectiveness of the proposed method was tested on a multi-machine power system for a wide range of loading conditions and disturbances. Eigenvalues analysis provided the satisfactory damping level for the system modes, especially the low frequency modes, for systems with the proposed COA-based tuned PSSs, especially according to the time domain-based multi-objective function. Time domain simulations showed that the oscillations of the synchronous machines can be quickly and effectively damped for power systems over a wide range of loading conditions using the proposed PSSs.

The same authors used this algorithm for an output feedback damping controller design of a static synchronous compensator (STATCOM) [98].

The same algorithm was used by Jalilvand et al. [99] to design a Unified Power Flow Controller (UPFC) based on damping controllers using COA. Their eigenvalues analysis and simulation results showed that it has good performance for damping low-frequency oscillations and can improve transient stability under different operating conditions.

The COLM algorithm was later improved by Hamaizia and Lozi [100] by locally averaging the global search, then performing a few steps of chaotic local search around every point obtained by the chaotic series. In another study, Bououden and Abdelouahab [101] used transformations to modify the density of the Lozi map and combined Lozi and logistic maps during global and local search [102].

### 4.5.2. Improved Lozi Map-Based Chaotic Optimization Algorithm (ILCOA)

The COLM algorithm (called the Lozi Chaotic Optimization Algorithm (LCOA)) was improved by Alizadeh et al. in [103]. ILCOA was employed to solve a single-machine Power System Stabilizer (PSS) design problem, and its promising features in comparison with LCOA were highlighted.

In ILCOA, the chaotic variable interval is partitioned into three areas, namely, $A_1 = [0, C_1]$, $A_2 = [C_1, C_2]$ and $A_3 = [C_2, 1]$, in which $0 < C_1 < 0.5$ and $0.5 < C_2 < 1$ are adjustable parameters.

During the chaotic global search, each time a value of the chaotic variable $z_i$ is generated from the Lozi map and its validity is evaluated. If the chaotic variable $z_i$ belongs to $A_1$ or $A_3$, it must be regenerated; otherwise, it is reserved and used for calculating the objective function and checking its optimum, as it certainly belongs to the area $A_2$.

The lower ($L_i$) and upper ($U_i$) limits of the solution variables have been reformed as well. Thus, in the chaotic global search subroutine of ILCOA, $U_i$ and $L_i$ are replaced by $U_i^{new}$ and $L_i^{new}$, respectively, where

$$U_i^{new} = \frac{1 - C_1}{C_2 - C_1} U_i + \frac{1 - C_2}{C_2 - C_1} L_i, \qquad (65)$$

$$L_i^{new} = \frac{-C_1}{C_2 - C_1} U_i + \frac{C_2}{C_2 - C_1} L_i, \qquad (66)$$

$$z_i^{new} = \frac{1}{C_2 - C_1} \times (z_i - C_1). \qquad (67)$$

and where, similar to the chaotic global search subroutine, $C_1 \le z_i \le C_2$; therefore, $0 \le z_i^{new} \le 1$.

There is another parameter in the chaotic local search subroutine, namely, the step size $\lambda$, which is an important parameter in the convergence behavior of optimization methods. It adjusts small ergodic ranges around $\bar{X}$. The selection of the step size $\lambda$ has a direct effect on the process of generating a new trial solution. Balance between the global and local exploration abilities is provided by an appropriate value of $\lambda$. In [103], the authors introduced a new selection strategy for step size $\lambda$, namely, the semi-exponential step size
(SESS), which focuses more on smaller values of $\lambda$ and contributes to better exploitation for refining the results:

$$ \left\{ \begin{array}{l} \lambda = \lambda_{max} - \frac{\lambda_{max} - \lambda_{mid}}{iter_{max}^{1}} \times \text{iteration}, \quad 1 \le \text{iteration} \le \text{iter}_{max}^{1} \\ \lambda = \left[ \lambda_{max} - \frac{\lambda_{max} - \lambda_{mid}}{iter_{max}^{1}} \right] \times e^{\frac{(iter_{max}^{1} - \text{iteration})}{k}}, \quad \text{iter}_{max}^{1} \le \text{iteration} \le \text{iter}_{max}^{2} \end{array} \right. \quad (68) $$

where $\lambda_{max}$ is the initial step size, $\lambda_{mid}$ is the step size of the linear section ending, $iter_{max}^{2}$ is the total number of searching generations in the local search subroutine, and $(iter_{max}^{2} = M_L)$ represents the generations in which the step size is linearly reduced. Different ending values of $\lambda$ can be achieved by adjusting $k$.

The results of the tests to solve a single-machine PSS design problem were promising, and showed that the ILCOA has features such as higher stability, better ability to escape from local minima, more robustness to search space size, and finer performance in comparison to LCOA, especially in high-dimensional problems with large search spaces.

In [104], Pourmousa et al. used ILCOA to evaluate the performance of photovoltaic panels. This evaluation inevitably involves having the exact model of solar cells. Different approaches to solar cell modeling have been proposed in the literature, and can generally be classified as either traditional or intelligent methods. To obtain an accurate model of such highly nonlinear systems, however, is a challenging task that continues to defy researchers. The authors suggested that the remarkable local and global search abilities of ILCOA grant it a distinct edge over other optimization methods, allowing it to sift through the entire search space for the global optimum. The efficacy of the proposed approach was highlighted by comparing its performance to those of three other algorithms: the Bird Mating Optimizer (BMO) [105], Improved Chaotic Whale Optimization Algorithm (ICWOA) [106], and LCOA in terms of Root Mean Square Error (RMSE), Relative Error, Mean Absolute Error (MAE), Normalized MAE, and Mean Bias Error.

The results showed that ILCOA performs better than the other algorithms and has higher convergence speed and stability. Moreover, it was able to reach a smaller value of the objective function compared with LCOA.

ILCOA was tested by Ebrahimi et al. [107] along with other algorithms to identify the parameter of the two chaotic Chen and Lorenz attractors as well as a permanent magnet synchronous motor system. Their simulation results ascertained the very good performance and high speed of this algorithm in identifying these systems in comparison with other algorithms.

# 5. Applications in Control Theory

Controlling chaos via different control schemes has received great interest after it was first introduced in 1990 by Ott, Grebogi, and Yorke [108]. The Lozi map is often used in this domain and in control theory-based chaos.

## 5.1. Observers

Observers are a primary tool used in control theory. Consider the problem of estimating the current state $x_k$ of a discrete-time nonlinear dynamical system described by a system of first-order difference equations:

$$ \left\{ \begin{array}{l} x_{k+1} = f(x_k), \\ y_{k+1} = h(x_k), \end{array} \right. \quad (69) $$

where, from past observations, $y_s$ and $s \le k$, where $k \in \{0, 1, 2, ...\}$.

The vector fields $f: \mathbb{R}^n \to \mathbb{R}^n$ and $h: \mathbb{R}^n \to \mathbb{R}^p$ are continuous with $p \le n$. The variable $x \in \mathbb{R}^n$ stands for the state of the system, while the variable $y \in \mathbb{R}^p$ represents the measurement output.
(SESS), which focuses more on smaller values of $\lambda$ and contributes to better exploitation for refining the results:

$$ egin{cases} 
\lambda = \lambda_{max} - \frac{\lambda_{max} - \lambda_{mid}}{iter_{max}^{1}} \times \text{iteration}, & 1 \le \text{iteration} \le \text{iter}_{max}^{1} \\ 
\lambda = \left[ \lambda_{max} - \frac{\lambda_{max} - \lambda_{mid}}{iter_{max}^{1}} \right] \times e^{\frac{(iter_{max}^{1} - \text{iteration})}{k}}, & \text{iter}_{max}^{1} \le \text{iteration} \le \text{iter}_{max}^{2} 
\end{cases} 
\quad (68) $$

where $\lambda_{max}$ is the initial step size, $\lambda_{mid}$ is the step size of the linear section ending, $iter_{max}^{2}$ is the total number of searching generations in the local search subroutine, and $(iter_{max}^{2} = M_L)$ represents the generations in which the step size is linearly reduced. Different ending values of $\lambda$ can be achieved by adjusting $k$.

The results of the tests to solve a single-machine PSS design problem were promising, and showed that the ILCOA has features such as higher stability, better ability to escape from local minima, more robustness to search space size, and finer performance in comparison to LCOA, especially in high-dimensional problems with large search spaces.

In [104], Pourmousa et al. used ILCOA to evaluate the performance of photovoltaic panels. This evaluation inevitably involves having the exact model of solar cells. Different approaches to solar cell modeling have been proposed in the literature, and can generally be classified as either traditional or intelligent methods. To obtain an accurate model of such highly nonlinear systems, however, is a challenging task that continues to defy researchers. The authors suggested that the remarkable local and global search abilities of ILCOA grant it a distinct edge over other optimization methods, allowing it to sift through the entire search space for the global optimum. The efficacy of the proposed approach was highlighted by comparing its performance to those of three other algorithms: the Bird Mating Optimizer (BMO) [105], Improved Chaotic Whale Optimization Algorithm (ICWOA) [106], and LCOA in terms of Root Mean Square Error (RMSE), Relative Error, Mean Absolute Error (MAE), Normalized MAE, and Mean Bias Error.

The results showed that ILCOA performs better than the other algorithms and has higher convergence speed and stability. Moreover, it was able to reach a smaller value of the objective function compared with LCOA.

ILCOA was tested by Ebrahimi et al. [107] along with other algorithms to identify the parameter of the two chaotic Chen and Lorenz attractors as well as a permanent magnet synchronous motor system. Their simulation results ascertained the very good performance and high speed of this algorithm in identifying these systems in comparison with other algorithms.

# 5. Applications in Control Theory

Controlling chaos via different control schemes has received great interest after it was first introduced in 1990 by Ott, Grebogi, and Yorke [108]. The Lozi map is often used in this domain and in control theory-based chaos.

## 5.1. Observers

Observers are a primary tool used in control theory. Consider the problem of estimating the current state $x_k$ of a discrete-time nonlinear dynamical system described by a system of first-order difference equations:

$$ egin{cases} 
x_{k+1} = f(x_k), \\ 
y_{k+1} = h(x_k), 
\end{cases} 
\quad (69) $$

where, from past observations, $y_s$ and $s \le k$, where $k \in \{0, 1, 2, ...\}$.

The vector fields $f: \mathbb{R}^n \to \mathbb{R}^n$ and $h: \mathbb{R}^n \to \mathbb{R}^p$ are continuous with $p \le n$. The variable $x \in \mathbb{R}^n$ stands for the state of the system, while the variable $y \in \mathbb{R}^p$ represents the measurement output.
An observer is a system driven by observations

$$ \hat{x}_{k+1} = \hat{f}(\hat{x}_k, y_k) \qquad (70) $$

such that the error $\tilde{x}_k = x_k - \hat{x}_k$ goes to zero as $k \to \infty$.

In [109], Xiao provided a direct method for constructing a change of coordinates for the design of nonlinear discrete-time observers along with an explicit expression of this change of coordinates. He used the Lozi (3) and Hénon (2) maps to perform simulations for chaotic systems to illustrate the method.

Instead, Voortman et al. [110] designed a data rate-constrained observer for a dynamical system which functions in both discrete time and continuous time. The system is connected to a remote location by a communication channel that can transmit limited amounts of data per unit of time. The objective of the observer is to provide estimates of the state at the remote location through messages that are sent via the channel. The observer is designed such that it is robust to losses in the communication channel. It was tested through simulations for the Lozi map and the Lorenz system. As the Lozi map is not differentiable, a smoothed version was used:

$$ \Phi_{\alpha} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 1 - a f_{\alpha}(x) + by, \\ x, \end{pmatrix} \qquad (71) $$

where

$$ f_{\alpha}(x) = \begin{cases} |x|, & \text{if } |x| \ge \alpha, \\ \frac{x^2}{2\alpha} + \frac{x}{2\alpha}, & \text{if } |x| < \alpha, \end{cases} \qquad (72) $$

with $\alpha \ll 1$.

By combining results from several other papers, the authors provided an analytical bound on the channel rate that depends on the Lyapunov dimension rather than the upper box dimension. These analytical bounds were computed for the smoothed Lozi map and the Lorenz system. Under the following Assumption 1, they proved Theorem 2 below.

**Assumption 1.** $a, b, \alpha > 0, 1 - a < b < 1, \alpha < (a + 1 - b)^{-1}$

**Theorem 2 ([110]).** Let Assumption 5 hold and let $S_0$ be a compact invariant set of the smoothed Lozi map (71). Then, the associated dynamical system is observable on the set $S_0$ via any communication channel with capacity

$$ c < \frac{\left[ 2\log_2\left(\frac{\sqrt{a^2+4b-a}}{2}\right) - \log_2 b \right] \left[ \log_2\left(\frac{\sqrt{a^2+4b-a}}{2}\right) \right]}{\log_2\left(\frac{\sqrt{a^2+4b-a}}{2}\right)}. $$

In addition, they computed the Lyapunov dimension for the smoothed Lozi map. Simulations of the observer on both systems proved that the theoretical rate is closely related to the actual rate required to implement the observer.

## 5.2. Control of Chaos

In this section, we consider three aspects of the control of chaos: stabilization, adaptive and predictive control methods, and the identification of chaotic systems.

### 5.2.1. Stabilization of Chaos

Orbits of a dynamical system originating near an unstable fixed point remain unstable throughout the evolution of the system. To obtain regular motion from such initial point, the method used by Saha and Kumra [111] is to modify the value of the parameters $a$ and $b$. As an example, they considered the case of $a = 1.6$ and $b = 0.5$ for which there exists an unstable fixed point $P^* = (0.47619, 0.238095)$. Taking the initial point $P = (0.5, 0.3)$ near $P^*$, they performed search using their previously defined method of asymptotic stability [112]
An observer is a system driven by observations

$$ \hat{x}_{k+1} = \hat{f}(\hat{x}_k, y_k) \qquad (70) $$

such that the error $\tilde{x}_k = x_k - \hat{x}_k$ goes to zero as $k \to \infty$.

In [109], Xiao provided a direct method for constructing a change of coordinates for the design of nonlinear discrete-time observers along with an explicit expression of this change of coordinates. He used the Lozi (3) and Hénon (2) maps to perform simulations for chaotic systems to illustrate the method.

Instead, Voortman et al. [110] designed a data rate-constrained observer for a dynamical system which functions in both discrete time and continuous time. The system is connected to a remote location by a communication channel that can transmit limited amounts of data per unit of time. The objective of the observer is to provide estimates of the state at the remote location through messages that are sent via the channel. The observer is designed such that it is robust to losses in the communication channel. It was tested through simulations for the Lozi map and the Lorenz system. As the Lozi map is not differentiable, a smoothed version was used:

$$ \Phi_{\alpha} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 1 - a f_{\alpha}(x) + by, \\ x, \end{pmatrix}, \qquad (71) $$

where

$$ f_{\alpha}(x) = \begin{cases} |x|, & \text{if } |x| \ge \alpha, \\ \frac{x^2}{2\alpha} + \frac{x}{2\alpha}, & \text{if } |x| < \alpha, \end{cases} \qquad (72) $$

with $\alpha \ll 1$.

By combining results from several other papers, the authors provided an analytical bound on the channel rate that depends on the Lyapunov dimension rather than the upper box dimension. These analytical bounds were computed for the smoothed Lozi map and the Lorenz system. Under the following Assumption 1, they proved Theorem 2 below.

**Assumption 1.** $a, b, \alpha > 0, 1 - a < b < 1, \alpha < (a + 1 - b)^{-1}$

**Theorem 2 ([110]).** Let Assumption 5 hold and let $S_0$ be a compact invariant set of the smoothed Lozi map (71). Then, the associated dynamical system is observable on the set $S_0$ via any communication channel with capacity

$$ c < \frac{\left[ 2\log_2\left(\frac{\sqrt{a^2+4b-a}}{2}\right) - \log_2 b \right] \left[ \log_2\left(\frac{\sqrt{a^2+4b-a}}{2}\right) \right]}{\log_2\left(\frac{\sqrt{a^2+4b-a}}{2}\right)}. $$

In addition, they computed the Lyapunov dimension for the smoothed Lozi map. Simulations of the observer on both systems proved that the theoretical rate is closely related to the actual rate required to implement the observer.

## 5.2. Control of Chaos

In this section, we consider three aspects of the control of chaos: stabilization, adaptive and predictive control methods, and the identification of chaotic systems.

### 5.2.1. Stabilization of Chaos

Orbits of a dynamical system originating near an unstable fixed point remain unstable throughout the evolution of the system. To obtain regular motion from such initial point, the method used by Saha and Kumra [111] is to modify the value of the parameters $a$ and $b$. As an example, they considered the case of $a = 1.6$ and $b = 0.5$ for which there exists an unstable fixed point $P^* = (0.47619, 0.238095)$. Taking the initial point $P = (0.5, 0.3)$ near $P^*$, they performed search using their previously defined method of asymptotic stability [112]
for the new parameter values, for which the solution is periodic with the same initial point. This method is based on the computation of the Lyapunov exponents. They found that $a = 1.52381$ and $b = 0.57619$.

Gokce et al. [113] investigated the stability and tracking performance of discrete-time chaotic systems in the presence of external disturbance and noise. They developed a neural network control scheme on the basis of a novel adaptive learning rate to stabilize the chaotic motion of discrete-time chaotic systems to a fixed point as well as to track the desired reference trajectory. The effectiveness of the proposed method was investigated through simulation studies on the Lozi map and a performance comparison was made with the backstepping control strategy. The backstepping method based on the Lyapunov stability is a recursive procedure that breaks the full system into a sequence of small subsystems. Simulation studies demonstrated that the proposed method can successfully stabilize the chaotic trajectory of the system to the fixed point and track the desired reference trajectory. It has been shown that the proposed method is less sensitive to noise and disturbance and has a more robust property than the backstepping control scheme if the neural network is trained properly with accurate training parameters.

Joshi et al. [42] studied the control of a complex fractional-ordered Lozi map (see Section 2.5.2). They used the Pyragas method [78], which is a feedback method that can be attempted without any detailed knowledge of the dynamical system under consideration. There are two ways in which delay can be introduced in the system, are formulated as follows:

$$x(n) = \gamma x(j-2) + x(0) + \frac{1}{\Gamma(\nu)} \sum_{j=1}^{n} \frac{\Gamma(n-j+\nu)}{\Gamma(n-j+1)} [-\alpha|x(j-1)| - x(j-1) + bx(j-2)], \quad (73)$$

and

$$x(n) = x(0) + \frac{1}{\Gamma(\nu)} \sum_{j=1}^{n} \frac{\Gamma(n-j+\nu)}{\Gamma(n-j+1)} [-\alpha|x(j-1)| - x(j-1) + bx(j-2) + \gamma x(j-3)], \quad (74)$$

where $\gamma$ is the strength of the delay. For their numerical simulations, they set $\gamma = 0.33$ for both (73) and (74).

They found that in both cases there is a reduction in the range over which chaos is observed. Period 2 is stabilized instead.

In Case 1 (73), Period 2 has an even more stable range, and the stability range of Period 1 is reduced, whereas for Case 2 (74) the range of Period 1 is significantly increased and that of Period 2 spans over a smaller range of parameters. Thus, chaos control can be achieved using the Pyragas method even in complex-order fractional maps.

### 5.2.2. Adaptive and Predictive Control

In [114], Chen and Dong presented interesting analysis and simulations on the control of chaotic dynamic systems using conventional feedback control strategies. The trajectories of the chaotic Lozi system are controlled to its equilibrium points using conventional feedback controls. The proposed controlled Lozi map is

$$\left\{ \begin{array}{l} x_{k+1} = 1 - a|x_k| + y_k, \\ y_{k+1} = bx_k - Kv_k. \end{array} \right. \qquad (75)$$

For this system, Chen and Dong proved the following.

**Proposition 2** ([114]). A necessary and sufficient condition for the local controllability of the chaotic Lozi system (3) to its equilibrium point $(x^*, y^*) = (\frac{1}{a-(b-1)}, \frac{b}{a-(b-1)})$ is $v_k \to 0$, as $k \to \infty$, where $u_k = -Kv_k$ is the feedback controller applied to the system shown in (75) and $v_k$ is the executive control input in the feedback $u_k = -Kv_k$.
for the new parameter values, for which the solution is periodic with the same initial point. This method is based on the computation of the Lyapunov exponents. They found that $a = 1.52381$ and $b = 0.57619$.

Gokce et al. [113] investigated the stability and tracking performance of discrete-time chaotic systems in the presence of external disturbance and noise. They developed a neural network control scheme on the basis of a novel adaptive learning rate to stabilize the chaotic motion of discrete-time chaotic systems to a fixed point as well as to track the desired reference trajectory. The effectiveness of the proposed method was investigated through simulation studies on the Lozi map and a performance comparison was made with the backstepping control strategy. The backstepping method based on the Lyapunov stability is a recursive procedure that breaks the full system into a sequence of small subsystems. Simulation studies demonstrated that the proposed method can successfully stabilize the chaotic trajectory of the system to the fixed point and track the desired reference trajectory. It has been shown that the proposed method is less sensitive to noise and disturbance and has a more robust property than the backstepping control scheme if the neural network is trained properly with accurate training parameters.

Joshi et al. [42] studied the control of a complex fractional-ordered Lozi map (see Section 2.5.2). They used the Pyragas method [78], which is a feedback method that can be attempted without any detailed knowledge of the dynamical system under consideration. There are two ways in which delay can be introduced in the system, are formulated as follows:

$$x(n) = \gamma x(j-2) + x(0) + \frac{1}{\Gamma(\nu)} \sum_{j=1}^{n} \frac{\Gamma(n-j+\nu)}{\Gamma(n-j+1)} [-\alpha|x(j-1)| - x(j-1) + bx(j-2)], \quad (73)$$

and

$$x(n) = x(0) + \frac{1}{\Gamma(\nu)} \sum_{j=1}^{n} \frac{\Gamma(n-j+\nu)}{\Gamma(n-j+1)} [-\alpha|x(j-1)| - x(j-1) + bx(j-2) + \gamma x(j-3)], \quad (74)$$

where $\gamma$ is the strength of the delay. For their numerical simulations, they set $\gamma = 0.33$ for both (73) and (74).

They found that in both cases there is a reduction in the range over which chaos is observed. Period 2 is stabilized instead.

In Case 1 (73), Period 2 has an even more stable range, and the stability range of Period 1 is reduced, whereas for Case 2 (74) the range of Period 1 is significantly increased and that of Period 2 spans over a smaller range of parameters. Thus, chaos control can be achieved using the Pyragas method even in complex-order fractional maps.

### 5.2.2. Adaptive and Predictive Control

In [114], Chen and Dong presented interesting analysis and simulations on the control of chaotic dynamic systems using conventional feedback control strategies. The trajectories of the chaotic Lozi system are controlled to its equilibrium points using conventional feedback controls. The proposed controlled Lozi map is

$$\left\{ \begin{array}{l} x_{k+1} = 1 - a|x_k| + y_k, \\ y_{k+1} = bx_k - Kv_k. \end{array} \right. \qquad (75)$$

For this system, Chen and Dong proved the following.

**Proposition 2** ([114]). A necessary and sufficient condition for the local controllability of the chaotic Lozi system (3) to its equilibrium point $(x^*, y^*) = (\frac{1}{a-(b-1)}, \frac{b}{a-(b-1)})$ is $v_k \to 0$, as $k \to \infty$, where $u_k = -Kv_k$ is the feedback controller applied to the system shown in (75) and $v_k$ is the executive control input in the feedback $u_k = -Kv_k$.
Computer simulations have been performed for $a = 1.8$, $b = -1$, $K = -0.195$; $a = 1.8$, $b = 0.4$, $K = 1.205$; $a = 1.8$, $b = 0.997$, $K = 1.802$; and $a = 1.8$, $b = 1.0$, $K = 1.805$.

Improving this method, Gonzalez [115] considered the Lozi map and showed that the trajectories can be driven to the fixed point as well as to other fixed reference points $(x^*, bx^*)$. The controlled the Lozi map defined as

$$ \begin{cases} x_{k+1} = 1 - a|x_k| + y_k + u_k, \\ y_{k+1} = bx_k. \end{cases} \qquad (76) $$

The following two control algorithms were proposed.

* Case 1: the value of $a$ is unknown and the value of $b$ is precisely known. In this case, the chosen control law is

$$ u_k = \tilde{a}_k |x_k| - bx_{k-1} - 1 + x^*, $$

with

$$ \tilde{a}_{k+1} = \tilde{a}_k + \frac{\alpha |x_k|}{\gamma + x_k^2} (x_{k+1} - (-\tilde{a} |x_k| + bx_{k-1} + 1 + u_k)) $$

and $\alpha > 0, \gamma \ge 0$.

With $a_k = \tilde{a}_k - a$, the closed loop system emerges:

$$ \begin{cases} x_{k+1} = a_k |x_k| + x^*, \\ a_{k+1} = a_k - \frac{\alpha |x_k|^2}{\gamma + x_k^2} a_k. \end{cases} $$

* Case 2: both values of $a$ and $b$ are known approximately. In this second case, the chosen control law is

$$ u_k = \tilde{a}_k |x_k| - 1 + x^*, \qquad (77) $$

where the parameter estimator is provided by

$$ \tilde{a}_{k+1} = \tilde{a}_k + \frac{\alpha |x_k|}{\gamma + x_k^2} (x_{k+1} - (-\tilde{a} |x_k| + 1 + u_k)) \qquad (78) $$

and $\alpha > 0, \gamma \ge 0$.

With $a_k = \tilde{a}_k - a$ and $z_k = x_{k-1}$, the closed loop systems emerges:

$$ \begin{cases} x_{k+1} = a_k |x_k| + bz_k + x^*, \\ z_{k+1} = x_{k-1}, \\ a_{k+1} = a_k - \frac{\alpha |x_k|^2}{\gamma + x_k^2} (\tilde{a}_k |x_k| + bz_k). \end{cases} \qquad (79) $$

In this case, there is now the law $b$. It was proved that for every real $a$ and $-1 < b < 1/2$, $b \ne 0$ there exist $\alpha$ and $\beta$ such that the fixed point is asymptotically stable for (79). This means that by applying the adaptive strategy (77) and (78) to the control system (76) and taking $(x_1(0), x_2(0), \tilde{a}_0)$ in the neighborhood of $(x^*, qx^*, a - sgn(x^*)b)$, we obtain $(x_1(k), x_2(k)) \to (x^*, qx^*)$ as $k \to +\infty$; at the same time, we obtain a better approximation to the values of $a$ and $b$.

These two proposed control laws can be applied when $0 < b < 1$, $a > b + 1$, and $2a + b < 4$, that is, for the regions in the parameter space where chaos can occur (see Figure 3). It has been shown that when using these control laws $(x, y)$ converges towards $(x^*, bx^*)$.

In [116], Park et al. investigated the possibility of controlling discrete-time chaotic dynamical systems through a generalized predictive control (GPC) method which does not require a precise mathematical model of the chaotic dynamics. GPC methods may be considered as a kind of adaptive control strategy. By combining the ARMAX (AutoRegressive Moving Average model with eXogenous noise input) modeling with the GPC
Computer simulations have been performed for $a = 1.8, b = -1, K = -0.195; a = 1.8, b = 0.4, K = 1.205; a = 1.8, b = 0.997, K = 1.802; \text{ and } a = 1.8, b = 1.0, K = 1.805$.

Improving this method, Gonzalez [115] considered the Lozi map and showed that the trajectories can be driven to the fixed point as well as to other fixed reference points $(x^*, bx^*)$. The controlled the Lozi map defined as

$$ \begin{cases} x_{k+1} = 1 - a|x_k| + y_k + u_k, \\ y_{k+1} = bx_k. \end{cases} \qquad (76) $$

The following two control algorithms were proposed.

* Case 1: the value of $a$ is unknown and the value of $b$ is precisely known. In this case, the chosen control law is

$$ u_k = \tilde{a}_k |x_k| - bx_{k-1} - 1 + x^*, $$

with

$$ \tilde{a}_{k+1} = \tilde{a}_k + \frac{\alpha |x_k|}{\gamma + x_k^2} (x_{k+1} - (-\tilde{a} |x_k| + bx_{k-1} + 1 + u_k)) $$

and $\alpha > 0, \gamma \ge 0$.

With $a_k = \tilde{a}_k - a$, the closed loop system emerges:

$$ \begin{cases} x_{k+1} = a_k |x_k| + x^*, \\ a_{k+1} = a_k - \frac{\alpha |x_k|^2}{\gamma + x_k^2} a_k. \end{cases} $$

* Case 2: both values of $a$ and $b$ are known approximately. In this second case, the chosen control law is

$$ u_k = \tilde{a}_k |x_k| - 1 + x^*, \qquad (77) $$

where the parameter estimator is provided by

$$ \tilde{a}_{k+1} = \tilde{a}_k + \frac{\alpha |x_k|}{\gamma + x_k^2} (x_{k+1} - (-\tilde{a} |x_k| + 1 + u_k)) \qquad (78) $$

and $\alpha > 0, \gamma \ge 0$.

With $a_k = \tilde{a}_k - a$ and $z_k = x_{k-1}$, the closed loop systems emerges:

$$ \begin{cases} x_{k+1} = a_k |x_k| + bz_k + x^*, \\ z_{k+1} = x_{k-1}, \\ a_{k+1} = a_k - \frac{\alpha |x_k|^2}{\gamma + x_k^2} (\tilde{a}_k |x_k| + bz_k). \end{cases} \qquad (79) $$

In this case, there is now the law $b$. It was proved that for every real $a$ and $-1 < b < 1/2, b \ne 0$ there exist $\alpha$ and $\beta$ such that the fixed point is asymptotically stable for (79). This means that by applying the adaptive strategy (77) and (78) to the control system (76) and taking $(x_1(0), x_2(0), \tilde{a}_0)$ in the neighborhood of $(x^*, qx^*, a - sgn(x^*)b)$, we obtain $(x_1(k), x_2(k)) \to (x^*, qx^*)$ as $k \to +\infty$; at the same time, we obtain a better approximation to the values of $a$ and $b$.

These two proposed control laws can be applied when $0 < b < 1, a > b + 1$, and $2a + b < 4$, that is, for the regions in the parameter space where chaos can occur (see Figure 3). It has been shown that when using these control laws $(x, y)$ converges towards $(x^*, bx^*)$.

In [116], Park et al. investigated the possibility of controlling discrete-time chaotic dynamical systems through a generalized predictive control (GPC) method which does not require a precise mathematical model of the chaotic dynamics. GPC methods may be considered as a kind of adaptive control strategy. By combining the ARMAX (AutoRegressive Moving Average model with eXogenous noise input) modeling with the GPC
technique, they developed a strategy for adaptive control of uncertain discrete-time chaotic systems. They investigated both the control performance and the system's sensitivity to initial conditions, and compared them with that of the conventional reference model adaptive control by means of numerical simulations on Lozi (for a = 1.8 and b = 0.4) and Hénon maps. Simulation results showed that this new controller yields faster settling time and more robust tracking performance.

For the same value of parameters of the Lozi map, Feng and Chen [117] have presented a new adaptive control algorithm. The basic idea behind their proposed approach is to represent the chaotic system as a Takagi-Sugeno fuzzy model (see Section 4.1.1), design a local linear adaptive controller in each local region, and then reassemble a global adaptive controller over the entire domain in such a way that the stability of the closed-loop adaptive control system is guaranteed.

### 5.2.3. Identification of Chaotic Systems

In [118,119], Chen et al. studied a feed-forward artificial neural network in a Wiener-type cascade dynamic model in which a three-layer feed-forward artificial neural network is employed as the nonlinear static subsystem and a simple linear plant is used as the dynamic subsystem. They presented theoretical results showing that various chaotic systems can be well identified by these two basic structures by training the neural network and choosing an appropriate linear subsystem. They demonstrated the approximation capability of the proposed system through simulations, where it was used to identify (synchronize) the chaotic continuous-time Duffing oscillator and the discrete-time Hénon and Lozi systems.

## 6. Cryptography

Cryptography is the primary means of protecting communications in the cyber-centric world in which humankind lives today. Modern technologies involve fast communication links between potentially billions of devices via complex networks (satellites, mobile phones, the internet, etc.). The primary concern posed by these complex and tangled networks is their protection against passive and active attacks that could compromise public safety and privacy. Cryptography has been around for over two thousand years, with the famous Caesar code being used by Emperor Julius Caesar. Today, the properties of chaotic attractors are recognized as the basis of many different methods of cryptography.

### 6.1. *Theoretical Algorithms*

The literature includes two articles in which the Lozi map is at the center of theoretical cryptographic algorithms. In the first considered article, Sira-Ramirez et al. [120] proposed the use of nonlinear exact chaotic system state reconstructors for the fast and efficient decoding of multiple discrete-time chaotic encrypted digital messages. After a general description of their method, in order to test the effectiveness of the exact reconstructor scheme in an actual message decoding problem, they applied it using the Lozi map. They considered transmission in a noise-free environment such as the internet.

They transmitted to a remote computer containing a reconstructor digital computer program, a message consisting of a digitized file of Van Gogh's self portrait. The encoding of the messages was carried out using the states z and w of the following unobservable but constructible Lozi system:

$$ egin{cases} x_{k+1} = z_k, \\ z_{k+1} = -a|z_k| + bx_k + 1, \\ w_{k+1} = x_k z_k, \\ y_k = x_k. \end{cases} \tag{80} $$
technique, they developed a strategy for adaptive control of uncertain discrete-time chaotic systems. They investigated both the control performance and the system's sensitivity to initial conditions, and compared them with that of the conventional reference model adaptive control by means of numerical simulations on Lozi (for $a = 1.8$ and $b = 0.4$) and Hénon maps. Simulation results showed that this new controller yields faster settling time and more robust tracking performance.

For the same value of parameters of the Lozi map, Feng and Chen [117] have presented a new adaptive control algorithm. The basic idea behind their proposed approach is to represent the chaotic system as a Takagi-Sugeno fuzzy model (see Section 4.1.1), design a local linear adaptive controller in each local region, and then reassemble a global adaptive controller over the entire domain in such a way that the stability of the closed-loop adaptive control system is guaranteed.

### 5.2.3. Identification of Chaotic Systems

In [118,119], Chen et al. studied a feed-forward artificial neural network in a Wiener-type cascade dynamic model in which a three-layer feed-forward artificial neural network is employed as the nonlinear static subsystem and a simple linear plant is used as the dynamic subsystem. They presented theoretical results showing that various chaotic systems can be well identified by these two basic structures by training the neural network and choosing an appropriate linear subsystem. They demonstrated the approximation capability of the proposed system through simulations, where it was used to identify (synchronize) the chaotic continuous-time Duffing oscillator and the discrete-time Hénon and Lozi systems.

## 6. Cryptography

Cryptography is the primary means of protecting communications in the cyber-centric world in which humankind lives today. Modern technologies involve fast communication links between potentially billions of devices via complex networks (satellites, mobile phones, the internet, etc.). The primary concern posed by these complex and tangled networks is their protection against passive and active attacks that could compromise public safety and privacy. Cryptography has been around for over two thousand years, with the famous Caesar code being used by Emperor Julius Caesar. Today, the properties of chaotic attractors are recognized as the basis of many different methods of cryptography.

### 6.1. *Theoretical Algorithms*

The literature includes two articles in which the Lozi map is at the center of theoretical cryptographic algorithms. In the first considered article, Sira-Ramirez et al. [120] proposed the use of nonlinear exact chaotic system state reconstructors for the fast and efficient decoding of multiple discrete-time chaotic encrypted digital messages. After a general description of their method, in order to test the effectiveness of the exact reconstructor scheme in an actual message decoding problem, they applied it using the Lozi map. They considered transmission in a noise-free environment such as the internet.

They transmitted to a remote computer containing a reconstructor digital computer program, a message consisting of a digitized file of Van Gogh's self portrait. The encoding of the messages was carried out using the states $z$ and $w$ of the following unobservable but constructible Lozi system:

$$ \begin{cases} x_{k+1} = z_k, \\ z_{k+1} = -a|z_k| + bx_k + 1, \\ w_{k+1} = x_k z_k, \\ y_k = x_k. \end{cases} \tag{80} $$
The exact reconstructor of system (80) for the non measured variable $z_k$ may be obtained trivially, as $x$ is a one-step delay of $z$. This was readily derived to be

$$ egin{cases} x_k = y_k, \\ z_k = -a|y_k| + by_{k-1} + 1, \\ w_{k+1} = y_{k-1}(-a|y_{k-1}| + by_{k-2} + 1). \end{cases} \quad (81) $$

The image and text files were chaotically encoded in a byte-by-byte fashion after a simple normalization of the carrier chaotic states by an appropriate constant factor. Thus, they defined

$$ I_k = K_1 z_k + i_k, T_k = K_2 w_k + t_k, i_k = I_k - K_1 \hat{z}_k, t_k = T_k - K_2 \hat{w}_k, $$

where $i_k$ and $t_k$ represent the $k$th byte of the image file, $I_k$ and $T_k$ are the corresponding $k$th bit of the chaotic encrypted image, and the factors $K_1$ and $K_2$ are suitable normalizing parameters, both taken as equal to 1000. The reconstructed states $\hat{z}$ and $\hat{w}$ exactly coincide with the original states $z$ and $w$.

The encryption and decryption programs were written in the C programming language and compiled with Borland C++ version 1.01. The size of the digital image file was approximately 120 KB. The process of encryption was completed in approximately two seconds, as was the decoding process.

In the second article, Aliwi and Ajeena [121] used an unconventional generalization of (3) based on the chaotic cosine map recently published in [122].

The Cosine Chaotic Map (CCM) is a chaotification method that enhances the chaotic complexity of existing chaotic maps. This method performs the cosine function alongside a chaotic map that cascades in the system under consideration. Thus, the results provide a new chaotic map having a wide chaotic range within the closed interval [-1, +1]. Theoretically, the CCM has properties based on the properties of the underlying seed maps. In the case of the Lozi map, Aliwi and Ajeena considered (3), changing 1 to 3 in the first component

$$ f\left(\begin{array}{c} x \\ y \end{array}\right)=\left(\begin{array}{c} 3-a|x|+y, \\ bx, \end{array}\right) \quad (82) $$

with $a = -1.8$ and $b = 0.25$. Then, they inserted (82) in CCM:

$$ f\left(\begin{array}{c} x \\ y \end{array}\right)=\left(\begin{array}{c} \cos \left(2^{\left(k+3-a\left|x\right|+y\right)}\right), \\ \cos \left(2^{\left(k+bx\right)}\right), \end{array}\right) \quad (83) $$

where $k \in [10, 24]$.

Starting with any initial point $(x_0, y_0)$ belonging to the basin of attraction of the Lozi map, the iterates randomly fulfill the square [-1, +1]² (see Figure 9).

Based on a Knapsack problem approach, a fuzzy chaos-based cryptography method for a chaotic cosine map was proposed in which the system works by designing the Takagi-Sugeno (TS) fuzzy model (see Section 4.1.1) on a chaotic map as a seed map in the cosine map. The basic idea of this method is to apply the chaotification method in order to use a cosine chaotic map approach on the outputs of the TS fuzzy model to compute a real super-increasing sequence applied in the Knapsack equation. Aliwi and Ajeena [121] explicitly described this method, which is too long to be included here. The algorithmic performance of the method was derived as well. In addition, numerical simulations were presented to demonstrate the effectiveness of the proposed method.
The exact reconstructor of system (80) for the non measured variable $z_k$ may be obtained trivially, as $x$ is a one-step delay of $z$. This was readily derived to be

$$ \left\{ \begin{array}{l} x_k = y_k, \\ z_k = -a|y_k| + by_{k-1} + 1, \\ w_{k+1} = y_{k-1}(-a|y_{k-1}| + by_{k-2} + 1). \end{array} \right. \qquad (81) $$

The image and text files were chaotically encoded in a byte-by-byte fashion after a simple normalization of the carrier chaotic states by an appropriate constant factor. Thus, they defined

$$ I_k = K_1 z_k + i_k, T_k = K_2 w_k + t_k, i_k = I_k - K_1 \hat{z}_k, t_k = T_k - K_2 \hat{w}_k, $$

where $i_k$ and $t_k$ represent the $k$th byte of the image file, $I_k$ and $T_k$ are the corresponding $k$th bit of the chaotic encrypted image, and the factors $K_1$ and $K_2$ are suitable normalizing parameters, both taken as equal to 1000. The reconstructed states $\hat{z}$ and $\hat{w}$ exactly coincide with the original states $z$ and $w$.

The encryption and decryption programs were written in the C programming language and compiled with Borland C++ version 1.01. The size of the digital image file was approximately 120 KB. The process of encryption was completed in approximately two seconds, as was the decoding process.

In the second article, Aliwi and Ajeena [121] used an unconventional generalization of (3) based on the chaotic cosine map recently published in [122].

The Cosine Chaotic Map (CCM) is a chaotification method that enhances the chaotic complexity of existing chaotic maps. This method performs the cosine function alongside a chaotic map that cascades in the system under consideration. Thus, the results provide a new chaotic map having a wide chaotic range within the closed interval [-1, +1]. Theoretically, the CCM has properties based on the properties of the underlying seed maps. In the case of the Lozi map, Aliwi and Ajeena considered (3), changing 1 to 3 in the first component

$$ f\left(\begin{array}{c} x \\ y \end{array}\right)=\left(\begin{array}{c} 3-a|x|+y, \\ bx, \end{array}\right) \qquad (82) $$

with $a = -1.8$ and $b = 0.25$. Then, they inserted (82) in CCM:

$$ f\left(\begin{array}{c} x \\ y \end{array}\right)=\left(\begin{array}{c} \cos \left(2^{\left(k+3-a|x|+y\right)}\right), \\ \cos \left(2^{\left(k+bx\right)}\right), \end{array}\right) \qquad (83) $$

where $k \in [10, 24]$.

Starting with any initial point $(x_0, y_0)$ belonging to the basin of attraction of the Lozi map, the iterates randomly fulfill the square [-1, +1]² (see Figure 9).

Based on a Knapsack problem approach, a fuzzy chaos-based cryptography method for a chaotic cosine map was proposed in which the system works by designing the Takagi-Sugeno (TS) fuzzy model (see Section 4.1.1) on a chaotic map as a seed map in the cosine map. The basic idea of this method is to apply the chaotification method in order to use a cosine chaotic map approach on the outputs of the TS fuzzy model to compute a real super-increasing sequence applied in the Knapsack equation. Aliwi and Ajeena [121] explicitly described this method, which is too long to be included here. The algorithmic performance of the method was derived as well. In addition, numerical simulations were presented to demonstrate the effectiveness of the proposed method.
<figure><img src="image_12.png" /><figcaption>Figure 9. Image on the (x, y)-plane of 300,000 iterates of the cosine Lozi chaotic map (83) for initial values x₀ = 0.1 and y₀ = 0.3 and parameter values a = −1.8, b = 0.25, and k = 12.0.</figcaption></figure>

## 6.2. Algorithms for Images

### 6.2.1. Image Encryption Algorithms

Several algorithms for image encryption based on the Lozi map have been developed.

An optical color image encryption scheme based on a fingerprint key and three-step phase-shifting digital holography was proposed by Su et al. [123]. In this scheme, the fingerprint directly serves as a secret key. The random phase masks generated from the fingerprint using a secure hash algorithm (SHA-256) and the chaotic Lozi map are used as interim variables. With the help of the fingerprint-based random phase masks located in the linear canonical transform domain and the three-step phase-shifting digital holography, the primary color image that is hidden in a grey-scale carrier image can be encrypted into three noise-like holograms. In addition, the parameters of the chaotic Lozi map and linear canonical transform can provide additional security to the proposed encryption scheme.

The generation process of the fingerprint-based random phase masks mainly comprises the following steps [123]:

* Step 1: Calculate the hash value of the fingerprint using the SHA-256; a 256-bit hash value expressed as a hexadecimal number array can be obtained as follows: $H = [h_1, h_2, \dots, h_{64}]$.

* Step 2: Use elements of the array $H$ to generate the initial values of the chaotic Lozi map in the following way:

$$ \begin{cases} x = x_0 + \text{hexa2dec}(H(h_i : h_{i+7})) \times 10^{-16}, \\ y = y_0 + \text{hexa2dec}(H(h_i : h_{i+7})) \times 10^{-16}, \end{cases} \quad (84) $$

where $i, j = 1, 2, \dots, 57$; $x_0, y_0$, and $x, y$ denote the common and generated initial values of the chaotic Lozi map, respectively; and $\text{hexa2dec}(x)$ is used to convert hexadecimal number $x$ to a decimal number.

* Step 3: Suppose that the size of fingerprint-based random phase mask to be generated is $M \times N$ pixels. First, construct the chaotic Lozi map with initial values $x$ and $y$, then iterate $\frac{M \times N}{2}$ times to obtain two random sequences $X = \{x_1, x_2, \dots, x_{\frac{M \times N}{2}}\}$ and $Y = \{y_1, y_2, \dots, y_{\frac{M \times N}{2}}\}$.
<figure><img src="image_13.png" /><figcaption>Figure 9. Image on the (x, y)-plane of 300,000 iterates of the cosine Lozi chaotic map (83) for initial values x₀ = 0.1 and y₀ = 0.3 and parameter values a = −1.8, b = 0.25, and k = 12.0.</figcaption></figure>

## 6.2. Algorithms for Images

### 6.2.1. Image Encryption Algorithms

Several algorithms for image encryption based on the Lozi map have been developed.

An optical color image encryption scheme based on a fingerprint key and three-step phase-shifting digital holography was proposed by Su et al. [123]. In this scheme, the fingerprint directly serves as a secret key. The random phase masks generated from the fingerprint using a secure hash algorithm (SHA-256) and the chaotic Lozi map are used as interim variables. With the help of the fingerprint-based random phase masks located in the linear canonical transform domain and the three-step phase-shifting digital holography, the primary color image that is hidden in a grey-scale carrier image can be encrypted into three noise-like holograms. In addition, the parameters of the chaotic Lozi map and linear canonical transform can provide additional security to the proposed encryption scheme.

The generation process of the fingerprint-based random phase masks mainly comprises the following steps [123]:

* Step 1: Calculate the hash value of the fingerprint using the SHA-256; a 256-bit hash value expressed as a hexadecimal number array can be obtained as follows: $H = [h_1, h_2, \dots, h_{64}]$.

* Step 2: Use elements of the array $H$ to generate the initial values of the chaotic Lozi map in the following way:

$$ \begin{cases} x = x_0 + \text{hexa2dec}(H(h_i : h_{i+7})) \times 10^{-16}, \\ y = y_0 + \text{hexa2dec}(H(h_i : h_{i+7})) \times 10^{-16}, \end{cases} \quad (84) $$

where $i, j = 1, 2, \dots, 57$; $x_0, y_0$, and $x, y$ denote the common and generated initial values of the chaotic Lozi map, respectively; and $\text{hexa2dec}(x)$ is used to convert hexadecimal number $x$ to a decimal number.

* Step 3: Suppose that the size of fingerprint-based random phase mask to be generated is $M \times N$ pixels. First, construct the chaotic Lozi map with initial values $x$ and $y$, then iterate $\frac{M \times N}{2}$ times to obtain two random sequences $X = \{x_1, x_2, \dots, x_{\frac{M \times N}{2}}\}$ and $Y = \{y_1, y_2, \dots, y_{\frac{M \times N}{2}}\}$.
* Step 4: Combine sequence X with sequence Y to form a new random sequence Z where $Z = \{x_1, x_2, \dots, x_{\frac{M \times N}{2}}; y_1, y_2, \dots, y_{\frac{M \times N}{2}}\}$.

* Step 5: Reshape the random sequence Z to form a two-dimensional random matrix RM with size $M \times N$, where $RM = reshape(Z, M, N)$.

* Step 6: The fingerprint-based random phase mask FRPM can be generated in the following way: $FRPM = exp(i \cdot \pi \cdot RM)$.

It is worth noting that a series of fingerprint-based random phase masks can be generated using different elements of the array H to generate the initial values of the chaotic Lozi map from Equation (84) and repetition of steps 3–6.

We do not describe the other parts of the algorithm that are not linked to the Lozi map here.

This algorithm was tested numerically with common initial values of $x_0 = 0.34$ and $y_0 = 0.0$ and the common control parameters $a = 1.75$ and $b = 0.33$. Numerical simulation results showed that the proposed encryption scheme has a high security level and can resist statistical attacks, data loss attacks, and noise attacks.

The same team from Hebei University developed a similar algorithm based on diffrac- tive imaging and the Lozi map in [124].

### 6.2.2. Image Authentication Algorithm

Suresh et al. [125] proposed a new approach to authenticate images using a combina- tion of different chaotic maps along with a newly developed kind of chaotic neural network. This authentication code is carried by the transmitted image itself at near-lossless image content quality, and does not require any extra memory requirement. The initial parameter sensitivity and aperiodicity of chaotic maps and its integration with other chaotic maps and a chaos-based neural network were utilized to create uncertainty in the predictability of authentication code in association with image pixels.

The proposed scheme for an image authentication scheme uses the Lozi map corre- sponding to (3) under the form

$$ \begin{cases} S_{n+1} = 1 - P|S_n| + y_n, \\ y_{n+1} = QS_n, \end{cases} \qquad (85) $$

where n is the iteration number and the logistic map

$$ X_{n+1} = vX_n(1 - X_n). \qquad (86) $$

is as follows [125]:

* Step 1: Generate the random initial condition under the defined range for the Lozi and logistic maps.

* Step 2: Generate more samples based on the number of image pixels using the Lozi map and logistic map (say, $L_i$ and $D_i$).

* Step 3: Remove the LSB value from all pixels $P_{ri}$ and normalize the pixels (say, $P_i$).

* Step 4: From the initial starting sample of the Lozi chaotic sequence, start adding samples with normalized pixels, i.e., $S_i = P_i + L_i$.

* Step 5: Keep the range of $S_i$ within [0, 1] by subtracting any values greater than 1 (say, $S_i$).

* Step 6: From the initial starting sample of the logistic chaotic sequence, start adding samples with $S_{ni}$, $Z_i = S_{ni} + D_i$.

* Step 7: Keep the range of $Z_i$ within [0, 1] by subtracting any values more than 1 (say, $Z_i$).

* Step 8: Take each value of $Z_i$ as an input to the ANN, which has weights trained using the Lozi map and generate the corresponding output (say, $AN_i$).

* Step 9: Evaluate the mean value of $AN_i$ (say, m) and apply thresholding with $AN_i \ge m$ to generate the authentication code (say, $A_i$).

* Step 10: With $P_{ri}$ add the authentication code $A_i$ to obtain the image with authentication code.
* Step 4: Combine sequence X with sequence Y to form a new random sequence Z where $Z = \{x_1, x_2, \dots, x_{\frac{M \times N}{2}}; y_1, y_2, \dots, y_{\frac{M \times N}{2}}\}$.

* Step 5: Reshape the random sequence Z to form a two-dimensional random matrix RM with size $M \times N$, where $RM = reshape(Z, M, N)$.

* Step 6: The fingerprint-based random phase mask FRPM can be generated in the following way: $FRPM = exp(i \cdot \pi \cdot RM)$.

It is worth noting that a series of fingerprint-based random phase masks can be generated using different elements of the array H to generate the initial values of the chaotic Lozi map from Equation (84) and repetition of steps 3–6.

We do not describe the other parts of the algorithm that are not linked to the Lozi map here.

This algorithm was tested numerically with common initial values of $x_0 = 0.34$ and $y_0 = 0.0$ and the common control parameters $a = 1.75$ and $b = 0.33$. Numerical simulation results showed that the proposed encryption scheme has a high security level and can resist statistical attacks, data loss attacks, and noise attacks.

The same team from Hebei University developed a similar algorithm based on diffrac- tive imaging and the Lozi map in [124].

### 6.2.2. Image Authentication Algorithm

Suresh et al. [125] proposed a new approach to authenticate images using a combina- tion of different chaotic maps along with a newly developed kind of chaotic neural network. This authentication code is carried by the transmitted image itself at near-lossless image content quality, and does not require any extra memory requirement. The initial parameter sensitivity and aperiodicity of chaotic maps and its integration with other chaotic maps and a chaos-based neural network were utilized to create uncertainty in the predictability of authentication code in association with image pixels.

The proposed scheme for an image authentication scheme uses the Lozi map corre- sponding to (3) under the form

$$ \begin{cases} S_{n+1} = 1 - P|S_n| + y_n, \\ y_{n+1} = QS_n, \end{cases} \qquad (85) $$

where n is the iteration number and the logistic map

$$ X_{n+1} = vX_n(1 - X_n). \qquad (86) $$

is as follows [125]:

* Step 1: Generate the random initial condition under the defined range for the Lozi and logistic maps.

* Step 2: Generate more samples based on the number of image pixels using the Lozi map and logistic map (say, $L_i$ and $D_i$).

* Step 3: Remove the LSB value from all pixels $P_{ri}$ and normalize the pixels (say, $P_i$).

* Step 4: From the initial starting sample of the Lozi chaotic sequence, start adding samples with normalized pixels, i.e., $S_i = P_i + L_i$.

* Step 5: Keep the range of $S_i$ within [0, 1] by subtracting any values greater than 1 (say, $S_i$).

* Step 6: From the initial starting sample of the logistic chaotic sequence, start adding samples with $S_{ni}$, $Z_i = S_{ni} + D_i$.

* Step 7: Keep the range of $Z_i$ within [0, 1] by subtracting any values more than 1 (say, $Z_i$).

* Step 8: Take each value of $Z_i$ as an input to the ANN, which has weights trained using the Lozi map and generate the corresponding output (say, $AN_i$).

* Step 9: Evaluate the mean value of $AN_i$ (say, m) and apply thresholding with $AN_i \ge m$ to generate the authentication code (say, $A_i$).

* Step 10: With $P_{ri}$ add the authentication code $A_i$ to obtain the image with authentication code.
For the Lozi map, the values of y are normalized in the range [0, 1] to each decision variable in the n-dimensional space of the optimization problem. This transformation is provided by

$$Z_k = \frac{y_k - \lambda_{min}}{\lambda_{max} - \lambda_{min}} \qquad (87)$$

where $y \in [-0.6418, 0.6716]$, $[\lambda_{max}, \lambda_{min}]$ equals [0.6418, -0.6716], $P = 1.7$, and $Q = 0.5$ (this is similar to the normalization used by Araujo and Coelho in (30)). For the logistic map, $v$ is defined with uniform randomness in the range of [3.65, 3.95].

Various experiments were carried out with initial conditions defined randomly in the range of [0, 0.5]. Extracting the right authentication code is only possible with the correct information about the initialization of both chaotic maps and neural network weights. It is impossible to extract this information from image pixels.

The proposed schemes completely destroy the relationship among pixels in the generation of authentication code. Experimental results show that this scheme has efficient in authentication and localization and does minimal damage to the original image; furthermore, it can be applied to both grayscale and color images.

### 6.2.3. Hardware-Implemented Cryptographic Algorithms

In addition to the previous algorithms, others have been directly implemented on hardware circuits, as we show in the three following examples.

In [126], Cicek implemented the Lozi, Tinkerbell, and Barnsley Fern two-dimensional discrete chaotic maps on a microcontroller. Then, he implemented a microcontroller-based random number generator (RNG) using these three different maps.

For RNG design, the outputs of the $x$ and $y$ state variables of the chaotic maps are first multiplied by 1000 to magnify the variables. Second, the output of these magnified variables is converted to a 32-bit single floating-point binary number, as in Figure 7 of [126]. After that, the first 24 bits (0, 1, 2, 3, ..., 23) of the 32 bits of this floating-point binary number are used for the preprocessing algorithm, which is shown schematically in Figures 8 and 9 of [126].

In the first case, the EXOR operation is performed mutually from the zeroth bit of the $x$ state variable of Lozi map and the twenty-third bit of the $x$ state variable of the Tinkerbell map ($0 \rightarrow 23, 1 \rightarrow 22, 2 \rightarrow 21, ...$). The result obtained here is applied to EXOR processing with the zeroth bit of $x$ from the Barnsley map. The other bits are applied to the same processing in order. In this way, the first 24-bit RNG output is obtained.

Then, a similar algorithm is applied for the second 24-bit RNG output. The subsequent 48-bit sequences are obtained from the chaotic map output values after every 95 iteration calculations. In this way, different random values are obtained simultaneously.

The implemented RNG design was run on the microcontroller and the obtained random numbers were tested for validity using the NIST SP800-22 test package.

Merah et al. [127] proposed a stream cipher based on chaotic synchronization of the Lozi map. Their technique consists of feeding back the ciphertext to the cryptosystem and incorporating it on the driver signal for synchronization.

The driver system (corresponding to (3)) is provided by

$$\begin{cases} x_{n+1} = 1 - a|x_n| + y_n, \\ y_{n+1} = bx_n, \end{cases} \qquad (88)$$

whereas the response system is provided by

$$\begin{cases} x'_{n+1} = 1 + y'_{n} - \alpha s(x_n), \\ y'_{n+1} = \beta x'_{n}, \end{cases} \qquad (89)$$

The driver signal $s(x_n) = |x_n|$ is used to drive the response system. Note that $e_1 = x_{n+1} - x'_{n+1}$ and $e_2 = y_{n+1} - y'_{n+1}$ are the synchronization errors. By assigning different initial conditions to the driver and response systems, $e_1$ and $e_2$ vanish after some number
For the Lozi map, the values of y are normalized in the range [0, 1] to each decision variable in the n-dimensional space of the optimization problem. This transformation is provided by

$$Z_k = \frac{y_k - \lambda_{min}}{\lambda_{max} - \lambda_{min}} \qquad (87)$$

where $y \in [-0.6418, 0.6716]$, $[\lambda_{max}, \lambda_{min}]$ equals [0.6418, -0.6716], $P = 1.7$, and $Q = 0.5$ (this is similar to the normalization used by Araujo and Coelho in (30)). For the logistic map, $v$ is defined with uniform randomness in the range of [3.65, 3.95].

Various experiments were carried out with initial conditions defined randomly in the range of [0, 0.5]. Extracting the right authentication code is only possible with the correct information about the initialization of both chaotic maps and neural network weights. It is impossible to extract this information from image pixels.

The proposed schemes completely destroy the relationship among pixels in the generation of authentication code. Experimental results show that this scheme has efficient in authentication and localization and does minimal damage to the original image; furthermore, it can be applied to both grayscale and color images.

### 6.2.3. Hardware-Implemented Cryptographic Algorithms

In addition to the previous algorithms, others have been directly implemented on hardware circuits, as we show in the three following examples.

In [126], Cicek implemented the Lozi, Tinkerbell, and Barnsley Fern two-dimensional discrete chaotic maps on a microcontroller. Then, he implemented a microcontroller-based random number generator (RNG) using these three different maps.

For RNG design, the outputs of the x and y state variables of the chaotic maps are first multiplied by 1000 to magnify the variables. Second, the output of these magnified variables is converted to a 32-bit single floating-point binary number, as in Figure 7 of [126]. After that, the first 24 bits (0, 1, 2, 3, ..., 23) of the 32 bits of this floating-point binary number are used for the preprocessing algorithm, which is shown schematically in Figures 8 and 9 of [126].

In the first case, the EXOR operation is performed mutually from the zeroth bit of the x state variable of Lozi map and the twenty-third bit of the x state variable of the Tinkerbell map ($0 \rightarrow 23, 1 \rightarrow 22, 2 \rightarrow 21, ...$). The result obtained here is applied to EXOR processing with the zeroth bit of x from the Barnsley map. The other bits are applied to the same processing in order. In this way, the first 24-bit RNG output is obtained.

Then, a similar algorithm is applied for the second 24-bit RNG output. The subsequent 48-bit sequences are obtained from the chaotic map output values after every 95 iteration calculations. In this way, different random values are obtained simultaneously.

The implemented RNG design was run on the microcontroller and the obtained random numbers were tested for validity using the NIST SP800-22 test package.

Merah et al. [127] proposed a stream cipher based on chaotic synchronization of the Lozi map. Their technique consists of feeding back the ciphertext to the cryptosystem and incorporating it on the driver signal for synchronization.

The driver system (corresponding to (3)) is provided by

$$\begin{cases} x_{n+1} = 1 - a|x_n| + y_n, \\ y_{n+1} = bx_n, \end{cases} \qquad (88)$$

whereas the response system is provided by

$$\begin{cases} x'_{n+1} = 1 + y'_{n} - \alpha s(x_n), \\ y'_{n+1} = \beta x'_{n}, \end{cases} \qquad (89)$$

The driver signal $s(x_n) = |x_n|$ is used to drive the response system. Note that $e_1 = x_{n+1} - x'_{n+1}$ and $e_2 = y_{n+1} - y'_{n+1}$ are the synchronization errors. By assigning different initial conditions to the driver and response systems, $e_1$ and $e_2$ vanish after some number
of iterations. Therefore, the coupled systems synchronize and the difference between the outputs of the master and slave systems tends to zero.

The authors took the driver signal $|x_n|$, which has 64 bits of length, then divided it into two parts, the first containing the lower bits $m$ and the second containing the upper bits $d$, where $d + m = 64$ bits. For encryption purposes, the $m$ lower bits are XORed with those of the plaintext, the resulting ciphertext (which has $m$ bits of length) is concatenated with the $d$ upper bits of $|x_n|$, and a new sample is made which is 64 bits in length ($d$ upper bits of the previous and $m$ lower bits of the ciphertext). The new $|x_n|$ is then fed back to chaotic system and sent to the receiver at the same time for synchronization and to bear the ciphertext.

On the receiver side, the response system synchronizes with the driver one, then the plaintext is recovered by XORing the $m$ lower bits of the driver signal with $m$ lower bits of the signal generated from the response of the chaotic system.

Simulation results showed that the proposed stream cipher has good confusion/diffusion properties and provides a strong key.

Xiao et al. [128] developed a time-frequency domain encryption technique based on multi-chaotics for physical layer security and a chaotic map (CM) for peak-to-average-power ratio (PAPR) reduction. They experimentally demonstrated its feasibility in an Orthogonal Frequency Division Multiplexing (OFDM)-Passive Optical Network (PON) system. The proposed scheme is based on the Lozi and logistic maps and can generate chaotic sequences to scramble the subcarrier in both the time and frequency domains to enhance physical layer security. In the experiment, an 8.9 GB/s encrypted OFDM signal was securely transmitted over 100 km of standard single-mode fiber.

The Lozi map was used to generate phase rotation sequences for frequency domain encryption and the logistic map to generate index sequences for time-domain interleaving. The combination of these two maps can enhance the key space and the security of an OFDM-PON system.

In frequency domain encryption, the chaotic sequences are generated by a Lozi map (88) with $a = 1.7$, $b = 0.5$. To generate phase rotation sequences for the frequency domain, $\{x_i\}$ and $\{y_i\}$ are postprocessed to be $\{Rx_i\}$ and $\{Iy_i\}$, as described by

$$ \begin{cases} R_{x_i} = \text{sign}(\text{mod}(\text{fix}(x_i \times 10^{14}) \cdot 256) - 128), \\ I_{y_i} = \text{sign}(\text{mod}(\text{fix}(y_i \times 10^{14}) \cdot 256) - 128), \end{cases} \quad (90) $$

where the function fix(x) returns the nearest integer rounded to zero. Then, the following phase rotation sequence $\{p_i\}$ is obtained:

$$ p_i = R_{x_i} + jI_{y_i}, \quad 1 \le i \le N. \quad (91) $$

Here, $N$ is the number of OFDM subcarriers. After constellation mapping, the input sequence becomes a matrix of $X = [X_0, X_1, \dots, X_{n-1}]^T$, where $X_k$ represents the corresponding subcarrier data. In a transmitter, the input sequence $X$ is expanded to $K$ identical sequences. According to the different initial conditions of the Lozi map, different phase rotation sequences $\{p_n\}$ are obtained. After calculating the PAPR values of the encrypted OFDM signal, the minimum PAPR signal is transmitted, then chaotic time domain interleaving is performed. Additionally, the logistic map is used to generate a permuted index sequence.

The experimental results showed that the proposed method can effectively prevent illegal users from eavesdropping. Moreover, the proposed scheme has low complexity, low cost, and is easy to implement.

## 7. Networks

During the past decades there has been growing interest in examining networks within the recent theory of complexity, which makes it possible to model complex systems encountered in real-life situations. In this section, we examine networks of chaotic attractors.
of iterations. Therefore, the coupled systems synchronize and the difference between the outputs of the master and slave systems tends to zero.

The authors took the driver signal $|x_n|$, which has 64 bits of length, then divided it into two parts, the first containing the lower bits $m$ and the second containing the upper bits $d$, where $d + m = 64$ bits. For encryption purposes, the $m$ lower bits are XORed with those of the plaintext, the resulting ciphertext (which has $m$ bits of length) is concatenated with the $d$ upper bits of $|x_n|$, and a new sample is made which is 64 bits in length ($d$ upper bits of the previous and $m$ lower bits of the ciphertext). The new $|x_n|$ is then fed back to chaotic system and sent to the receiver at the same time for synchronization and to bear the ciphertext.

On the receiver side, the response system synchronizes with the driver one, then the plaintext is recovered by XORing the $m$ lower bits of the driver signal with $m$ lower bits of the signal generated from the response of the chaotic system.

Simulation results showed that the proposed stream cipher has good confusion/diffusion properties and provides a strong key.

Xiao et al. [128] developed a time-frequency domain encryption technique based on multi-chaotics for physical layer security and a chaotic map (CM) for peak-to-average-power ratio (PAPR) reduction. They experimentally demonstrated its feasibility in an Orthogonal Frequency Division Multiplexing (OFDM)-Passive Optical Network (PON) system. The proposed scheme is based on the Lozi and logistic maps and can generate chaotic sequences to scramble the subcarrier in both the time and frequency domains to enhance physical layer security. In the experiment, an 8.9 GB/s encrypted OFDM signal was securely transmitted over 100 km of standard single-mode fiber.

The Lozi map was used to generate phase rotation sequences for frequency domain encryption and the logistic map to generate index sequences for time-domain interleaving. The combination of these two maps can enhance the key space and the security of an OFDM-PON system.

In frequency domain encryption, the chaotic sequences are generated by a Lozi map (88) with $a = 1.7$, $b = 0.5$. To generate phase rotation sequences for the frequency domain, $\{x_i\}$ and $\{y_i\}$ are postprocessed to be $\{Rx_i\}$ and $\{Iy_i\}$, as described by

$$ \begin{cases} R_{x_i} = \text{sign}(\text{mod}(\text{fix}(x_i \times 10^{14}) \cdot 256) - 128), \\ I_{y_i} = \text{sign}(\text{mod}(\text{fix}(y_i \times 10^{14}) \cdot 256) - 128), \end{cases} \quad (90) $$

where the function fix(x) returns the nearest integer rounded to zero. Then, the following phase rotation sequence $\{p_i\}$ is obtained:

$$ p_i = R_{x_i} + jI_{y_i}, \quad 1 \le i \le N. \quad (91) $$

Here, $N$ is the number of OFDM subcarriers. After constellation mapping, the input sequence becomes a matrix of $X = [X_0, X_1, \dots, X_{n-1}]^T$, where $X_k$ represents the corresponding subcarrier data. In a transmitter, the input sequence $X$ is expanded to $K$ identical sequences. According to the different initial conditions of the Lozi map, different phase rotation sequences $\{p_n\}$ are obtained. After calculating the PAPR values of the encrypted OFDM signal, the minimum PAPR signal is transmitted, then chaotic time domain interleaving is performed. Additionally, the logistic map is used to generate a permuted index sequence.

The experimental results showed that the proposed method can effectively prevent illegal users from eavesdropping. Moreover, the proposed scheme has low complexity, low cost, and is easy to implement.

## 7. Networks

During the past decades there has been growing interest in examining networks within the recent theory of complexity, which makes it possible to model complex systems encountered in real-life situations. In this section, we examine networks of chaotic attractors.
## 7.1. Synchronization

Based on the definition of chaos, two trajectories of a chaotic system starting from two close but different initial conditions become exponentially separated in time because their evolution strongly depends on the initial conditions. Pecora and Carroll [129] demonstrated that coupled (even different) chaotic systems with different initial values can be synchronized; subsequently, there has been increasing interest in the study of chaos synchronization and its applications in various fields, including physics, biology, chemistry, mathematics, and engineering.

Several different methods are used to synchronize chaotic attractors. Synchronization can involve either two attractors or several in complex networks, as we discuss in the cases of chimera, solitary states, and cellular neural networks.

### 7.1.1. Observer-Based Synchronization

In addition to the observers described in Section 5.1, Zhang et al. [130] proposed a switching observer-based synchronization scheme for piecewise linear discrete chaotic systems such as skew tent maps and Lozi maps based on piecewise quadratic Lyapunov functions. In this scheme, the observer switches according to the output of the drive system.

For piecewise linear chaotic systems of the form

$$ \begin{cases} x(t+1) = A_l x(t) + \alpha_l & \text{for } x(t) \in R_l, \\ y(t) = C x(t), \end{cases} \quad (92) $$

the observer is defined as follows:

$$ \begin{cases} \hat{x}(t+1) = A_l \hat{x}(t) + \alpha_l + K_l(y(t) - \hat{y}(t)) & \text{for } x(t) \in R_l, \\ \hat{y}(t) = C \hat{x}(t), \end{cases} \quad (93) $$

where $\hat{x}(t) \in \mathbb{R}^n$ is the observer state, $\{R_l\}_{l \in L} \subseteq \mathbb{R}^n$ denotes a partition of the state space of the chaotic system into a number of closed polyhedral subspaces, $y(t) \in \mathbb{R}^m$ and $\hat{y}(t) \in \mathbb{R}^m$ are the outputs of the chaotic system and observer, respectively, and $K_l \in \mathbb{R}^{n \times m}$ is a switching gain matrix that is yet to be determined.

For synchronization, $y(t) = Cx(t)$ is the synchronization signal to be sent to the receiver (observer) and the observer switches the dynamics according to the state of chaotic systems. However, it is well known that not all of the state variables are measurable in practice. Thus, Zhang et al. divided the space of the output $y(t)$ into several polyhedral cells $\{S_l\}, l \in L \subseteq \mathbb{R}^m$ to meet the following two requirements:

(1) The number of cells of the space of the output $y(t)$ is exactly equal to that of the state of the chaotic drive system.

(2) The observer is guaranteed to acquire the information about which cell in the state of chaotic system is from the synchronization signal $y(t)$; in other words, the output $y(t)$ is in cell $S_l$ if and only if $x(t)$ is in cell $R_l$.

To prove the synchronization property, Zhang et al. defined the synchronization error as $e(t) = x(t) - \hat{x}(t)$. Combining (92) and (93), they obtained to the following error dynamics:

$$ e(t+1) = (A_l - K_l C)e(t) = F_l e(t) \quad \text{for } y \in S_l. \quad (94) $$

Then, they obtained the following switching feedback stabilization result for the synchronization error dynamics (94).

**Theorem 3** ([130]). The synchronization error dynamics (94) are asymptotically stable with the switching feedback gain $K_l, l \in L$ if there exists a set of positive matrices $P_l, l \in L$ such that the following linear matrix inequalities are satisfied:

$$ (A_l - K_l C)^T P_l (A_l - K_l C) - P_l < 0, \quad l \in L \quad (95) $$
## 7.1. Synchronization

Based on the definition of chaos, two trajectories of a chaotic system starting from two close but different initial conditions become exponentially separated in time because their evolution strongly depends on the initial conditions. Pecora and Carroll [129] demonstrated that coupled (even different) chaotic systems with different initial values can be synchronized; subsequently, there has been increasing interest in the study of chaos synchronization and its applications in various fields, including physics, biology, chemistry, mathematics, and engineering.

Several different methods are used to synchronize chaotic attractors. Synchronization can involve either two attractors or several in complex networks, as we discuss in the cases of chimera, solitary states, and cellular neural networks.

### 7.1.1. Observer-Based Synchronization

In addition to the observers described in Section 5.1, Zhang et al. [130] proposed a switching observer-based synchronization scheme for piecewise linear discrete chaotic systems such as skew tent maps and Lozi maps based on piecewise quadratic Lyapunov functions. In this scheme, the observer switches according to the output of the drive system.

For piecewise linear chaotic systems of the form

$$ \begin{cases} x(t+1) = A_l x(t) + \alpha_l & \text{for } x(t) \in R_l, \\ y(t) = C x(t), \end{cases} \qquad (92) $$

the observer is defined as follows:

$$ \begin{cases} \hat{x}(t+1) = A_l \hat{x}(t) + \alpha_l + K_l(y(t) - \hat{y}(t)) & \text{for } x(t) \in R_l, \\ \hat{y}(t) = C \hat{x}(t), \end{cases} \qquad (93) $$

where $\hat{x}(t) \in \mathbb{R}^n$ is the observer state, $\{R_l\}_{l \in L} \subseteq \mathbb{R}^n$ denotes a partition of the state space of the chaotic system into a number of closed polyhedral subspaces, $y(t) \in \mathbb{R}^m$ and $\hat{y}(t) \in \mathbb{R}^m$ are the outputs of the chaotic system and observer, respectively, and $K_l \in \mathbb{R}^{n \times m}$ is a switching gain matrix that is yet to be determined.

For synchronization, $y(t) = Cx(t)$ is the synchronization signal to be sent to the receiver (observer) and the observer switches the dynamics according to the state of chaotic systems. However, it is well known that not all of the state variables are measurable in practice. Thus, Zhang et al. divided the space of the output $y(t)$ into several polyhedral cells $\{S_l\}, l \in L \subseteq \mathbb{R}^m$ to meet the following two requirements:

(1) The number of cells of the space of the output $y(t)$ is exactly equal to that of the state of the chaotic drive system.

(2) The observer is guaranteed to acquire the information about which cell in the state of chaotic system is from the synchronization signal $y(t)$; in other words, the output $y(t)$ is in cell $S_l$ if and only if $x(t)$ is in cell $R_l$.

To prove the synchronization property, Zhang et al. defined the synchronization error as $e(t) = x(t) - \hat{x}(t)$. Combining (92) and (93), they obtained to the following error dynamics:

$$ e(t+1) = (A_l - K_l C)e(t) = F_l e(t) \quad \text{for } y \in S_l. \qquad (94) $$

Then, they obtained the following switching feedback stabilization result for the synchronization error dynamics (94).

**Theorem 3** ([130]). The synchronization error dynamics (94) are asymptotically stable with the switching feedback gain $K_l, l \in L$ if there exists a set of positive matrices $P_l, l \in L$ such that the following linear matrix inequalities are satisfied:

$$ (A_l - K_l C)^T P_l (A_l - K_l C) - P_l < 0, \quad l \in L \qquad (95) $$
$$ (A_j - K_j C)^T P_j (A_j - K_j C) - P_j < 0, \quad l, j \in \Omega \cap L \qquad (96) $$

where $\Omega$ is the set that represents all possible transitions from one region to another, that is,

$$ \{\Omega = l, j | y(t) \in S_j, y(t-1) \in S_l, j \neq l\}. \qquad (97) $$

They used the values $a = 1.7$ and $b = 0.5$ with initial conditions $x_1 = -0.2, x_2 = 0.2$ for the Lozi map to illustrate the synchronization of two maps.

### 7.1.2. Dead-Beat Synchronization

“Dead-beat” synchronization, first introduced by De Angeli et al. [131], corresponds to the situation in which the error between the drive and response systems converges to zero in finite steps.

Grassi [132] studied the propagation of projective synchronization in a series connection of N chaotic discrete-time drive systems and N response systems. By exploiting an observer-based approach, he demonstrated that dead-beat projective synchronization (i.e., exact synchronization in finite time for any scaling factor) is achieved between the nth drive and nth response systems. In particular, he showed that projective synchronization starts from the innermost (Nth) drive–response system pair and propagates toward the outermost (first) drive–response system pair. He illustrated the propagation of different types of chaos synchronization in a series connection consisting of three chaotic attractors: the Gingerbread man map, the Lozi map and the generalized Hénon map.

The Gingerbread man map [133] (a conservative version of the Lozi map) is defined as

$$ \begin{bmatrix} x_{1,1}(k+1) \\ x_{1,2}(k+1) \end{bmatrix} = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} x_{1,1}(k) \\ x_{1,2}(k) \end{bmatrix} + \begin{bmatrix} 1 \\ 0 \end{bmatrix} |x_{1,1}(k)| + \begin{bmatrix} 1 \\ 0 \end{bmatrix} + \begin{bmatrix} 0 \\ 1 \end{bmatrix} \epsilon_1 \varphi_1(k). \qquad (98) $$

The Hénon map is generalized as follows:

$$ \begin{bmatrix} x_{2,1}(k+1) \\ x_{2,2}(k+1) \\ x_{2,3}(k+1) \end{bmatrix} = \begin{bmatrix} 0 & 0 & -0.1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{bmatrix} \begin{bmatrix} x_{2,1}(k) \\ x_{2,2}(k) \\ x_{2,3}(k) \end{bmatrix} + \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} (x_{2,2}(k))^2 + \begin{bmatrix} 1.76 \\ 0 \\ 0 \end{bmatrix} + \begin{bmatrix} -1 \\ 0 \\ 1 \end{bmatrix} \epsilon_1 \varphi_2(k). \qquad (99) $$

and the Lozi map for $a = 1.4$ and $b = 0.3$ is

$$ \begin{bmatrix} x_{3,1}(k+1) \\ x_{3,2}(k+1) \end{bmatrix} = \begin{bmatrix} 0 & 1 \\ 0.3 & 0 \end{bmatrix} \begin{bmatrix} x_{3,1}(k) \\ x_{3,2}(k) \end{bmatrix} + \begin{bmatrix} -1.4 \\ 0 \end{bmatrix} |x_{3,1}(k)| + \begin{bmatrix} 1 \\ 0 \end{bmatrix} + \begin{bmatrix} -1.4 \\ 0 \end{bmatrix} \epsilon_3 \varphi_3(k), \qquad (100) $$

where $\varphi_1$ is an arbitrary perturbation

$$ \varphi_2(k) = |x_{1,1}(k)| + [0 \quad -1][x_{1,1}(k) \quad x_{1,2}(k)]^T + \epsilon_1 \varphi_1(k) $$

and

$$ \varphi_3(k) = (x_{2,2}(k))^2 + [0 \quad 0 \quad 1][x_{2,1}(k) \quad x_{2,2}(k) \quad x_{2,3}(k)]^T + \epsilon_2 \varphi_2(k). $$

Synchronization was obtained after few iterations.

In [134], Lian et al. analyzed various scalar coupling signals with several classical chaotic attractors (Logistic, Hénon, Lozi, etc.) to show that the connected systems achieve dead-beat synchronization.

### 7.1.3. Impulsive Synchronization

In the case of two coupled attractors, Zheng, Nian, and Liu [135] presented a theorem on impulsive synchronization using synchronization impulses with varying impulse intervals. As an example and application of the theorem, they derived sufficient conditions for the synchronization of two chaotic Lozi maps via impulse control.
$$ (A_j - K_j C)^T P_j (A_j - K_j C) - P_j < 0, \quad l, j \in \Omega \cap L \quad (96) $$

where $\Omega$ is the set that represents all possible transitions from one region to another, that is,

$$ \{\Omega = l, j | y(t) \in S_j, y(t-1) \in S_l, j \neq l\}. \quad (97) $$

They used the values $a = 1.7$ and $b = 0.5$ with initial conditions $x_1 = -0.2, x_2 = 0.2$ for the Lozi map to illustrate the synchronization of two maps.

### 7.1.2. Dead-Beat Synchronization

“Dead-beat” synchronization, first introduced by De Angeli et al. [131], corresponds to the situation in which the error between the drive and response systems converges to zero in finite steps.

Grassi [132] studied the propagation of projective synchronization in a series connection of N chaotic discrete-time drive systems and N response systems. By exploiting an observer-based approach, he demonstrated that dead-beat projective synchronization (i.e., exact synchronization in finite time for any scaling factor) is achieved between the nth drive and nth response systems. In particular, he showed that projective synchronization starts from the innermost (Nth) drive–response system pair and propagates toward the outermost (first) drive–response system pair. He illustrated the propagation of different types of chaos synchronization in a series connection consisting of three chaotic attractors: the Gingerbread man map, the Lozi map and the generalized Hénon map.

The Gingerbread man map [133] (a conservative version of the Lozi map) is defined as

$$ \begin{bmatrix} x_{1,1}(k+1) \\ x_{1,2}(k+1) \end{bmatrix} = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} x_{1,1}(k) \\ x_{1,2}(k) \end{bmatrix} + \begin{bmatrix} 1 \\ 0 \end{bmatrix} |x_{1,1}(k)| + \begin{bmatrix} 1 \\ 0 \end{bmatrix} + \begin{bmatrix} 0 \\ 1 \end{bmatrix} \epsilon_1 \varphi_1(k). \quad (98) $$

The Hénon map is generalized as follows:

$$ \begin{bmatrix} x_{2,1}(k+1) \\ x_{2,2}(k+1) \\ x_{2,3}(k+1) \end{bmatrix} = \begin{bmatrix} 0 & 0 & -0.1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{bmatrix} \begin{bmatrix} x_{2,1}(k) \\ x_{2,2}(k) \\ x_{2,3}(k) \end{bmatrix} + \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} (x_{2,2}(k))^2 + \begin{bmatrix} 1.76 \\ 0 \\ 0 \end{bmatrix} + \begin{bmatrix} -1 \\ 0 \\ 1 \end{bmatrix} \epsilon_1 \varphi_2(k). \quad (99) $$

and the Lozi map for $a = 1.4$ and $b = 0.3$ is

$$ \begin{bmatrix} x_{3,1}(k+1) \\ x_{3,2}(k+1) \end{bmatrix} = \begin{bmatrix} 0 & 1 \\ 0.3 & 0 \end{bmatrix} \begin{bmatrix} x_{3,1}(k) \\ x_{3,2}(k) \end{bmatrix} + \begin{bmatrix} -1.4 \\ 0 \end{bmatrix} |x_{3,1}(k)| + \begin{bmatrix} 1 \\ 0 \end{bmatrix} + \begin{bmatrix} -1.4 \\ 0 \end{bmatrix} \epsilon_3 \varphi_3(k), \quad (100) $$

where $\varphi_1$ is an arbitrary perturbation

$$ \varphi_2(k) = |x_{1,1}(k)| + [0 \quad -1][x_{1,1}(k) \quad x_{1,2}(k)]^T + \epsilon_1 \varphi_1(k) $$

and

$$ \varphi_3(k) = (x_{2,2}(k))^2 + [0 \quad 0 \quad 1][x_{2,1}(k) \quad x_{2,2}(k) \quad x_{2,3}(k)]^T + \epsilon_2 \varphi_2(k). $$

Synchronization was obtained after few iterations.

In [134], Lian et al. analyzed various scalar coupling signals with several classical chaotic attractors (Logistic, Hénon, Lozi, etc.) to show that the connected systems achieve dead-beat synchronization.

### 7.1.3. Impulsive Synchronization

In the case of two coupled attractors, Zheng, Nian, and Liu [135] presented a theorem on impulsive synchronization using synchronization impulses with varying impulse intervals. As an example and application of the theorem, they derived sufficient conditions for the synchronization of two chaotic Lozi maps via impulse control.
Letting $w_n^T = (x_n, y_n)$, they rewrote (88) in the following matrix form:

$$w_{n+1} = A\hat{w}_n + \Phi(w_n), \quad (101)$$

where

$$A = \begin{pmatrix} 0 & b \\ 1 & 0 \end{pmatrix}, \quad \Phi(w_n) = \begin{pmatrix} 1 - a|x_n| \\ 0 \end{pmatrix}.$$

In the impulsive synchronization configuration, the master system is provided by (101) and the slave system is provided by the following impulsive equation:

$$\left\{ \begin{aligned} \dot{w}_{n+1} &= A\hat{w}_n + \Phi(\hat{w}_n), & n \neq n_i, \\ \Delta\hat{w}_{n_i} &= \hat{w}_{n_i^+} - \hat{w}_{n_i^-} = -Be_{n_i^-} = -Be_{n_i}, & n = n_i, & i = 1, 2, \dots \end{aligned} \right. \quad (102)$$

where $B$ is a $2 \times 2$ synchronization matrix and $e_n = w_n - \hat{w}_n = (x_n - \hat{x}_n, y_n - \hat{y}_n)$ is the synchronization error.

They proved the following theorem to guarantee that the impulsive synchronization of the Lozi maps is asymptotically stable.

**Theorem 4** ([135]). Let $d$ be the largest eigenvalue of $(I + B^T)(I + B)$, where $B$ is a $2 \times 2$ synchronization matrix and $I$ denotes the $2 \times 2$ identity matrix. We assume that the impulse are equidistant from each other and are separated by an integer $\Delta$ if there exists an $r > 1$ such that

$$0 \le \ln(1+a) \le -\frac{1}{2\Delta}\lnEI(rd); \quad (103)$$

Then, the impulsive synchronization of the two Lozi maps provided in (102) is asymptotically stable.

In addition, they carried out a number of numerical simulations to illustrate this theorem.

Zhang and Jiang [136] generalized the impulsive synchronization method and illustrated it using Hénon and Lozi maps.

### 7.1.4. Fuzzy Synchronization

Another kind of synchronization involves using a fuzzy model-based adaptive approach to synchronize two different discrete-time chaotic systems. In [137], Vasegh and Majd considered a Takagi-Sugeno (TS) fuzzy model (see Section 4.1.1) employed to represent a chaotic drive and response system. As the parameters of the drive system are assumed to be unknown, an adaptive law was derived to estimate its unknown parameters. They proposed a control law to stabilize the error dynamics. This design procedure was illustrated using the two examples of the Lozi and Hénon maps to demonstrate the effectiveness of the proposed methodology.

They defined the Lozi fuzzy model as

$$y(k+1) = -A|y(k)| + By(k-1) + C + Du(k) \quad (104)$$

and chose two membership functions

$$\mu_1 y(k) = \frac{\mathrm{sign}(y(k))}{d}, \quad \mu_2 y(k) = 1 - \frac{\mathrm{sign}(y(k))}{d}, \quad (105)$$

where $d$ is a large enough constant to cover the range of the output. This model can be expressed as follows:

* Rule 1: If $y(k)$ is $\mu_1$, then $y(k+1) = a_{11}y(k) + a_{12}y(k-1) + b_1u(k) + c_1\xi(k)$

* Rule 2: If $y(k)$ is $\mu_2$, then $y(k+1) = a_{21}y(k) + a_{22}y(k-1) + b_2u(k) + c_2\xi(k)$

with the following parameters: $a_{11} = -Ad, a_{21} = 0, a_{12} = a_{22} = B, b_1 = b_2 = D, c_1 = c_2 = C$, and $\xi(k) = 1$.
Letting $w_n^T = (x_n, y_n)$, they rewrote (88) in the following matrix form:

$$w_{n+1} = A\hat{w}_n + \Phi(w_n), \qquad (101)$$

where

$$A = \begin{pmatrix} 0 & b \\ 1 & 0 \end{pmatrix}, \quad \Phi(w_n) = \begin{pmatrix} 1 - a|x_n| \\ 0 \end{pmatrix}.$$

In the impulsive synchronization configuration, the master system is provided by (101) and the slave system is provided by the following impulsive equation:

$$\left\{ \begin{aligned} \dot{w}_{n+1} &= A\hat{w}_n + \Phi(\hat{w}_n), & n \neq n_i, \\ \Delta\hat{w}_{n_i} &= \hat{w}_{n_i^+} - \hat{w}_{n_i^-} = -Be_{n_i^-} = -Be_{n_i}, & n = n_i, & i = 1, 2, \dots \end{aligned} \right. \qquad (102)$$

where B is a 2 × 2 synchronization matrix and $e_n = w_n - \hat{w}_n = (x_n - \hat{x}_n, y_n - \hat{y}_n)$ is the synchronization error.

They proved the following theorem to guarantee that the impulsive synchronization of the Lozi maps is asymptotically stable.

**Theorem 4** ([135]). Let d be the largest eigenvalue of $(I + B^T)(I + B)$, where B is a 2 × 2 synchronization matrix and I denotes the 2 × 2 identity matrix. We assume that the impulse are equidistant from each other and are separated by an integer $\Delta$ if there exists an $r > 1$ such that

$$0 \le \ln(1+a) \le -\frac{1}{2\Delta}\lnEI; \qquad (103)$$

Then, the impulsive synchronization of the two Lozi maps provided in (102) is asymptotically stable.

In addition, they carried out a number of numerical simulations to illustrate this theorem.

Zhang and Jiang [136] generalized the impulsive synchronization method and illustrated it using Hénon and Lozi maps.

### 7.1.4. Fuzzy Synchronization

Another kind of synchronization involves using a fuzzy model-based adaptive approach to synchronize two different discrete-time chaotic systems. In [137], Vasegh and Majd considered a Takagi-Sugeno (TS) fuzzy model (see Section 4.1.1) employed to represent a chaotic drive and response system. As the parameters of the drive system are assumed to be unknown, an adaptive law was derived to estimate its unknown parameters. They proposed a control law to stabilize the error dynamics. This design procedure was illustrated using the two examples of the Lozi and Hénon maps to demonstrate the effectiveness of the proposed methodology.

They defined the Lozi fuzzy model as

$$y(k + 1) = -A|y(k)| + By(k - 1) + C + Du(k) \qquad (104)$$

and chose two membership functions

$$\mu_1 y(k) = \frac{\mathrm{sign}(y(k))}{d}, \quad \mu_2 y(k) = 1 - \frac{\mathrm{sign}(y(k))}{d}, \qquad (105)$$

where d is a large enough constant to cover the range of the output. This model can be expressed as follows:

* Rule 1: If $y(k)$ is $\mu_1$, then $y(k + 1) = a_{11}y(k) + a_{12}y(k - 1) + b_1u(k) + c_1\xi(k)$

* Rule 2: If $y(k)$ is $\mu_2$, then $y(k + 1) = a_{21}y(k) + a_{22}y(k - 1) + b_2u(k) + c_2\xi(k)$

with the following parameters: $a_{11} = -Ad$, $a_{21} = 0$, $a_{12} = a_{22} = B$, $b_1 = b_2 = D$, $c_1 = c_2 = C$, and $\xi(k) = 1$.
For two different chaotic systems where one is used as the drive system and the other as the response system, the task of synchronization is to force the trajectory of the response system to become identical to that of the drive system, where the response system receives the driving signals from the drive system. Vasegh and Majd looked into this problem for uncertain chaotic systems and proved several theorems. As an illustrative example, they considered the Hénon fuzzy map as a drive system with the numerical values

$$y(k + 1) = -y(k)^2 + 0.3y(k - 1) + 1.4, \quad (106)$$

which was modeled using fuzzy rules similar to the above rules used for the Lozi map, with the membership functions

$$\mu_1 y(k) = 0.5 \left( 1 - \frac{y(k)}{d} \right), \quad \mu_2 y(k) = 0.5 \left( 1 + \frac{y(k)}{d} \right), \quad (107)$$

and parameters $a_{11} = -a_{12} = Ad$, $a_{12} = a_{22} = B$, $b_1 = b_2 = D$, $c_1 = c_2 = C$, and $\xi(k) = 1$.

The Lozi fuzzy map was used as the response system with the following fixed parameter values:

$$x(k + 1) = -1.8|x(k)| + 0.4x(k - 1) + 1. \quad (108)$$

The simulation results showed the effectiveness of the proposed method.

### 7.1.5. Chaotic Synchronization

Dmitriev et al. [138] considered a synchronization scheme for two-dimensional maps that is based on transmission of information about the state of drive system by unidirectional coupling.

In the proposed scheme (see Figure 3 of [138]), the signal $x_n$ produced by iteration of the Lozi map is transformed by a two-level quantizer (threshold device) into a symbolic sequence $X_n = sign(x_n)$, i.e., into -1 if $x_n < 0$ and into +1 if $x_n \ge 0$. Then, this binary signal is sent through the channel, where it is corrupted by noise ($z_n = X_n + w_n$). The quantizer transforms it back to the binary signal $\hat{X}_n = sign(z_n)$. The ±1 values of the binary sequence correspond to the symbols L and R of the symbolic sequence. This symbolic sequence is used in the reconstruction block to restore the chaotic sequence. Thus, in the absence of noise, the chaotic sequence reconstructed with a high precision is fed to the input of the response system (or an estimate of this chaotic sequence if there is noise).

The reconstruction algorithm provided by Dmitriev et al. is defined as follows.

For a given symbolic sequence $X_1, \dots, X_n, \dots, X_{n+k}, \dots, X_N$, where $X_k \in \{-1, 1\}$: N is the number of elements in the sequence, it is possible to evaluate $|x_n|$ at the first step for all elements other than the first and the last (i.e., for $n = 2, \dots, N-1$) in the following way:

$$x_n^1 = X_n |x_n^1| = X_n \left| \frac{\alpha - 1}{\alpha} - \frac{1}{\alpha} X_{n+1} + \frac{\beta}{\alpha} X_{n-1} \right|. \quad (109)$$

The elements of the symbolic sequence are used at this stage as an initial approximation for the estimate. At the ith step of the procedure, the estimates obtained at the previous (i - 1)th step are used:

$$x_n^i = X_n |x_n^i| = X_n \left| \frac{\alpha - 1}{\alpha} - \frac{1}{\alpha} x_{n+1}^{i-1} + \frac{\beta}{\alpha} x_{n-1}^{i-1} \right|, \quad (110)$$

where $n = 2, \dots, N-1$. The process is complete when the difference between the estimates of $x_i$ in the ith and i + 1th iterations is less than some small $\epsilon$.

It has been shown that in order to achieve synchronization of both hyperbolic and non-hyperbolic maps it is sufficient to transmit information contained in the chaotic sequence and then utilize this information to reconstruct the chaotic sequence. The precision of reconstruction depends on the length of the sequence and the number of elements
For two different chaotic systems where one is used as the drive system and the other as the response system, the task of synchronization is to force the trajectory of the response system to become identical to that of the drive system, where the response system receives the driving signals from the drive system. Vasegh and Majd looked into this problem for uncertain chaotic systems and proved several theorems. As an illustrative example, they considered the Hénon fuzzy map as a drive system with the numerical values

$$y(k + 1) = -y(k)^2 + 0.3y(k - 1) + 1.4, \quad (106)$$

which was modeled using fuzzy rules similar to the above rules used for the Lozi map, with the membership functions

$$\mu_1 y(k) = 0.5 \left( 1 - \frac{y(k)}{d} \right), \quad \mu_2 y(k) = 0.5 \left( 1 + \frac{y(k)}{d} \right), \quad (107)$$

and parameters $a_{11} = -a_{12} = Ad$, $a_{12} = a_{22} = B$, $b_1 = b_2 = D$, $c_1 = c_2 = C$, and $\xi(k) = 1$.

The Lozi fuzzy map was used as the response system with the following fixed parameter values:

$$x(k + 1) = -1.8|x(k)| + 0.4x(k - 1) + 1. \quad (108)$$

The simulation results showed the effectiveness of the proposed method.

### 7.1.5. Chaotic Synchronization

Dmitriev et al. [138] considered a synchronization scheme for two-dimensional maps that is based on transmission of information about the state of drive system by unidirectional coupling.

In the proposed scheme (see Figure 3 of [138]), the signal $x_n$ produced by iteration of the Lozi map is transformed by a two-level quantizer (threshold device) into a symbolic sequence $X_n = sign(x_n)$, i.e., into -1 if $x_n < 0$ and into +1 if $x_n \ge 0$. Then, this binary signal is sent through the channel, where it is corrupted by noise ($z_n = X_n + w_n$). The quantizer transforms it back to the binary signal $\hat{X}_n = sign(z_n)$. The ±1 values of the binary sequence correspond to the symbols L and R of the symbolic sequence. This symbolic sequence is used in the reconstruction block to restore the chaotic sequence. Thus, in the absence of noise, the chaotic sequence reconstructed with a high precision is fed to the input of the response system (or an estimate of this chaotic sequence if there is noise).

The reconstruction algorithm provided by Dmitriev et al. is defined as follows.

For a given symbolic sequence $X_1, \dots, X_n, \dots, X_{n+k}, \dots, X_N$, where $X_k \in \{-1, 1\}$: N is the number of elements in the sequence, it is possible to evaluate $|x_n|$ at the first step for all elements other than the first and the last (i.e., for $n = 2, \dots, N-1$) in the following way:

$$x_n^1 = X_n |x_n^1| = X_n \left| \frac{\alpha - 1}{\alpha} - \frac{1}{\alpha} X_{n+1} + \frac{\beta}{\alpha} X_{n-1} \right|. \quad (109)$$

The elements of the symbolic sequence are used at this stage as an initial approximation for the estimate. At the ith step of the procedure, the estimates obtained at the previous (i-1)th step are used:

$$x_n^i = X_n |x_n^i| = X_n \left| \frac{\alpha - 1}{\alpha} - \frac{1}{\alpha} x_{n+1}^{i-1} + \frac{\beta}{\alpha} x_{n-1}^{i-1} \right|, \quad (110)$$

where $n = 2, \dots, N-1$. The process is complete when the difference between the estimates of $x_i$ in the ith and i+1th iterations is less than some small $\epsilon$.

It has been shown that in order to achieve synchronization of both hyperbolic and non-hyperbolic maps it is sufficient to transmit information contained in the chaotic sequence and then utilize this information to reconstruct the chaotic sequence. The precision of reconstruction depends on the length of the sequence and the number of elements
of the symbolic sequence that are directly used to reconstruct the chaotic sample. The reconstructed sequence of samples ensures high-quality synchronization of drive and response systems.

It has been shown that the necessary amount of information that must be delivered to the response system is determined by the rate of information production by the dynamical system. When this threshold is increased, however, arbitrarily precise synchronization can be achieved. The only price to pay for this is increased complexity of the reconstruction algorithm and delay of the reconstructed signal. This is contrary to the common belief that the synchronization error necessarily monotonically increases with the noise.

Another method based on evolutionary optimization, called the Tribes optimization approach, was used by Coelho and Bernert [139] for synchronization of two identical discrete chaotic systems subject to different initial conditions. The Tribes algorithm, first introduced by Clerc [140], is an adaptive and robust parameter searching technique. Moreover, Tribes is a parameter-free particle swarm system paradigm in which the population is divided in subpopulations that each maintaining their own order and structure. The Tribes algorithm is inspired by the social behavior of bird flocking, and is an adaptive optimization procedure that does not require sociometric or swarm size parameter tuning.

Coelho and Bernert evaluated the classical Tribes, Mtribes (a modified Tribes optimization algorithm based on truncated chaotic Zaslavskii map), and PSO approaches (see Section 4.1.2) to tune PID controller gains in a synchronization application of two Lozi systems with different initial conditions.

The principles of Tribes are [139]:

(i) The swarm is divided in tribes, and each particle belongs to a single tribe

(ii) Initially, the swarm is composed of only one particle

(iii) Particles are added or removed according to tribes' behaviors

(iv) Displacement strategies are adapted according to the performance of the particles

"Good" tribes may benefit from removal of their weakest member, as they already possess good problem solutions and may be able to afford to reduce their population; "bad" tribes, on the other hand, may benefit from the addition of a new member, as this increases the possibility of improvement. New particles are randomly generated.

Mtribes uses a truncated chaotic Zaslavskii map

$$ \begin{cases} w_n = \text{mod}[w_{n-1} + v + az_n] & (1), \\ z_n = \cos(2\pi w_{n-1}) + e^{-r}z_{n-1}, & \end{cases} \quad (111) $$

where $n$ is the sample and $\text{mod}$ is the modulus after division by 1. The Zaslavskii map shows a strange attractor with the largest Lyapunov exponent for $v = 400$, $r = 3$, and $a = 12.6695$. In this case, the values of $z_n \in [-1.0512, 1.0512]$. The choice of the chaotic Zaslavskii map in the Mtribes design is justified theoretically by its unpredictability, i.e., by its spread-spectrum characteristic and large Lyapunov exponent.

Simulation results based on synchronization of two Lozi systems (with $a = 1.7$ and $b = 0.5$) with different initial value conditions have demonstrated the effectiveness and efficiency of the proposed Mtribes method.

### 7.1.6. Synchronization of Complex Fractional Lozi Maps

Joshi et al. [42] studied the synchronization of complex fractional-ordered Lozi maps (see Section 2.5.3). They defined a system of two symmetrically coupled maps

$$ \left\{ \begin{aligned} f(n) &= 1 - a|x(n)| + bx(n-1), \\ g(n) &= 1 - a|y(n)| + by(n-1), \\ x(n) &= x(0) + \frac{1}{\Gamma(v)} \sum_{j=1}^{n} \frac{\Gamma(n-j+v)}{\Gamma(n-j+1)} [(1-\epsilon)f(j-1) + \epsilon g(j-1) - x(j-1)], \\ y(n) &= y(0) + \frac{1}{\Gamma(v)} \sum_{j=1}^{n} \frac{\Gamma(n-j+v)}{\Gamma(n-j+1)} [(1-\epsilon)g(j-1) + \epsilon f(j-1) - y(j-1)]. \end{aligned} \right. \quad (112) $$

For this, they found numerically that there is a significant range over which synchronized chaos is observed with positive coupling for $r = 0.01$, where $v = v_0 e^{ir\pi/2}$ with
of the symbolic sequence that are directly used to reconstruct the chaotic sample. The reconstructed sequence of samples ensures high-quality synchronization of drive and response systems.

It has been shown that the necessary amount of information that must be delivered to the response system is determined by the rate of information production by the dynamical system. When this threshold is increased, however, arbitrarily precise synchronization can be achieved. The only price to pay for this is increased complexity of the reconstruction algorithm and delay of the reconstructed signal. This is contrary to the common belief that the synchronization error necessarily monotonically increases with the noise.

Another method based on evolutionary optimization, called the Tribes optimization approach, was used by Coelho and Bernert [139] for synchronization of two identical discrete chaotic systems subject to different initial conditions. The Tribes algorithm, first introduced by Clerc [140], is an adaptive and robust parameter searching technique. Moreover, Tribes is a parameter-free particle swarm system paradigm in which the population is divided in subpopulations that each maintaining their own order and structure. The Tribes algorithm is inspired by the social behavior of bird flocking, and is an adaptive optimization procedure that does not require sociometric or swarm size parameter tuning.

Coelho and Bernert evaluated the classical Tribes, Mtribes (a modified Tribes optimization algorithm based on truncated chaotic Zaslavskii map), and PSO approaches (see Section 4.1.2) to tune PID controller gains in a synchronization application of two Lozi systems with different initial conditions.

The principles of Tribes are [139]:

(i) The swarm is divided in tribes, and each particle belongs to a single tribe

(ii) Initially, the swarm is composed of only one particle

(iii) Particles are added or removed according to tribes' behaviors

(iv) Displacement strategies are adapted according to the performance of the particles

"Good" tribes may benefit from removal of their weakest member, as they already possess good problem solutions and may be able to afford to reduce their population; "bad" tribes, on the other hand, may benefit from the addition of a new member, as this increases the possibility of improvement. New particles are randomly generated.

Mtribes uses a truncated chaotic Zaslavskii map

$$ \begin{cases} w_n = \text{mod}[w_{n-1} + v + az_n] & (1), \\ z_n = \cos(2\pi w_{n-1}) + e^{-r}z_{n-1}, & \end{cases} \quad (111) $$

where $n$ is the sample and $\text{mod}$ is the modulus after division by 1. The Zaslavskii map shows a strange attractor with the largest Lyapunov exponent for $v = 400$, $r = 3$, and $a = 12.6695$. In this case, the values of $z_n \in [-1.0512, 1.0512]$. The choice of the chaotic Zaslavskii map in the Mtribes design is justified theoretically by its unpredictability, i.e., by its spread-spectrum characteristic and large Lyapunov exponent.

Simulation results based on synchronization of two Lozi systems (with $a = 1.7$ and $b = 0.5$) with different initial value conditions have demonstrated the effectiveness and efficiency of the proposed Mtribes method.

### 7.1.6. Synchronization of Complex Fractional Lozi Maps

Joshi et al. [42] studied the synchronization of complex fractional-ordered Lozi maps (see Section 2.5.3). They defined a system of two symmetrically coupled maps

$$ \begin{cases} f(n) = 1 - a|x(n)| + bx(n-1), \\ g(n) = 1 - a|y(n)| + by(n-1), \\ x(n) = x(0) + \frac{1}{\Gamma(v)} \sum_{j=1}^{n} \frac{\Gamma(n-j+v)}{\Gamma(n-j+1)} [(1-\epsilon)f(j-1) + \epsilon g(j-1) - x(j-1)], \\ y(n) = y(0) + \frac{1}{\Gamma(v)} \sum_{j=1}^{n} \frac{\Gamma(n-j+v)}{\Gamma(n-j+1)} [(1-\epsilon)g(j-1) + \epsilon f(j-1) - y(j-1)]. \end{cases} \quad (112) $$

For this, they found numerically that there is a significant range over which synchronized chaos is observed with positive coupling for $r = 0.01$, where $v = v_0 e^{ir\pi/2}$ with
$0 < v_0 < 1$ (see Section 2.5.3). In addition, they found a significant range over which $Re(x(t) - y(t)) \to 0$ (with the imaginary part going to zero as well). They demonstrated synchronization in a detailed manner for $a = 1.4$ and noted that synchronization is observed for stronger coupling. Even for weaker couplings, certain correlations are seen. They expressed $Re(y(t))$ as a function of $Re(x(t))$ after discarding transients for $\epsilon = 0.05$ and $\epsilon = 0.02$.

Furthermore, they observed many other relationships between the variables $x$ and $y$ which we do not report due to the limited extent of this survey.

### 7.1.7. Synchronization of Globally Coupled Maps

Roy and Acharyya [141] considered a network of Lozi maps periodically synchronized by the periodic variation of one of the parameters. Interestingly, it was observed that, due to the variation of the parameter with time, the transition from a desynchronized state to a synchronized state occurs at lower values of the global coupling strength.

The system transition to the synchronized state in globally coupled Lozi maps under parametric excitation can provide a better way to achieve the synchronized state in the case of various natural and well known examples of synchronizing systems.

The global coupling in the Lozi maps (3)

$$ \begin{cases} x_{t+1} = 1 - a(t)|x_t| + y_t \equiv f(x_t, y_t), \\ y_{t+1} = bx_t, \end{cases} \qquad (113) $$

is introduced by the term $\epsilon h_t$, and is described by the equation

$$ \begin{cases} x_{t+1}^i = (1 - \epsilon)f(x_t^i, y_t^i) + \epsilon h_t \\ y_{t+1}^i = bx_t^i, \end{cases} \qquad (114) $$

where

$$ h_t = \frac{1}{N} \sum_{j=1}^{N} f(x_t^j, y_t^j). \qquad (115) $$

Here, the parameter $\epsilon$ represents the strength of the global coupling of the maps, $x_t^i$ and $y_t^i$ are the values of the state variables at the $i^{th}$ lattice point $i = (1, 2, \dots, N)$ at discrete time $t$, and the parameter $\epsilon$ represents the strength of the global coupling of the maps.

Synchronization in this system of equations at iterate $n$ arises when $(x_n^i, y_n^i) = (x_n^j, y_n^j), \forall (i, j)$. Note that synchronization of the $x$ variable implies synchronization of the $y$ variable.

Note that in a previous work Cano and Cosenza studied the behavior of a large number ($N = 1000$) of globally coupled Lozi maps when the parameter $a$ of the Lozi maps remains constant over time [142]. They observed four desynchronized states, chimeras (see Section 7.2), clusters, and synchronized states in increasing order of the coupling strength $\epsilon$. Although the parameters $a$ and $b$ of the map were chosen such that the individual mapping was in the chaotic range, depending on the strength of the global coupling parameter $\epsilon$ the behavior of the equilibrium state will be one of four states.

Roy and Acharyya, instead, were only interested in observing how the synchronization of globally coupled Lozi maps changes when time dependence of the parameter is introduced.

Time dependence of the parameter $a(t)$ is introduced by considering periodically kicked $a(t)$ of time period $T$, i.e., $a(t)$ varies with time as follows:

$a(t) = a_2$ always except at times, $t = T, 2T, 3T, 4T, \dots$, etc. when $a$ takes the value $a_2$. Therefore, the periodically kicked $a(t)$ of time period ($T$) follows:

* $a(t) = a_2$ at every $t = nT$

* $a(t) = a_1$ otherwise

where $n \in \mathbb{Z}^+$, $T \in \mathbb{Z}^+$.
$0 < v_0 < 1$ (see Section 2.5.3). In addition, they found a significant range over which $Re(x(t) - y(t)) \to 0$ (with the imaginary part going to zero as well). They demonstrated synchronization in a detailed manner for $a = 1.4$ and noted that synchronization is observed for stronger coupling. Even for weaker couplings, certain correlations are seen. They expressed $Re(y(t))$ as a function of $Re(x(t))$ after discarding transients for $\epsilon = 0.05$ and $\epsilon = 0.02$.

Furthermore, they observed many other relationships between the variables $x$ and $y$ which we do not report due to the limited extent of this survey.

### 7.1.7. Synchronization of Globally Coupled Maps

Roy and Acharyya [141] considered a network of Lozi maps periodically synchronized by the periodic variation of one of the parameters. Interestingly, it was observed that, due to the variation of the parameter with time, the transition from a desynchronized state to a synchronized state occurs at lower values of the global coupling strength.

The system transition to the synchronized state in globally coupled Lozi maps under parametric excitation can provide a better way to achieve the synchronized state in the case of various natural and well known examples of synchronizing systems.

The global coupling in the Lozi maps (3)

$$ \begin{cases} x_{t+1} = 1 - a(t)|x_t| + y_t \equiv f(x_t, y_t), \\ y_{t+1} = bx_t, \end{cases} \qquad (113) $$

is introduced by the term $\epsilon h_t$, and is described by the equation

$$ \begin{cases} x_{t+1}^i = (1 - \epsilon)f(x_t^i, y_t^i) + \epsilon h_t \\ y_{t+1}^i = bx_t^i, \end{cases} \qquad (114) $$

where

$$ h_t = \frac{1}{N} \sum_{j=1}^{N} f(x_t^j, y_t^j). \qquad (115) $$

Here, the parameter $\epsilon$ represents the strength of the global coupling of the maps, $x_t^i$ and $y_t^i$ are the values of the state variables at the $i^{th}$ lattice point $i = (1, 2, \dots, N)$ at discrete time $t$, and the parameter $\epsilon$ represents the strength of the global coupling of the maps.

Synchronization in this system of equations at iterate $n$ arises when $(x_n^i, y_n^i) = (x_n^j, y_n^j), \forall (i, j)$. Note that synchronization of the $x$ variable implies synchronization of the $y$ variable.

Note that in a previous work Cano and Cosenza studied the behavior of a large number ($N = 1000$) of globally coupled Lozi maps when the parameter $a$ of the Lozi maps remains constant over time [142]. They observed four desynchronized states, chimeras (see Section 7.2), clusters, and synchronized states in increasing order of the coupling strength $\epsilon$. Although the parameters $a$ and $b$ of the map were chosen such that the individual mapping was in the chaotic range, depending on the strength of the global coupling parameter $\epsilon$ the behavior of the equilibrium state will be one of four states.

Roy and Acharyya, instead, were only interested in observing how the synchronization of globally coupled Lozi maps changes when time dependence of the parameter is introduced.

Time dependence of the parameter $a(t)$ is introduced by considering periodically kicked $a(t)$ of time period $T$, i.e., $a(t)$ varies with time as follows:

$a(t) = a_2$ always except at times, $t = T, 2T, 3T, 4T, \dots$, etc. when $a$ takes the value $a_2$. Therefore, the periodically kicked $a(t)$ of time period ($T$) follows:

* $a(t) = a_2$ at every $t = nT$

* $a(t) = a_1$ otherwise

where $n \in \mathbb{Z}^+$, $T \in \mathbb{Z}^+$.
The values of $a_1$ and $a_2$ were chosen such that the map was in the chaotic region for $a_1$ and in the fixed point region for $a_2$. Thus, the parameter $a$ of the Lozi map was forced to change its value from $a_1 = 1.4$ to $a_2 = 1.0$ after a regular interval of time.

They found that the transition from a desynchronized to a synchronized state changes as the value of $\epsilon$ crosses $\epsilon_c$ from below, and that this change occurs at lower values of coupling strength ($\epsilon_c$) compared to the case of time-independent $a$. They observed that the value of $\epsilon_c$ increases as the time period ($T$) of the periodically kicked $a(t)$ increases and approaches the value that corresponds to the value of constant $a_1 = 1.4$.

## 7.2. Chimeras

In addition to the general synchronization presented above, the Lozi map can be used to construct networks of chaotic attractors, either alone or with the Hénon map, which present unusual behaviors such as chimera and solitary states.

In networks of chaotic attractors, chimeras consist of the coexistence of one or more clusters and a subset of desynchronized elements. In 2020, Parastesh et al. wrote a survey on chimeras [143]. In this section, we include more recent results.

It is important to note that chimeras are not easy to find. In 2015 and 2016, Semenova et al. [144,145] studied the following network of Lozi maps:

$$ \left\{ \begin{array}{l} x_{n+1}^i = f(x_n^i, y_n^i) + \frac{\sigma}{2P} \sum_{j=i-p}^{j=i+P} [f(x_n^j, y_n^j) - f(x_n^i, y_n^i)], \quad i=1,2,\dots,N, \\ y_{n+1}^i = b x_n^i, \end{array} \right. \qquad (116) $$

with

$$ f(x_n, y_n) = 1 - a|x_n| + y_n, \qquad (117) $$

where $N$ is the number of elements in the ensemble of coupled equations. The nonlocal coupling is characterized by the coupling strength $\sigma$, number of neighbours $2P$ ($P$ neighbors on the either side of the $i$th element), and coupling range $r = P/N$.

They hypothesized that chimeric states can be obtained only in networks of chaotic non-hyperbolic systems, and are not found in networks of hyperbolic systems such as (116). This hypothesis was supported by their numerical simulations.

However, they revisited this hypothesis later and found it to be wrong due to insufficient previous simulations (see Section 7.3.1).

As mentioned above, in 2017 Cano and Cosenza [142] considered the autonomous system of globally coupled Lozi maps described by Equations (113) and (114), with $a(t)$ remaining constant $a$:

$$ \left\{ \begin{array}{l} x_{n+1}^i = (1 - \epsilon)f(x_n^i, y_n^i) + \epsilon h_n, \\ y_{n+1}^i = b x_n^i, \end{array} \right. \qquad (118) $$

with $f$ defined by (117) and

$$ h_n = \frac{1}{N} \sum_{j=1}^{N} f(x_n^i, y_n^i). \qquad (119) $$

In addition to synchronization, they observed the following collective states in this globally coupled system:

(i) **Clustering**: a dynamical cluster is defined as a subset of elements that are synchronized among themselves. In a clustered state, the elements in the system segregate into $K$ distinct subsets that evolve in time, i.e., $x_n^i = x_n^j = X_n^v, \forall (i, j)$ in the $v$th cluster with $v = 1, \dots, K$.

(ii) A chimeric state consists of the coexistence of one or more clusters and a subset of desynchronized elements.

(iii) A desynchronized or incoherent state occurs when $x_n^i \neq x_n^j, \forall (i, j)$.
The values of $a_1$ and $a_2$ were chosen such that the map was in the chaotic region for $a_1$ and in the fixed point region for $a_2$. Thus, the parameter $a$ of the Lozi map was forced to change its value from $a_1 = 1.4$ to $a_2 = 1.0$ after a regular interval of time.

They found that the transition from a desynchronized to a synchronized state changes as the value of $\epsilon$ crosses $\epsilon_c$ from below, and that this change occurs at lower values of coupling strength ($\epsilon_c$) compared to the case of time-independent $a$. They observed that the value of $\epsilon_c$ increases as the time period ($T$) of the periodically kicked $a(t)$ increases and approaches the value that corresponds to the value of constant $a_1 = 1.4$.

## 7.2. Chimeras

In addition to the general synchronization presented above, the Lozi map can be used to construct networks of chaotic attractors, either alone or with the Hénon map, which present unusual behaviors such as chimera and solitary states.

In networks of chaotic attractors, chimeras consist of the coexistence of one or more clusters and a subset of desynchronized elements. In 2020, Parastesh et al. wrote a survey on chimeras [143]. In this section, we include more recent results.

It is important to note that chimeras are not easy to find. In 2015 and 2016, Semenova et al. [144,145] studied the following network of Lozi maps:

$$ \left\{ \begin{array}{l} x_{n+1}^i = f(x_n^i, y_n^i) + \frac{\sigma}{2P} \sum_{j=i-p}^{j=i+P} [f(x_n^j, y_n^j) - f(x_n^i, y_n^i)], \quad i=1,2,\dots,N, \\ y_{n+1}^i = b x_n^i, \end{array} \right. \qquad (116) $$

with

$$ f(x_n, y_n) = 1 - a|x_n| + y_n, \qquad (117) $$

where $N$ is the number of elements in the ensemble of coupled equations. The nonlocal coupling is characterized by the coupling strength $\sigma$, number of neighbours $2P$ ($P$ neighbors on the either side of the $i$th element), and coupling range $r = P/N$.

They hypothesized that chimeric states can be obtained only in networks of chaotic non-hyperbolic systems, and are not found in networks of hyperbolic systems such as (116). This hypothesis was supported by their numerical simulations.

However, they revisited this hypothesis later and found it to be wrong due to insufficient previous simulations (see Section 7.3.1).

As mentioned above, in 2017 Cano and Cosenza [142] considered the autonomous system of globally coupled Lozi maps described by Equations (113) and (114), with $a(t)$ remaining constant $a$:

$$ \left\{ \begin{array}{l} x_{n+1}^i = (1 - \epsilon)f(x_n^i, y_n^i) + \epsilon h_n, \\ y_{n+1}^i = b x_n^i, \end{array} \right. \qquad (118) $$

with $f$ defined by (117) and

$$ h_n = \frac{1}{N} \sum_{j=1}^{N} f(x_n^i, y_n^i). \qquad (119) $$

In addition to synchronization, they observed the following collective states in this globally coupled system:

(i) **Clustering**: a dynamical cluster is defined as a subset of elements that are synchronized among themselves. In a clustered state, the elements in the system segregate into $K$ distinct subsets that evolve in time, i.e., $x_n^i = x_n^j = X_n^v, \forall (i, j)$ in the $v$th cluster with $v = 1, \dots, K$.

(ii) A chimeric state consists of the coexistence of one or more clusters and a subset of desynchronized elements.

(iii) A desynchronized or incoherent state occurs when $x_n^i \neq x_n^j, \forall (i, j)$.
They considered the system of nonlocally coupled Lozi maps described by

$$ \left\{ \begin{array}{l} x_{n+1}^i = (1 - \epsilon)f(x_n^i, y_n^i) + \epsilon h_n^i, \\ y_{n+1}^i = b x_n^i, \end{array} \right. \qquad (120) $$

with

$$ h_n^i = \frac{1}{2k} \sum_{j=i-k}^{j=i+k} [f(x_n^i, y_n^i) - f(x_n^j, y_n^j)], \qquad (121) $$

where the elements $i = 1, \dots, N$ are located on a ring with periodic boundary conditions, $\epsilon$ is the coupling parameter, $k$ is the number of neighbors coupled on either side of site $i$, and $h_n^i$ is the local field acting on element $i$.

The presence of chimeric states in globally coupled networks of identical oscillators at first seemed counterintuitive because of the perfect symmetry of such a system. However, such networks are among the simplest extended systems that can exhibit chimeric behavior. Cano and Cosenza identified that the presence of global interactions can indeed allow for the emergence of chimera states in networks of coupled elements possessing chaotic hyperbolic attractors (such as Lozi maps) in cases where such states do not form with local interactions.

Both chimeras and clusters can be interpreted as manifestations of the multistability of the resulting drive-response dynamics at the local level in systems with global interactions. The above-mentioned results suggest that chimeric states, as with other collective behaviors, arise from the interplay between the local dynamics and the network topology, and that either ingredient can prevent or induce its occurrence.

Other example of chimeric states have been exhibited by Anishchenko et al. [146], who numerically explored the dynamics of two coupled one-dimensional ensembles, namely, an ensemble of Hénon maps and an ensemble of Lozi maps. Both networks were considered under conditions of nonlocal coupling. The ensemble of Lozi maps was characterized by a hyperbolic attractor of the individual elements, while the ensemble of Hénon maps was characterized by a non-hyperbolic attractor. They revealed the features involved in realizing chimeric states in the coupled system, which were caused by the mutual influence of two ensembles with fundamentally different dynamics without coupling (see Section 7.3.2 for a presentation of these results).

## 7.3. Solitary States

In addition to chimeric states, solitary states exemplify another important type of partial synchronization patterns. They are defined as network states for which a single element or several elements behave differently compared with the majority of units, which can either demonstrate coherent dynamics or be fully synchronized.

### 7.3.1. Existence of Solitary States

Semenova et al. [147], revisiting their previous work, studied (116).

They showed that the ensemble of nonlocally coupled Lozi maps demonstrates the solitary state for specific values of the coupling parameters. Coupling changes the properties of partial elements and leads to the bistability, though the Lozi map does not have this property in its uncoupled form. The emergence of solitary states is accompanied by a second attracting set arising for the ensemble element.

Using a ring network of nonlocally coupled Lozi maps as an example, Rybalova and Strelkova [148] explored the robustness of solitary states against heterogeneity in the parameters of the local dynamics or coupling strength. They found that if these network parameters are continuously modulated by noise, then solitary states are suppressed as the noise intensity increases. However, these states may persist in the case of static randomly distributed system parameters for a wide range of the distribution width. Domains of solitary state existence were constructed in the parameter plane of coupling strength and noise intensity using a cross-correlation coefficient.
They considered the system of nonlocally coupled Lozi maps described by

$$ \begin{cases} x_{n+1}^i = (1 - \epsilon)f(x_n^i, y_n^i) + \epsilon h_n^i, \\ y_{n+1}^i = bx_n^i, \end{cases} \qquad (120) $$

with

$$ h_n^i = \frac{1}{2k} \sum_{j=i-k}^{j=i+k} [f(x_n^i, y_n^i) - f(x_n^j, y_n^j)], \qquad (121) $$

where the elements $i = 1, \dots, N$ are located on a ring with periodic boundary conditions, $\epsilon$ is the coupling parameter, $k$ is the number of neighbors coupled on either side of site $i$, and $h_n^i$ is the local field acting on element $i$.

The presence of chimeric states in globally coupled networks of identical oscillators at first seemed counterintuitive because of the perfect symmetry of such a system. However, such networks are among the simplest extended systems that can exhibit chimeric behavior. Cano and Cosenza identified that the presence of global interactions can indeed allow for the emergence of chimera states in networks of coupled elements possessing chaotic hyperbolic attractors (such as Lozi maps) in cases where such states do not form with local interactions.

Both chimeras and clusters can be interpreted as manifestations of the multistability of the resulting drive-response dynamics at the local level in systems with global interactions. The above-mentioned results suggest that chimeric states, as with other collective behaviors, arise from the interplay between the local dynamics and the network topology, and that either ingredient can prevent or induce its occurrence.

Other example of chimeric states have been exhibited by Anishchenko et al. [146], who numerically explored the dynamics of two coupled one-dimensional ensembles, namely, an ensemble of Hénon maps and an ensemble of Lozi maps. Both networks were considered under conditions of nonlocal coupling. The ensemble of Lozi maps was characterized by a hyperbolic attractor of the individual elements, while the ensemble of Hénon maps was characterized by a non-hyperbolic attractor. They revealed the features involved in realizing chimeric states in the coupled system, which were caused by the mutual influence of two ensembles with fundamentally different dynamics without coupling (see Section 7.3.2 for a presentation of these results).

## 7.3. Solitary States

In addition to chimeric states, solitary states exemplify another important type of partial synchronization patterns. They are defined as network states for which a single element or several elements behave differently compared with the majority of units, which can either demonstrate coherent dynamics or be fully synchronized.

### 7.3.1. Existence of Solitary States

Semenova et al. [147], revisiting their previous work, studied (116).

They showed that the ensemble of nonlocally coupled Lozi maps demonstrates the solitary state for specific values of the coupling parameters. Coupling changes the properties of partial elements and leads to the bistability, though the Lozi map does not have this property in its uncoupled form. The emergence of solitary states is accompanied by a second attracting set arising for the ensemble element.

Using a ring network of nonlocally coupled Lozi maps as an example, Rybalova and Strelkova [148] explored the robustness of solitary states against heterogeneity in the parameters of the local dynamics or coupling strength. They found that if these network parameters are continuously modulated by noise, then solitary states are suppressed as the noise intensity increases. However, these states may persist in the case of static randomly distributed system parameters for a wide range of the distribution width. Domains of solitary state existence were constructed in the parameter plane of coupling strength and noise intensity using a cross-correlation coefficient.
They numerically studied the spatiotemporal dynamics of a one-dimensional ensemble (a ring network) of nonlocally coupled discrete-time systems. This network is described by the following system of equations, a generalization of (116):

$$ \left\{ \begin{array}{l} x_i^{n+1} = F(x_i^n, y_i^n) + \frac{\sigma_i^n}{2R} \sum_{j=i-P}^{j=i+P} [F(x_j^n, y_j^n) - F(x_i^n, y_i^n)], \\ y_i^{n+1} = G(x_i^n, y_i^n), \end{array} \right. \qquad (122) $$

where $x_i^n, y_i^n$ are dynamical variables, $i = 1, 2, \dots, N$ denotes the element number, $N = 1000$ is the total number of elements in the ensemble, and $n$ is the discrete time. The local dynamics of every element $i$ in the ring (122) are described by the two-dimensional Lozi map

$$ \left\{ \begin{array}{l} F(x_i^n, y_i^n) = 1 - \alpha_i^n |x_i^n| + y_i^n, \\ G(x_i^n, y_i^n) = \beta x_i^n, \end{array} \right. \qquad (123) $$

where $\alpha_i^n > 0$ and $\beta > 0$ are the control parameters. The Lozi map demonstrates a quasihyperbolic chaotic attractor which follows directly from a border-collision bifurcation developed from a stable four-periodic orbit [10,12] and exists within the parameter range $\alpha \in [1.3, 1.8]$. In their simulations, Rybalova and Strelkova fixed $\beta = 0.3$ for all the network elements.

The heterogeneities in the parameters $\alpha_i^n$ and $\sigma_i^n$ were realized using two independent noise sources with a uniform distribution and a standard normal (Gaussian) distribution, while noise-modulated parameters were determined by the following expressions:

$$ \alpha_i^n = \alpha_0 + D_\alpha \xi_i^n, \qquad (124) $$

$$ \sigma_i^n = \sigma_0 + D_\sigma \xi_i^n, \qquad (125) $$

where $\alpha_0 = 1.4$ is the mean value of the local dynamics parameter, $\sigma_0$ is the mean value of the coupling strength, which was varied in the numerical experiments, $\xi_i^n$ is a noise generator with a uniform distribution in the interval $[-1, 1]$, and $D$ is the noise intensity (i.e., the width of the interval within which random source values are distributed).

Calculations were performed for fifty different realizations of the initial conditions $x_i^0$ and $y_i^0$ randomly distributed in the interval $[-0.5, 0.5]$. For each set of initial conditions, different uniform noise realizations were used for parameter modulation or to realize an inhomogeneous distribution of the system parameters. This allowed satisfactory statistical data to be obtained.

They considered many different situations, including noise-modulated local dynamics parameters, noise-modulated coupling strengths, the case of randomly distributed local dynamics parameters, the case of randomly distributed coupling strengths, and localized inhomogeneity of the network parameters.

Their findings provide insights into the influence of noise-modulated parameters on the existence and robustness of solitary states, and show the possibilities of controlling them (either by suppression or inducement and persistence) by introducing parameter inhomogeneities.

### 7.3.2. Solitary State Chimeras

Solitary states can additionally be found in the unexpected form of solitary state chimeras (SSC).

Rybalova et al. [149] showed that this chimeric state can appear in a two-layer network of nonlocally coupled nonlinear chaotic maps. One of the coupled networks represents a ring of nonlocally coupled Lozi maps that individually shows a special spatiotemporal pattern, called the solitary state, when there is no coupling between the subnetworks. The authors analyzed the bifurcation mechanism of the solitary state appearance in the network of coupled Lozi maps, showing that this particular state is responsible for the formation of the solitary state chimera in the two-layer network.
They numerically studied the spatiotemporal dynamics of a one-dimensional ensemble (a ring network) of nonlocally coupled discrete-time systems. This network is described by the following system of equations, a generalization of (116):

$$ \left\{ \begin{array}{l} x_i^{n+1} = F(x_i^n, y_i^n) + \frac{\sigma_i^n}{2R} \sum_{j=i-P}^{j=i+P} [F(x_j^n, y_j^n) - F(x_i^n, y_i^n)], \\ y_i^{n+1} = G(x_i^n, y_i^n), \end{array} \right. \qquad (122) $$

where $x_i^n, y_i^n$ are dynamical variables, $i = 1, 2, \dots, N$ denotes the element number, $N = 1000$ is the total number of elements in the ensemble, and $n$ is the discrete time. The local dynamics of every element $i$ in the ring (122) are described by the two-dimensional Lozi map

$$ \left\{ \begin{array}{l} F(x_i^n, y_i^n) = 1 - \alpha_i^n |x_i^n| + y_i^n, \\ G(x_i^n, y_i^n) = \beta x_i^n, \end{array} \right. \qquad (123) $$

where $\alpha_i^n > 0$ and $\beta > 0$ are the control parameters. The Lozi map demonstrates a quasihyperbolic chaotic attractor which follows directly from a border-collision bifurcation developed from a stable four-periodic orbit [10,12] and exists within the parameter range $\alpha \in [1.3, 1.8]$. In their simulations, Rybalova and Strelkova fixed $\beta = 0.3$ for all the network elements.

The heterogeneities in the parameters $\alpha_i^n$ and $\sigma_i^n$ were realized using two independent noise sources with a uniform distribution and a standard normal (Gaussian) distribution, while noise-modulated parameters were determined by the following expressions:

$$ \alpha_i^n = \alpha_0 + D_\alpha \xi_i^n, \qquad (124) $$

$$ \sigma_i^n = \sigma_0 + D_\sigma \xi_i^n, \qquad (125) $$

where $\alpha_0 = 1.4$ is the mean value of the local dynamics parameter, $\sigma_0$ is the mean value of the coupling strength, which was varied in the numerical experiments, $\xi_i^n$ is a noise generator with a uniform distribution in the interval $[-1, 1]$, and $D$ is the noise intensity (i.e., the width of the interval within which random source values are distributed).

Calculations were performed for fifty different realizations of the initial conditions $x_i^0$ and $y_i^0$ randomly distributed in the interval $[-0.5, 0.5]$. For each set of initial conditions, different uniform noise realizations were used for parameter modulation or to realize an inhomogeneous distribution of the system parameters. This allowed satisfactory statistical data to be obtained.

They considered many different situations, including noise-modulated local dynamics parameters, noise-modulated coupling strengths, the case of randomly distributed local dynamics parameters, the case of randomly distributed coupling strengths, and localized inhomogeneity of the network parameters.

Their findings provide insights into the influence of noise-modulated parameters on the existence and robustness of solitary states, and show the possibilities of controlling them (either by suppression or inducement and persistence) by introducing parameter inhomogeneities.

### 7.3.2. Solitary State Chimeras

Solitary states can additionally be found in the unexpected form of solitary state chimeras (SSC).

Rybalova et al. [149] showed that this chimeric state can appear in a two-layer network of nonlocally coupled nonlinear chaotic maps. One of the coupled networks represents a ring of nonlocally coupled Lozi maps that individually shows a special spatiotemporal pattern, called the solitary state, when there is no coupling between the subnetworks. The authors analyzed the bifurcation mechanism of the solitary state appearance in the network of coupled Lozi maps, showing that this particular state is responsible for the formation of the solitary state chimera in the two-layer network.
They first considered the single-layer network (116) with $N = 1000$. Their numerical studies showed that the solitary state's emergence is related to the structural changes in the chaotic attractor of the individual elements in this ensemble. In turn, these changes are caused by the impact of the 2P nodes which the solitary state oscillator is coupled with. It can be argued that the solitary state regime is related to the formation of a new attractor in the phase space, which the network oscillators can reach (or not) depending on their initial conditions.

Their numerical simulations revealed the mechanism of solitary state appearance in a ring of nonlocally coupled Lozi maps (116). It was shown that the nonlocal coupling induces the bistability mode in the individual nodes, at which two chaotic attractors with different basins of attraction co-exist in the phase space of this network. The first attractor corresponds to the Lozi attractor in the individual element without coupling. The second attractor emerges only in the presence of nonlocal coupling; thus, it promotes the appearance of solitary states. Randomly chosen initial conditions provide the possibility of the initial states of a group of oscillators belonging to the basin of attraction of the second attractor; in this way, solitary state regimes can be realized.

Next, they considered a complex system consisting of three coupled rings of nonlocally coupled chaotic maps. This multilayer network is described by the following equations:

$$ 
\left\{ 
\begin{aligned}
x_{n+1}^i &= f(x_n^i, y_n^i) + \frac{\sigma_1}{2P} \sum_{j=i-P}^{j=i+P} [f(x_n^j, y_n^j) - f(x_n^i, y_n^i)] + \gamma_1 F_n^i, \\
y_{n+1}^i &= b x_n^i, \\
u_{n+1}^i &= f(u_n^i, v_n^i) + \frac{\sigma_2}{2R} \sum_{j=i-R}^{j=i+R} [f(u_n^j, v_n^j) - f(u_n^i, v_n^i)] + \gamma_2 G_n^i, \\
v_{n+1}^i &= b x_n^i, \\
z_{n+1}^i &= f(z_n^i, s_n^i) + \frac{\sigma_3}{2T} \sum_{j=i-T}^{j=i+T} [f(z_n^j, s_n^j) - f(z_n^i, s_n^i)] + \gamma_3 H_n^i, \\
s_{n+1}^i &= b x_n^i,
\end{aligned}
\right. 
\quad (126) $$

The first system of equations in (126) specifies a ring network of nonlocally coupled Hénon maps

$$ f(x_n, y_n) = 1 - ax_n^2 + y_n, \quad (127) $$

with $a = 1.4$, $b = 0.3$, $\sigma_1 = 0.72$, and $P = 320$. The second pair of equations corresponds to the ring of nonlocally coupled Lozi maps (with $f$ defined by (117)) and was analyzed for $a = 1.4$, $b = 0.3$, $\sigma_2 = 0.206$, and $R = 180$. The third pair of equations determines the ring of nonlocally coupled Hénon maps with $a = 1.4$, $b = 0.3$, $\sigma_1 = 0.295$, and $T = 320$.

The first two rings are coupled inertially via the coupling functions $F_n^i = -G_n^i = u_n^i - x_n^i$ with the coupling coefficients $\gamma_1$ and $\gamma_2$. The third ring node is connected unidirectionally to the first ring units by the coupling term $\gamma_3 H_n^i$, where

$$ H_n^i = f(x_n^i, y_n^i) - f(z_n^i, s_n^i), \quad (128) $$

defines the diffusive coupling with coupling coefficient $\gamma_3$. The coupling parameters $\sigma_{1,2,3}$ and $P, R, T$ have the same meaning as in the description of system (116). The initial conditions were chosen to be randomly distributed in the interval $[-0.5, 0]$ for all the variables of the network (126).

Using numerical simulations, the authors demonstrated that a network of two symmetrically coupled ensembles of Hénon and Lozi maps can show a novel type of chimeric state, namely, a solitary state chimera (SSC), when the coupling between them is weak. This special structure emerges in the case of a Lozi ensemble that exhibits a developed regime of solitary states. The SSC is fairly stable, and is observed within a finite range of parameter variation. If the two-layer network of nonlocally coupled Hénon and Lozi maps in the solitary state chimera is unidirectionally coupled to the third ring of nonlocally coupled Hénon maps, then the effect of external synchronization can be observed in a finite range of the coupling coefficient $\gamma_3$.
They first considered the single-layer network (116) with $N = 1000$. Their numerical studies showed that the solitary state's emergence is related to the structural changes in the chaotic attractor of the individual elements in this ensemble. In turn, these changes are caused by the impact of the 2P nodes which the solitary state oscillator is coupled with. It can be argued that the solitary state regime is related to the formation of a new attractor in the phase space, which the network oscillators can reach (or not) depending on their initial conditions.

Their numerical simulations revealed the mechanism of solitary state appearance in a ring of nonlocally coupled Lozi maps (116). It was shown that the nonlocal coupling induces the bistability mode in the individual nodes, at which two chaotic attractors with different basins of attraction co-exist in the phase space of this network. The first attractor corresponds to the Lozi attractor in the individual element without coupling. The second attractor emerges only in the presence of nonlocal coupling; thus, it promotes the appearance of solitary states. Randomly chosen initial conditions provide the possibility of the initial states of a group of oscillators belonging to the basin of attraction of the second attractor; in this way, solitary state regimes can be realized.

Next, they considered a complex system consisting of three coupled rings of nonlocally coupled chaotic maps. This multilayer network is described by the following equations:

$$ 
\left\{ 
\begin{aligned}
x_{n+1}^i &= f(x_n^i, y_n^i) + \frac{\sigma_1}{2P} \sum_{j=i-P}^{j=i+P} [f(x_n^j, y_n^j) - f(x_n^i, y_n^i)] + \gamma_1 F_n^i, \\
y_{n+1}^i &= b x_n^i, \\
u_{n+1}^i &= f(u_n^i, v_n^i) + \frac{\sigma_2}{2R} \sum_{j=i-R}^{j=i+R} [f(u_n^j, v_n^j) - f(u_n^i, v_n^i)] + \gamma_2 G_n^i, \\
v_{n+1}^i &= b x_n^i, \\
z_{n+1}^i &= f(z_n^i, s_n^i) + \frac{\sigma_3}{2T} \sum_{j=i-T}^{j=i+T} [f(z_n^j, s_n^j) - f(z_n^i, s_n^i)] + \gamma_3 H_n^i, \\
s_{n+1}^i &= b x_n^i,
\end{aligned}
\right. 
\quad (126) $$

The first system of equations in (126) specifies a ring network of nonlocally coupled Hénon maps

$$ f(x_n, y_n) = 1 - ax_n^2 + y_n, \quad (127) $$

with $a = 1.4$, $b = 0.3$, $\sigma_1 = 0.72$, and $P = 320$. The second pair of equations corresponds to the ring of nonlocally coupled Lozi maps (with $f$ defined by (117)) and was analyzed for $a = 1.4$, $b = 0.3$, $\sigma_2 = 0.206$, and $R = 180$. The third pair of equations determines the ring of nonlocally coupled Hénon maps with $a = 1.4$, $b = 0.3$, $\sigma_1 = 0.295$, and $T = 320$.

The first two rings are coupled inertially via the coupling functions $F_n^i = -G_n^i = u_n^i - x_n^i$ with the coupling coefficients $\gamma_1$ and $\gamma_2$. The third ring node is connected unidirectionally to the first ring units by the coupling term $\gamma_3 H_n^i$, where

$$ H_n^i = f(x_n^i, y_n^i) - f(z_n^i, s_n^i), \quad (128) $$

defines the diffusive coupling with coupling coefficient $\gamma_3$. The coupling parameters $\sigma_{1,2,3}$ and $P, R, T$ have the same meaning as in the description of system (116). The initial conditions were chosen to be randomly distributed in the interval $[-0.5, 0]$ for all the variables of the network (126).

Using numerical simulations, the authors demonstrated that a network of two symmetrically coupled ensembles of Hénon and Lozi maps can show a novel type of chimeric state, namely, a solitary state chimera (SSC), when the coupling between them is weak. This special structure emerges in the case of a Lozi ensemble that exhibits a developed regime of solitary states. The SSC is fairly stable, and is observed within a finite range of parameter variation. If the two-layer network of nonlocally coupled Hénon and Lozi maps in the solitary state chimera is unidirectionally coupled to the third ring of nonlocally coupled Hénon maps, then the effect of external synchronization can be observed in a finite range of the coupling coefficient $\gamma_3$.
These results were preceded by studies by Buckh et al. [150] (from the same group of researchers around Vadim Anishchenko at Saratov University) and by Anishchenko et al. [146]. These studies investigated a complex system with only two coupled rings of nonlocally coupled chaotic maps (one ring of coupled Hénon maps and one ring of coupled Lozi maps) presenting SSCs. These SSCs are characterized by the coexistence of an incoherence cluster with uncorrelated chaotic oscillations of the cluster elements and a coherence cluster with synchronous chaotic oscillations. Additionally, these authors constructed the region of synchronization for a selected spatiotemporal structure in the plane of coupling parameters of the network under study; their results provide corroboration of the fact that the effect of synchronization is realized in double-layer systems.

In [151], Rybalova et al. further studied the model of three coupled rings from [149] by permuting the rings of attractors. They studied a Hénon–Lozi–Hénon system and a Lozi–Hénon–Lozi system, concluding that complete synchronization does not occur in the Hénon–Lozi–Hénon network. Instead, complete synchronization takes place in the Lozi–Hénon–Lozi network for almost the whole range of the intralayer coupling $\sigma_2$, starting with its very small value when the elements in the relay layer are almost desynchronized.

### 7.3.3. Noise Induced and Controlled Chimera

In [152], Rybalova et al. studied (116) in more detail, establishing that the SSC regime can be induced by multiplicative noise

$$ \sigma = \sigma_0(1 + \sqrt{2D\xi^i}), \qquad (129) $$

introduced into the nonlocal coupling coefficient of this network, where $\xi^i$ is a random process distributed uniformly in the interval $[-1, 1]$ and $D$ is the noise intensity.

In this case, bistability emerges in the phase space of each individual oscillator. Because the initial conditions are set randomly for all the ensemble elements, the appearance of solitary states and SSCs becomes possible. However, their detailed studies showed that bistability can be caused by both the noise perturbation and by the signals from $P$ neighboring oscillators on each individual oscillator of the ensemble (the second term in (116)).

More importantly, they established that the structure of the basins of attraction essentially depends on how far the ensemble oscillator in the solitary state regime is located from the phase chimera. The basin of attraction of the attractor corresponding to the solitary state regime gradually narrows as the distance between this oscillator and the phase chimera grows. Moreover, when an amplitude chimera exists in the considered network, the ridling effect takes place, which increases the probability of switching the phase trajectories into the solitary state and the SSC. The fact that the basins of attraction for the SSC decrease with the distance from the SSC cluster explains the existence of a finite incoherence cluster in space.

Instead of multiplicative noise, Rybalova et al. [153] numerically studied the spatiotemporal dynamics of ring networks of coupled discrete-time systems in the presence of additive noise. They explored the robustness of chimeric states with respect to noise perturbations for two ensembles in which the individual elements are described by either logistic maps or Hénon maps in the chaotic regime. The influence of noise on the behavior of solitary states was investigated for a ring of nonlocally coupled Lozi maps

$$ \left\{ \begin{array}{l} x_{n+1}^i = f(x_n^i, y_n^i) + \frac{\sigma}{2P} \sum_{j=i-P}^{j=i+P} [f(x_n^j, y_n^j) - f(x_n^i, y_n^i)] + D\xi_n^i, \quad i=1,2,\dots,N, \\ y_{n+1}^i = bx_n^i, \end{array} \right. \qquad (130) $$

where $\xi_n^i$ is a random process distributed uniformly in the interval $[-1, 1]$ and $D$ is the noise intensity.

Numerical simulations were performed for a set of different noise realizations and random initial conditions to provide reliable statistical data. The type of dynamics of the considered networks was quantified using a cross-correlation coefficient. They found that
These results were preceded by studies by Buckh et al. [150] (from the same group of researchers around Vadim Anishchenko at Saratov University) and by Anishchenko et al. [146]. These studies investigated a complex system with only two coupled rings of nonlocally coupled chaotic maps (one ring of coupled Hénon maps and one ring of coupled Lozi maps) presenting SSCs. These SSCs are characterized by the coexistence of an incoherence cluster with uncorrelated chaotic oscillations of the cluster elements and a coherence cluster with synchronous chaotic oscillations. Additionally, these authors constructed the region of synchronization for a selected spatiotemporal structure in the plane of coupling parameters of the network under study; their results provide corroboration of the fact that the effect of synchronization is realized in double-layer systems.

In [151], Rybalova et al. further studied the model of three coupled rings from [149] by permuting the rings of attractors. They studied a Hénon–Lozi–Hénon system and a Lozi–Hénon–Lozi system, concluding that complete synchronization does not occur in the Hénon–Lozi–Hénon network. Instead, complete synchronization takes place in the Lozi–Hénon–Lozi network for almost the whole range of the intralayer coupling $\sigma_2$, starting with its very small value when the elements in the relay layer are almost desynchronized.

### 7.3.3. Noise Induced and Controlled Chimera

In [152], Rybalova et al. studied (116) in more detail, establishing that the SSC regime can be induced by multiplicative noise

$$ \sigma = \sigma_0(1 + \sqrt{2D\xi^i}), \qquad (129) $$

introduced into the nonlocal coupling coefficient of this network, where $\xi^i$ is a random process distributed uniformly in the interval $[-1, 1]$ and $D$ is the noise intensity.

In this case, bistability emerges in the phase space of each individual oscillator. Because the initial conditions are set randomly for all the ensemble elements, the appearance of solitary states and SSCs becomes possible. However, their detailed studies showed that bistability can be caused by both the noise perturbation and by the signals from P neighboring oscillators on each individual oscillator of the ensemble (the second term in (116)).

More importantly, they established that the structure of the basins of attraction essentially depends on how far the ensemble oscillator in the solitary state regime is located from the phase chimera. The basin of attraction of the attractor corresponding to the solitary state regime gradually narrows as the distance between this oscillator and the phase chimera grows. Moreover, when an amplitude chimera exists in the considered network, the ridling effect takes place, which increases the probability of switching the phase trajectories into the solitary state and the SSC. The fact that the basins of attraction for the SSC decrease with the distance from the SSC cluster explains the existence of a finite incoherence cluster in space.

Instead of multiplicative noise, Rybalova et al. [153] numerically studied the spatiotemporal dynamics of ring networks of coupled discrete-time systems in the presence of additive noise. They explored the robustness of chimeric states with respect to noise perturbations for two ensembles in which the individual elements are described by either logistic maps or Hénon maps in the chaotic regime. The influence of noise on the behavior of solitary states was investigated for a ring of nonlocally coupled Lozi maps

$$ \left\{ \begin{array}{l} x_{n+1}^i = f(x_n^i, y_n^i) + \frac{\sigma}{2P} \sum_{j=i-P}^{j=i+P} [f(x_n^j, y_n^j) - f(x_n^i, y_n^i)] + D\xi_n^i, \quad i=1,2,\dots,N, \\ y_{n+1}^i = bx_n^i, \end{array} \right. \qquad (130) $$

where $\xi_n^i$ is a random process distributed uniformly in the interval $[-1, 1]$ and $D$ is the noise intensity.

Numerical simulations were performed for a set of different noise realizations and random initial conditions to provide reliable statistical data. The type of dynamics of the considered networks was quantified using a cross-correlation coefficient. They found that
there is a finite and sufficiently wide region with respect to the coupling strength and the noise intensity in which the probability of observing chimeras and solitary states is high.

### 7.4. Cellular Neural Networks

In 1988, L.O. Chua introduced the concept of Cellular Neural Networks [154]. Similar to other neural networks, these are large-scale nonlinear analog circuits that process signals in real time.

Similar to cellular automata, they are made up of a massive aggregation of regularly-spaced circuit clones (called cells) which communicate directly which each other only through their nearest neighbors. Each cell consists of a linear capacitor, a nonlinear voltage-controlled current source, and a few resistive linear circuit elements.

Such neural networks are uniquely suited for high-speed parallel signal processing. Applications of cellular neural networks in areas such as image processing and pattern recognition have been demonstrated [155].

In [156], Chang and Juang considered the following generalized version of the Lozi map which arises in the study of the complexity of a set of bounded stable stationary solutions of one-dimensional cellular neural networks:

$$T(x, y) = (y, F(y) - bx), \qquad (131)$$

with

$$F(y) = \begin{pmatrix} a_1 y + a_0 - a_1 + c_1 & \text{if } y \ge 1, \\ a_0 y + c_1 & \text{if } |y| \le 1, \\ a_{-1} y + a_{-1} - a_0 + c_1 & \text{if } y \le -1. \end{pmatrix}. \qquad (132)$$

Here, $a_0 < 0$, $a1, a_1 > 1$, $b > 0$, and $c_1 \in \mathbb{R}$ is a biased term.

They first proved a theorem which states that a semi-conjugate condition for $T$ implies the existence of a Smale horseshoe. Second, the theorem was applied to show the spatial chaos of one-dimensional cellular neural networks.

In [157], they studied this map in more detail and provided results on its entropy in the context of cellular neural networks.

## 8. Miscellaneous

As a complement to the principal applications of the Lozi chaotic map, a few other miscellaneous applications can be mentioned.

### 8.1. The Use of the Lozi Map as a Benchmark

In certain cases, the Lozi map has been used as a benchmark to test numerical methods, for example, in [158], in which the goal of Skrynnkye et al. was to demonstrate the effectiveness of the cycle search averaged predictive control method on certain dynamical systems widely referred to in technical reference sources. Their article demonstrated the efficiency of the large-length cycle searching method based on the stabilization of unstable and a priori unknown periodic orbits of these systems using several well-known examples of discrete-time nonlinear systems. Research in this area has mainly focused on Lozi systems.

As an example, they computed two different cycles of period 1001 and one cycle of period 1111 for $a = 1.7$ and $b = 0.5$.

They suggested that the proposed algorithm with the phenomenological rule of the choice of control parameter makes it possible to hope for the possibility of finding a large number of giga- and even tera-cycles.

In [159], Shang et al. proposed a Multi-Dimensional Scaling (MDS) method based on the Complexity-Invariant Distance (CID) and Generalized Complexity-Invariant Distance (GCID) to analyze and classify complex time series such as traffic signals and financial stock indexes.
there is a finite and sufficiently wide region with respect to the coupling strength and the noise intensity in which the probability of observing chimeras and solitary states is high.

### 7.4. Cellular Neural Networks

In 1988, L.O. Chua introduced the concept of Cellular Neural Networks [154]. Similar to other neural networks, these are large-scale nonlinear analog circuits that process signals in real time.

Similar to cellular automata, they are made up of a massive aggregation of regularly-spaced circuit clones (called cells) which communicate directly which each other only through their nearest neighbors. Each cell consists of a linear capacitor, a nonlinear voltage-controlled current source, and a few resistive linear circuit elements.

Such neural networks are uniquely suited for high-speed parallel signal processing. Applications of cellular neural networks in areas such as image processing and pattern recognition have been demonstrated [155].

In [156], Chang and Juang considered the following generalized version of the Lozi map which arises in the study of the complexity of a set of bounded stable stationary solutions of one-dimensional cellular neural networks:

$$T(x, y) = (y, F(y) - bx), \qquad (131)$$

with

$$F(y) = \begin{pmatrix} a_1 y + a_0 - a_1 + c_1 & \text{if } y \ge 1, \\ a_0 y + c_1 & \text{if } |y| \le 1, \\ a_{-1} y + a_{-1} - a_0 + c_1 & \text{if } y \le -1. \end{pmatrix}. \qquad (132)$$

Here, $a_0 < 0$, $a1, a_1 > 1$, $b > 0$, and $c_1 \in \mathbb{R}$ is a biased term.

They first proved a theorem which states that a semi-conjugate condition for $T$ implies the existence of a Smale horseshoe. Second, the theorem was applied to show the spatial chaos of one-dimensional cellular neural networks.

In [157], they studied this map in more detail and provided results on its entropy in the context of cellular neural networks.

## 8. Miscellaneous

As a complement to the principal applications of the Lozi chaotic map, a few other miscellaneous applications can be mentioned.

### 8.1. The Use of the Lozi Map as a Benchmark

In certain cases, the Lozi map has been used as a benchmark to test numerical methods, for example, in [158], in which the goal of Skrynnkyte et al. was to demonstrate the effectiveness of the cycle search averaged predictive control method on certain dynamical systems widely referred to in technical reference sources. Their article demonstrated the efficiency of the large-length cycle searching method based on the stabilization of unstable and a priori unknown periodic orbits of these systems using several well-known examples of discrete-time nonlinear systems. Research in this area has mainly focused on Lozi systems.

As an example, they computed two different cycles of period 1001 and one cycle of period 1111 for $a = 1.7$ and $b = 0.5$.

They suggested that the proposed algorithm with the phenomenological rule of the choice of control parameter makes it possible to hope for the possibility of finding a large number of giga- and even tera-cycles.

In [159], Shang et al. proposed a Multi-Dimensional Scaling (MDS) method based on the Complexity-Invariant Distance (CID) and Generalized Complexity-Invariant Distance (GCID) to analyze and classify complex time series such as traffic signals and financial stock indexes.
Shang et al. adopted a new method for measuring dissimilarity called the CID along with an advanced version called the GCID. Both methods were applied in MDS for the classification of time series, including artificial and real-world time series. First, in order to test the efficiency of the MDS method based on CID and GCID, they used three types of artificial time series: the α map model, the 2D Hénon map model, and the Lozi map model (with $a = 0.95$ and $b = 0.12$ for the first pair of parameters and $a = 1.0$, $b = 0.14$ for the second), which they used to observe the spatial distributions created by the MDS with the CID and GCID and examine their plausibility by calculating the stress. Additionally, three more sets of artificial time series were created with longer data length to demonstrate that their method is not likely to be affected by data length.

Jacob et al. [160] proposed a novel measure of degree heterogeneity for unweighted and undirected complex networks which requires only the degree distribution of the network for its computation. They showed that the proposed measure can be applied to all types of network topology with ease and that it increases with the diversity of the node degrees in the network.

As a specific application, they showed that the proposed measure can be used to compare the heterogeneity of recurrence networks constructed from time series of several low-dimensional chaotic attractors, thereby providing a single index to compare the structural complexity of chaotic attractors.

They compared the Lorenz, Rössler, and Duffing continuous attractors and the Hénon, Lozi, and Cat maps.

Their results indicated that the Lorenz attractor is the most structurally complex among the continuous systems, while in the case of two-dimensional discrete systems the Lozi attractor is the most diverse in terms of probability density variations.

In [161], Amigo and Kennel showed that the topological permutation entropy rate ($h_{top}^*(f)$) for expansive maps equals the conventional topological entropy rate ($h_{top}(f)$) from symbolic dynamics. After providing rigorous theoretical proofs of this equality, they sought to check it numerically. This task was found to be quite difficult; they stated that "Estimation of topological entropies from naive numerical simulation of long orbits is notoriously difficult. Metric entropy by itself can be quite tricky and difficult, requiring very long data sets for increasing L (the number of the joints distributions of a finite-state stochastic process), but topological entropy is worse yet, because it weights each pattern equally. This means that patterns which are exceptionally infrequent on the natural measure of the attractor can still have a significant influence on the result. Attempting to estimate the same quantities using empirical occurrences of order patterns is even more difficult, requiring more data than would a good, low-alphabet generating partition for ordinary symbolic dynamics."

As such, they found that the Lozi map is the only known map that satisfies all of the following criteria: a continuous system in more than one dimension, with a natural chaotic attractor, and with a topological entropy that can be found by independent rigorous means, for which there exist parameter values with reasonably low entropies.

In particular, they observed that $a = 6/5$ and $b = -2/15$ yield a low-entropy chaotic attractor (roughly 0.3 bits/iteration) and that the topological entropy for these parameters can be bounded rigorously via computer-assisted analytical computations. After defining clever strategies to compute ($h_{top}^*(\mathcal{L}_{a,b})$) and ($h_{top}(\mathcal{L}_{a,b})$) for these values, their numerical simulations conformed to the proven theorems.

Meckler [162] proposed an algorithm to allow automatic calculation of $D_2$, the correlation dimension of reconstructed attractor of Electroencephalogram (EEG), and processing of large accumulations of experimental data. This algorithm was adopted for EEG signals specifically. The algorithm was tested on the Tent, Lozi, and Hénon maps and on the Rössler and Lorenz attractors.

Yan and Duan [163], working in the domain of Intelligent Transportation Systems (ITS), proposed a novel co-evolutionary Artificial Bee Colony (ABC) algorithm improved by the Differential Evolution (DE) optimization algorithm combined with a traffic flow prediction
Shang et al. adopted a new method for measuring dissimilarity called the CID along with an advanced version called the GCID. Both methods were applied in MDS for the classification of time series, including artificial and real-world time series. First, in order to test the efficiency of the MDS method based on CID and GCID, they used three types of artificial time series: the α map model, the 2D Hénon map model, and the Lozi map model (with $a = 0.95$ and $b = 0.12$ for the first pair of parameters and $a = 1.0$, $b = 0.14$ for the second), which they used to observe the spatial distributions created by the MDS with the CID and GCID and examine their plausibility by calculating the stress. Additionally, three more sets of artificial time series were created with longer data length to demonstrate that their method is not likely to be affected by data length.

Jacob et al. [160] proposed a novel measure of degree heterogeneity for unweighted and undirected complex networks which requires only the degree distribution of the network for its computation. They showed that the proposed measure can be applied to all types of network topology with ease and that it increases with the diversity of the node degrees in the network.

As a specific application, they showed that the proposed measure can be used to compare the heterogeneity of recurrence networks constructed from time series of several low-dimensional chaotic attractors, thereby providing a single index to compare the structural complexity of chaotic attractors.

They compared the Lorenz, Rössler, and Duffing continuous attractors and the Hénon, Lozi, and Cat maps.

Their results indicated that the Lorenz attractor is the most structurally complex among the continuous systems, while in the case of two-dimensional discrete systems the Lozi attractor is the most diverse in terms of probability density variations.

In [161], Amigo and Kennel showed that the topological permutation entropy rate ($h_{top}^*(f)$) for expansive maps equals the conventional topological entropy rate ($h_{top}(f)$) from symbolic dynamics. After providing rigorous theoretical proofs of this equality, they sought to check it numerically. This task was found to be quite difficult; they stated that “Estimation of topological entropies from naive numerical simulation of long orbits is notoriously difficult. Metric entropy by itself can be quite tricky and difficult, requiring very long data sets for increasing $L$ (the number of the joints distributions of a finite-state stochastic process), but topological entropy is worse yet, because it weights each pattern equally. This means that patterns which are exceptionally infrequent on the natural measure of the attractor can still have a significant influence on the result. Attempting to estimate the same quantities using empirical occurrences of order patterns is even more difficult, requiring more data than would a good, low-alphabet generating partition for ordinary symbolic dynamics.”

As such, they found that the Lozi map is the only known map that satisfies all of the following criteria: a continuous system in more than one dimension, with a natural chaotic attractor, and with a topological entropy that can be found by independent rigorous means, for which there exist parameter values with reasonably low entropies.

In particular, they observed that $a = 6/5$ and $b = -2/15$ yield a low-entropy chaotic attractor (roughly 0.3 bits/iteration) and that the topological entropy for these parameters can be bounded rigorously via computer-assisted analytical computations. After defining clever strategies to compute ($h_{top}^*(\mathcal{L}_{a,b})$) and ($h_{top}(\mathcal{L}_{a,b})$) for these values, their numerical simulations conformed to the proven theorems.

Meckler [162] proposed an algorithm to allow automatic calculation of $D_2$, the correlation dimension of reconstructed attractor of Electroencephalogram (EEG), and processing of large accumulations of experimental data. This algorithm was adopted for EEG signals specifically. The algorithm was tested on the Tent, Lozi, and Hénon maps and on the Rössler and Lorenz attractors.

Yan and Duan [163], working in the domain of Intelligent Transportation Systems (ITS), proposed a novel co-evolutionary Artificial Bee Colony (ABC) algorithm improved by the Differential Evolution (DE) optimization algorithm combined with a traffic flow prediction
model trained by an Extreme Learning Machine (ELM) neural network. Accurate and rapid short-term traffic flow prediction is important to the performance of many advanced applications of Intelligent Transportation Systems (ITS). Forecasting results can help road users to find better travel routes, reduce traffic congestion, and improve the capacity of existing road networks.

The proposed algorithm inherited the better generalization performance and lower training time consumption of the standard ELM, achieved a more balanced search strategy with the optimized weights and biases to overcome the random initialization deficiency of the typical ELM, and successfully obtained higher prediction accuracy compared to state-of-the-art methods.

To verify the efficiency of the proposed algorithm, the authors applied it to Lozi and Tent chaotic time series simulations, and measured traffic flow time series experiments; their simulation and experimental results demonstrated that the proposed model has superior performance and competitive computational efficiency.

## 8.2. Economy

Tang et al. [164] carried out an analysis of Parrondo's game with different chaotic switching strategies. The performance on Parrondo's game was compared with random and periodic switching strategies. The main idea of Parrondo's paradox, first presented in 1996, is that two individually losing games can be combined to win via deterministic or non-deterministic mixing of the games [165]. In [164], a fair way to compare random and chaotic Parrondo's games was generalized. The logistic, tent, sinusoidal, and Gaussian 1D maps were considered together with the Hénon and Lozi maps.

To play a chaotic Parrondo's game where one of these chaotic generator is chosen, consider a sequence generated from an initial value. Then, every *n*th iterate of such a sequence determines whether Game A or Game B is played. Of course, the outcomes of Parrondo's game is affected by the different switching strategies applied and the chosen initial value. The relative proportions of Games A and B should be equal for all switching strategies in a fair comparison.

In conclusion, the authors found that a chaotic Parondo's game can provide a higher rate of winning compared to random switching strategies.

This result recalls the remark made by Caponetto et al. in [56] (see Section 4) that chaotic sequences can improve the performance of evolutionary algorithms compared to random sequences.

## 8.3. Education

Surprisingly, chaotic attractors can be involved in new education methods, as demonstrated by the following example. Al-Hilli [166] studied the use of software and technology to solve mathematics problems in order to speed up the learning process. She investigated a case study in which the dynamic properties that lead to transformations into chaos were studied, including the point of resistance and the points of proximity. She focused on studying the Lozi map (3), highlighting simple properties such as the sensitive dependence on initial conditions. She discussed solving this problem by applying Matlab software to vary the initial points and parameter values and display the resulting graphs. Due to the simple definition of this map, it can be easily understood by students with even a low level of mathematics.

The subjects of study were 88 primary school teachers in Baghdad who were studying in the second semester. A questionnaire developed by the researcher was used to collect the required information. It contained 24 closed-ended questions in the domain of the five studied variables, including attitude change, reasoning, and creativity (with five questions each) and fixation (with four questions).

The results showed that the use of information and communication technology is effective in changing students' attitudes towards mathematics. Thus, it was shown that the
model trained by an Extreme Learning Machine (ELM) neural network. Accurate and rapid short-term traffic flow prediction is important to the performance of many advanced applications of Intelligent Transportation Systems (ITS). Forecasting results can help road users to find better travel routes, reduce traffic congestion, and improve the capacity of existing road networks.

The proposed algorithm inherited the better generalization performance and lower training time consumption of the standard ELM, achieved a more balanced search strategy with the optimized weights and biases to overcome the random initialization deficiency of the typical ELM, and successfully obtained higher prediction accuracy compared to state-of-the-art methods.

To verify the efficiency of the proposed algorithm, the authors applied it to Lozi and Tent chaotic time series simulations, and measured traffic flow time series experiments; their simulation and experimental results demonstrated that the proposed model has superior performance and competitive computational efficiency.

## 8.2. Economy

Tang et al. [164] carried out an analysis of Parrondo's game with different chaotic switching strategies. The performance on Parrondo's game was compared with random and periodic switching strategies. The main idea of Parrondo's paradox, first presented in 1996, is that two individually losing games can be combined to win via deterministic or non-deterministic mixing of the games [165]. In [164], a fair way to compare random and chaotic Parrondo's games was generalized. The logistic, tent, sinusoidal, and Gaussian 1D maps were considered together with the Hénon and Lozi maps.

To play a chaotic Parrondo's game where one of these chaotic generator is chosen, consider a sequence generated from an initial value. Then, every *n*th iterate of such a sequence determines whether Game A or Game B is played. Of course, the outcomes of Parrondo's game is affected by the different switching strategies applied and the chosen initial value. The relative proportions of Games A and B should be equal for all switching strategies in a fair comparison.

In conclusion, the authors found that a chaotic Parondo's game can provide a higher rate of winning compared to random switching strategies.

This result recalls the remark made by Caponetto et al. in [56] (see Section 4) that chaotic sequences can improve the performance of evolutionary algorithms compared to random sequences.

## 8.3. Education

Surprisingly, chaotic attractors can be involved in new education methods, as demonstrated by the following example. Al-Hilli [166] studied the use of software and technology to solve mathematics problems in order to speed up the learning process. She investigated a case study in which the dynamic properties that lead to transformations into chaos were studied, including the point of resistance and the points of proximity. She focused on studying the Lozi map (3), highlighting simple properties such as the sensitive dependence on initial conditions. She discussed solving this problem by applying Matlab software to vary the initial points and parameter values and display the resulting graphs. Due to the simple definition of this map, it can be easily understood by students with even a low level of mathematics.

The subjects of study were 88 primary school teachers in Baghdad who were studying in the second semester. A questionnaire developed by the researcher was used to collect the required information. It contained 24 closed-ended questions in the domain of the five studied variables, including attitude change, reasoning, and creativity (with five questions each) and fixation (with four questions).

The results showed that the use of information and communication technology is effective in changing students' attitudes towards mathematics. Thus, it was shown that the
students learning and teaching efficacy can be increased by about 20% percent for the solved case study in Matlab software compared to conventional teaching and learning systems.

## 9. Conclusions

This article is a review of applications of the two-dimensional piece-wise linear chaotic Lozi map developed over the four last decades. These applications concern many areas (cryptography, electronic devices, memristors, optimization, synchronization, secure communications, AI with swarm intelligence, chimeras, solitary states, and more). As new applications have been regularly discovered at an accelerated pace over the last decade, there was a need to render them more easily accessible, which is the goal of this survey.

The Lozi map was initially discovered in the limited context of chaotic dynamical systems as they existed in the 1970s, when it was used as a toy model to rigorously prove the existence of a strange (and chaotic) attractor for plane mapping, which seemed inaccessible for the Hénon map. When I first began to study this model in June 1977, (without, of course all of the modern mathematical tools developed since then), I was convinced that a few weeks would be enough to explain the structure of such an attractor, as it was basically composed of line segments. However, the task turned out to be more difficult than I expected, mainly because unlike Michal Misiurewicz, I did not limit the range of the parameter values for the study. Following Misiurewicz's work, hundreds of papers were published on countless aspects of this chaotic attractor.

Today, if we go back to the late 1970s, life was very different from now in many ways. There were no personal computers (Michel Hénon used one of the only two computers at the University of Nice, an IBM 7040, to plot the figure of his original paper), no internet, and no mobile phones. Communications between researchers took place through slow post office mail and air travel was very expensive, limiting in-person contact between researchers. Technological advancements over the past fifty years have been spectacular in all aspects of life. It took almost my entire professional life as a mathematician to see the results I expected to be published in a few weeks.

However, amazingly, I was able to see the great unity of mathematics over these decades. For example, since 1975, when I was a young associate professor, I have been dis-appointed for years in the search for a classical global optimization algorithm able to avoid pseudo-minima. It is funny to notice how the map I introduced for a completely different purpose is now commonly used to solve this problem through a metaheuristic algorithms.

Thus, it is difficult to predict what the next areas of application of this map will be in the future. Perhaps in the field of biological and medical sciences, where research using chaos seems promising? Perhaps more advances are coming in control theory and cryptography? It can be observed that the Lozi map belongs to the class of chaotic and hyperchaotic systems which possess Input-to-State Stability (ISS)-like features. These systems are now involved in this field (see [167]).

With new applications being regularly discovered at an accelerated pace over the past decade, there is no reason to think that this phenomenon will stop.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The author declares no conflict of interest.

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