# Fixed-Time Synchronization of the New Single-Parameter Chaotic System

Authors: Yifeng Wei; Xia Qing; Xie Chengrong; Yuhua Xu
Journal: Complexity
 (2020)

---

## Abstract

This paper advances a new single-parameter chaotic system with a simple structure, and some basic dynamic behavior of the new single-parameter system is discussed, such as equilibria, dissipativity, the existence of an attractor, Lyapunov exponent, Poincaré map, spectrum map, and bifurcation diagram. Moreover, a new fixed-time convergence theorem is proposed for general chaotic systems based on finite-time control theory, and a fixed-time controller is also put to achieve synchronization of the new chaotic system. Simulation results are presented to show the effectiveness of the theoretical results. The conclusion of the paper is useful for the nonlinear economics and engineering application.


---

## Body

## 1. Introduction

After the first chaotic attractor was discovered by Lorenz, people began to search for new chaotic attractors with great enthusiasm. In recent years, for autonomous three-dimensional ordinary differential equations, the search for new chaotic attractors has attracted more and more attention. For example, Lorenz system [1], Rossler system [2], Chen and Ueta system [3], Lü and Chen system [4], and Liu et al. system [5]. In a three-dimensional autonomous system, all kinds of chaotic systems with distinct features were put forward, such as chaotic systems with only five terms [6], chaotic systems with linear terms [7], and chaotic systems with only one equilibrium point [8]. In practical engineering applications, we hope that the structure of the chaotic system is as simple as possible based on the consideration implementation; for example, Lü et al. have proposed a unified chaotic system with a single parameter with a simple structure, and it has been well applied in practice [9, 10]. Therefore, how to construct a simple chaotic system is still a meaningful work in engineering.Furthermore, research on synchronization and control of the chaos system is particularly topical for decades [11–13]. And, people came to know about the value of achieving control of chaos systems in finite time in some domains of engineering. [14–20]. It is necessary to give the initial conditions of the network in advance, owing to that the establishment time of the finite-time control depends on the initial state of the network to a great extent. However, the initial conditions may be random, so it is not easy to solve practical problems. In order to compensate for this, Polyakov produced the fixed-time stability of the system [21], which ensures that the system is without respect to the initial conditions in a bounded time. In the past few years, a lot of fixed-time stability theories have been raised to study fixed-time stability of the system [22–24]. Inspired by the existing research results, we study fixed-time synchronization of the chaos system. The aim of this paper is to present a new single-parameter chaotic system with a simple structure, and some basic dynamic behavior of the new system is discussed. The synchronization of the new chaotic system is realized by using the new fixed-time controller in this paper. Compared with the existing finite-time synchronization methods of chaotic systems, the fixed-time synchronization method proposed in this paper is more general.This paper is organized as follows: Section2 gives the new single-parameter chaos system. Section 3 discusses some basic characters of the new chaotic system. Section 4 puts forward the fixed-time control condition for a general chaos system. Section 5 gives an illustrative example. The conclusions are given in Section 6.


## 2. The New Single-Parameter Chaos System

Consider the following system:(1)x˙=y−x,y˙=−xz,z˙=xy−a,which is chaotic for 0.16<a<4.8 (see Figures 1–4 for a=0.5).Figure 1
A new chaotic attractor.Figure 2
x-y phase plane of the chaotic attractor.Figure 3
x-z phase plane of the chaotic attractor.Figure 4
y-z phase plane of the chaotic attractor.


## 3. Some Basic Dynamic Properties of New Chaotic Systems

In this section, some basic characteristics of the new chaotic system are analyzed.


### 3.1. Equilibria

Let(2)y−x=0,−xz=0,xy−a=0.Whena>0, the two equilibrium points of the system can be expressed as E1=a,a,0 and E2=−a,−a,0.Leta=0.5, and it is easy to obtain the following form of system equilibrium E1=2/2,2/2,0 and E2=−2/2,−2/2,0.ForE1=2/2,2/2,0, system (1) is linearized, and the Jacobian matrix is defined as J=−110−z0−xyx0E1.LetλI−J=0 to get the eigenvalues of matrix J.Therefore, whenE1=2/2,2/2,0, the eigenvalues of matrix J are(3)λ1=−1.2442,λ2=0.1221+0.8882i,λ2=0.1221−0.8882i.Similarly, whenE2=−2/2,−2/2,0, the eigenvalues of matrix J are(4)λ1=−1.2442,λ2=0.1221+0.8882i,λ2=0.1221−0.8882i.According to equations (3) and (4), it can be determined that both E1 and E2 are saddle focus nodes.


### 3.2. Dissipativity and the Existence of Attractor

For dynamical system (1), we can obtain(5)∇⋅V=∂x˙∂x+∂y˙∂y+∂z˙∂z=−1,where −1 is a negative value. System (1) is one dissipative system, and an exponential contraction of system (1) is dV/dt=e−1. This means that every volume containing the trajectory of the dynamic system will be reduced to zero as t⟶∞ at the exponential rate −1, and the asymptotic motion is located on the attractor of system (1).


### 3.3. Lyapunov Exponent

It is generally true that a system is chaotic as long as at least one of its Lyapunov exponents is positive. When the initial value is (1, 1, 1), the Lyapunov exponent of system (1) is calculated to be L1=0.7125, L2=0.0082, and L3=−0.4486, and its dimension is(6)DL=j+1Lj+1∑i=1jLi=2+L1+L2L3=2+0.7207−0.4486=3.6066.It implies that system (1) is a dissipative system. Moreover, its Lyapunov dimension is fractional. The fractal character of the attractor means that the system has aperiodic orbits, and the orbits near it are divergent.


### 3.4. Poincaré Map, Spectrum Map, and Bifurcation Diagram

In Figure5, the continuous broadband feature is exhibited for the spectrum of system (1). For z = 0, x = 0, and y = 0, Figures 6(a)–6(c)visualize the Poincaré map in planes, respectively, and several sheets of the attractors are displayed. This can better sum up all the possible behaviors by the bifurcation graph when the parameters on a graph are changing. For 0≤a≤8, Figure 7 is a bifurcation diagram of system (1), which shows its complex bifurcation phenomena.Figure 5
An apparently continuous broadband frequency spectrum.Figure 6
Poincaré maps for (a)z = 0, (b) x = 0, and (c) y = 0.
(a)(b)(c)Figure 7
Bifurcation diagram of system (1) for 0≤a≤8.


## 3.1. Equilibria

Let(2)y−x=0,−xz=0,xy−a=0.Whena>0, the two equilibrium points of the system can be expressed as E1=a,a,0 and E2=−a,−a,0.Leta=0.5, and it is easy to obtain the following form of system equilibrium E1=2/2,2/2,0 and E2=−2/2,−2/2,0.ForE1=2/2,2/2,0, system (1) is linearized, and the Jacobian matrix is defined as J=−110−z0−xyx0E1.LetλI−J=0 to get the eigenvalues of matrix J.Therefore, whenE1=2/2,2/2,0, the eigenvalues of matrix J are(3)λ1=−1.2442,λ2=0.1221+0.8882i,λ2=0.1221−0.8882i.Similarly, whenE2=−2/2,−2/2,0, the eigenvalues of matrix J are(4)λ1=−1.2442,λ2=0.1221+0.8882i,λ2=0.1221−0.8882i.According to equations (3) and (4), it can be determined that both E1 and E2 are saddle focus nodes.


## 3.2. Dissipativity and the Existence of Attractor

For dynamical system (1), we can obtain(5)∇⋅V=∂x˙∂x+∂y˙∂y+∂z˙∂z=−1,where −1 is a negative value. System (1) is one dissipative system, and an exponential contraction of system (1) is dV/dt=e−1. This means that every volume containing the trajectory of the dynamic system will be reduced to zero as t⟶∞ at the exponential rate −1, and the asymptotic motion is located on the attractor of system (1).


## 3.3. Lyapunov Exponent

It is generally true that a system is chaotic as long as at least one of its Lyapunov exponents is positive. When the initial value is (1, 1, 1), the Lyapunov exponent of system (1) is calculated to be L1=0.7125, L2=0.0082, and L3=−0.4486, and its dimension is(6)DL=j+1Lj+1∑i=1jLi=2+L1+L2L3=2+0.7207−0.4486=3.6066.It implies that system (1) is a dissipative system. Moreover, its Lyapunov dimension is fractional. The fractal character of the attractor means that the system has aperiodic orbits, and the orbits near it are divergent.


## 3.4. Poincaré Map, Spectrum Map, and Bifurcation Diagram

In Figure5, the continuous broadband feature is exhibited for the spectrum of system (1). For z = 0, x = 0, and y = 0, Figures 6(a)–6(c)visualize the Poincaré map in planes, respectively, and several sheets of the attractors are displayed. This can better sum up all the possible behaviors by the bifurcation graph when the parameters on a graph are changing. For 0≤a≤8, Figure 7 is a bifurcation diagram of system (1), which shows its complex bifurcation phenomena.Figure 5
An apparently continuous broadband frequency spectrum.Figure 6
Poincaré maps for (a)z = 0, (b) x = 0, and (c) y = 0.
(a)(b)(c)Figure 7
Bifurcation diagram of system (1) for 0≤a≤8.


## 4. Fixed-Time Synchronization of the General Chaotic System

Considering the following chaotic system in a general form:(7)x˙=fx,t,where x=x1,…,xnT and fx:Rm⟶Rm is a continuous function.So, the controlled chaos system is given by(8)y˙=fy,t+rx,y,t,where y=y1,…,ynT and rx,y,t is the controller.Then, the error system can be(9)e˙=fy,t−fx,t+rx,y,t.Assumptions 1.
We also assume thatf is Lipschitz with respect to its argument, i.e.,(10)ft,y−ft,x≤ηe,η∈R,where e=e1,e2,…,enT=y1−x1,y2−x2,…,yn−xnT.Lemma 1 (see [21]).
Forδi≥0,i=1,…,n, 0<ς≤1, and θ>1, then(11)∑i=1nδiς≥∑i=1nδiς,∑i=1nδiθ≥n1−θ∑i=1nδiθ.Lemma 2 (see [21]).
Assume that a continuous and a positive-definite functionυt satisfies(12)υ˙t≤−μυpt,t≥t0,υt0≥0,where μ>0 and 0<p<1 are all constants. Then, for any given t0, υt satisfies(13)υ1−pt≤υ1−pt0−μ1−pt−t0,t0≤t≤t1,(14)υt=0,t≥T1,with T1 given by(15)T1=t0+υ1−pt0μ1−p.In the following, the new fixed-time controller based on Lemma2 is proposed.Theorem 1.
Assuming that hypothesis 1 holds, two chaotic systems (7) and (8) can be synchronized by the following fixed-time controllers:(16)rt=−k1e2e−k2e1+k3signe−1,e≠0,0,e=0,and the time is bounded as(17)T≤1k1+1k2n−k3/222+k3/2−2η2k3,where k1>2η,k2>2η, and 0<k3<1.Proof.
Let(18)υt=12eTtet.
Then,(19)υ˙t=eTte˙t=eTtfvt−fut+rt≤ηeTtet−k1−k2eTte1+k3signe−1=ηeTtet−k1−k2eTte1+k3,e>1,ηeTtet−k1−k2eTte1−k3,e<1.
Obviously,(20)−k2eTe1+k3≤−k2n−k3/2eTe2+k3/2=−k2n−k3/222+k3/2υ2+k3/2,−k2eTe1−k3≤−k2eTe2−k3/2=−k222−k3/2υ2−k3/2.
So,(21)υ˙≤2ηυ−k1−k2n−k3/222+k3/2υ2+k3/2,e>1,2ηυ−k1−k222−k3/2υ2−k3/2,e<1.
Letϖυ=υ2, then(22)ϖ˙υ=2υυ˙≤2υ2ηυ−k1−k2n−k3/222+k3/2υ2+k3/2,e>1,2υ2ηυ−k1−k222−k3/2υ2−k3/2,e<1.
Whenk1>2ηandk2>2η, from Lemma 2, we have(23)2ηυ−k1−k2n−k3/222+k3/2υ2+k3/2≤−κ12/2+k3−22/2+k3η2/2+k3υ2/2+k3+2k22/2+k3n−k3/2+k3υ2+k3/2,(24)2ηυ−k1−k222−k3/2υ2−k3/2≤−k12/2−k3−22/2−k3η2/2−k3υ2/2−k3+2k22/2−k3υ2−k3/2.
By (23)-(24),(25)ϖ˙υ≤−2κ12/2+k3−22/2+k3η2/2+k3ϖ1/2+k3+2k22/2+k3n−k32+k3ϖ1/22+k3/2ϖ1/2,e>1,−2k12/2−k3−22/2−k3η2/2−k3ϖ1/2−k3+2k22/2−k3ϖ1/22−k3/2ϖ1/2,e<1.
Ifk1>2η,k2>2η, and ϖ˙υ≤−2k1ϖ1/2,e>1,−24−k3/2k2ϖ4−k3/4,e<1, it is known from Lemma 2 that the zero solution of system (9) is fixed-time stable.
From (21),(26)dυdt≤2ηυ−k1−k2n−k3/222+k3/2υ2+k3/2,e>1,2ηυ−k1−k222−k3/2υ2−k3/2,e<1.
So,(27)dt≤dυ2ηυ−k1−k2n−k3/222+k3/2υ2+k3/2,e>1,dυ2ηυ−k1−k222−k3/2υ2−k3/2,e<1.
WhenT, υt⟶0, the settling time T can be estimated as(28)T≤∫01dυk1−2ηυ+k222−k3/2υ2−k3/2+∫1υ0dυk1−2ηυ+k2n−k3/222+k3/2υ2+k3/2≤∫01dυk1+∫1υ0dυk2n−k3/222+k3/2−2ηυ2+k3/2.
Whenυ2+k3/2>1, ∫1υ0dυ/υ2+k3/2<2/k3, so(29)T≤1k1+1k2n−k3/222+k3/2−2η2k3.
The proof is completed.Remark 1.
In [12–14], different finite-time controllers are designed for different chaotic systems. The fixed-time controllers designed in this paper are more general and can be used in other different chaotic systems.Remark 2.
For the complex dynamic network satisfying Assumption1, the method of Theorem 1 can be used to control the complex network.


## 5. Simulation and Results

We assume that the new chaotic system is the drive system, that is,(30)u˙1=u2−u1,u˙2=−u1u3,u˙3=u1u2.And, the response system is(31)v˙1=v2−v1−k1e12+e22+e32e1−k2e11+k3signe12+e22+e32−1,v˙2=−v1v3−k1e12+e22+e32e2−k2e21+k3signe12+e22+e32−1,v˙3=v1v2−a−k1e12+e22+e32e3−k2e31+k3signe12+e22+e32−1.The initial conditions of the master and slave systems are (1, 2, 1) and (2, 1, 3), respectively. Through simulation calculation,η≈58.2516, and let k1=117, k2=118, and k3=0.5. Synchronization of two chaotic systems cannot be achieved without a controller as shown in Figure 8. The synchronization evolution of two chaos systems is shown in Figure 9. Numerical simulations illustrate the validity of Theorem 1.Figure 8
Error evolution without the controller.Figure 9
Synchronization errors.


## 6. Conclusion

The paper has presented a new chaotic system with a single parameter and discussed some basic dynamic behavior of the new system. Base on finite-time control theory, the fixed-time synchronization criterion of the new chaotic system has also been obtained. Finally, the result of numerical simulation proves that they are feasible. In the future, we will discuss the fixed-time synchronization of general complex networks.



---

*Source: 1067863-2020-08-03.xml*