Based on the heteroclinic Shil’nikov theorem and switching control, a kind of multipiecewise linear chaotic system is constructed in this paper. Firstly, two fundamental linear systems are constructed via linearization of a chaotic system at its two equilibrium points. Secondly, a two-piecewise linear chaotic system which satisfies the Shil’nikov theorem is generated by constructing heteroclinic loop between equilibrium points of the two fundamental systems by switching control. Finally, another multipiecewise linear chaotic system that also satisfies the Shil’nikov theorem is obtained via alternate translation of the two fundamental linear systems and heteroclinic loop construction of adjacent equilibria for the multipiecewise linear system. Some basic dynamical characteristics, including divergence, Lyapunov exponents, and bifurcation diagrams of the constructed systems, are analyzed. Meanwhile, computer simulation and circuit design are used for the proposed chaotic systems, and they are demonstrated to be effective for the method of chaos anticontrol.
In 1994, Schiff et al. first put forward the concept of chaos anticontrol [
Shil’nikov theorem is a beneficial tool to analyze the chaotic behavior of a nonlinear system [
In 2004, Zhou et al. firstly constructed a three-dimensional autonomous chaotic system which has only one equilibrium point [
In this paper, we use the heteroclinic Shil’nikov theorem to construct a kind of chaotic systems with both nonlinearities of two-piecewise linear and multipiecewise linear based on switching control method. Some basic dynamical characteristics of the constructed systems are analyzed theoretically and verified experimentally. Results of simulations and circuit experiments demonstrate the validity for the method of chaos anticontrol.
Consider the following three-dimensional autonomous dynamical system:
Given the three-dimensional autonomous system shown in ( Both There is a heteroclinic loop,
Both the original system (
In the following study, we will use the switching controller to construct piecewise linear chaotic systems based on this theorem.
For simplicity, we use the well-known Lorenz system to generate fundamental linear systems which are used to construct the piecewise linear chaotic systems.
The Lorenz system is described by
From (
Equations (
The eigenvectors corresponding to the equilibrium points
At point
Similarly, the space straight line
Next, we will study the existence conditions of the heteroclinic loop connecting the equilibrium points of system (
After translational transform for equilibrium points
Let the equilibrium points of system (
The eigenspaces corresponding to
From (
Similarly, from (
Let
If
In the following parts, the conditions that the coordinates of the equilibrium point must meet will be analyzed when the heteroclinic loop of system (
From (
Equation (
To make the coordinates of the equilibria of system (
According to the above analysis and by introducing switching function
Two-wing chaotic attractor of system (
Based on (
The mathematical expression for the multilinear chaotic systems is described as
When
The
In this section, some basic dynamics of system (
It is well known that a chaotic system is a dissipative system. According to (
Lyapunov exponent is the most direct evidence to determine a chaotic system. Figure
The diagram of LE versus parameter
Figure
The bifurcation diagram versus parameter
According to (
Module-based circuit diagram for realizing system (
Module-based circuit diagram for realizing functions
According to Figure
Experimental observation of multiwing and grid wing chaotic attractors of system (
Based on the heteroclinic Shil’nikov theorem, a method for constructing a kind of multipiecewise linear chaotic system has been proposed via switching control in this paper. By using this method, a two-piecewise linear and a multipiecewise linear chaotic systems are constructed.
It is worth pointing that this method of constructing chaotic system has general significance. It does not only suit Lorenz system in this paper but also can be applied to the three-dimensional autonomous system which has saddle focus equilibrium points
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was supported by the National Natural Science Foundation of China (Grant nos. 61271064 and 60971046), the Natural Science Foundation of Zhejiang Province, China (Grant no. LZ12F01001), the Research Foundation of Binzhou University (Grant no. BZXYG1501), and the Program for Zhejiang Leading Team of S & T Innovation, China (Grant no. 2010R50010-07).