Generation Method of Multipiecewise Linear Chaotic Systems Based on the Heteroclinic Shil ’ nikov Theorem and Switching Control

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.


Introduction
In 1994, Schiff et al. first put forward the concept of chaos anticontrol [1], which involves discrete-time and continuoustime system for the chaos anticontrol.In the study of chaos anticontrol for continuous-time system, a lot of progress has been made and some methods of generating chaotic and hyperchaotic systems were found, such as the methods of time-delay feedback to chaos [2][3][4], topological conjugate to chaos [5], pulse control to chaos [6], parameter perturbation control to hyperchaos [7][8][9], and state feedback control to hyperchaos [10][11][12][13].However, these methods of chaos anticontrol, which are based on the parameter "try," numerical simulation, and the Lyapunov index calculation, rely on the experience, lacking more theoretical basis.Therefore, it is still a challenge to construct a general approach of generating chaotic system.Shil'nikov theorem is a beneficial tool to analyze the chaotic behavior of a nonlinear system [14], and it is also one of the most common decision theorems to determine whether chaos exists or not.
In 2004, Zhou et al. firstly constructed a threedimensional autonomous chaotic system which has only one equilibrium point [15] by using the homoclinic orbit Shil'nikov theorem.Then, Li and Chen proposed a method to construct a two-piecewise linear chaotic system by using the heteroclinic Shil'nikov theorem [16,17].Recently, Yu et al. also proposed a method for constructing multipiecewise linear chaotic systems [18,19].However, the above studies lack further theoretical analysis and appropriate experimental verification.
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
Both the original system (1) and its perturbed varieties exhibit the Smale horseshoe chaos.
In the following study, we will use the switching controller to construct piecewise linear chaotic systems based on this theorem.

Two Fundamental Linear Systems
For simplicity, we use the well-known Lorenz system to generate fundamental linear systems which are used to construct the piecewise linear chaotic systems.

Construction of a Two-Piecewise Linear Chaotic System
Next, we will study the existence conditions of the heteroclinic loop connecting the equilibrium points of system (4) and system (5), choosing the controller as (, , ) and the switching plane as  = {(, , ) |  = 0}.

Basic Dynamics of Two-Piecewise Linear Chaotic System
In this section, some basic dynamics of system (18) will be analyzed, including the dissipation, the Lyapunov exponents, and the bifurcation diagram.

Dissipation of System.
It is well known that a chaotic system is a dissipative system.According to (17) and (18), we can get the dissipation character of the two-piecewise linear system from the following equation:

Lyapunov Exponent Spectrum and Bifurcation Diagram.
Lyapunov exponent is the most direct evidence to determine a chaotic system.Figure 4 shows the Lyapunov exponent spectrum versus parameter  in system (18), from which we can see one of the Lyapunov exponents is always positive when the parameter  changes in a large range, indicating that system (18) is chaotic.Larger range of chaotic parameter can provide larger space of key which can increase the difficulty of unmasking signals and improve the security of communications.Therefore, system (18) has significance if it is applied to secure communications as pseudorandom signal source.
Figure 5 shows the bifurcation diagram versus parameter  of system (18), from which we can see that system (18) is chaotic apart from a few tiny periodic windows.From Figure 5, we also can see the road of system (18) to chaos.System (18) evolves into chaotic state not through perioddoubling bifurcation but directly from stable state or from periodic motion, while the system parameters vary.

Circuit Implementation of the Multipiecewise Linear Chaotic System
According to ( 19)-( 20), a circuit of realizing system (19) is shown in Figure 6, which can generate 6-wing chaotic attractor and 6 × 2-grid wing chaotic attractor.The modular circuits for implementing functions ±( −  1 ), ±( −  2 ), ( −  3 ), and  are designed and shown in Figure 7, which consist of adders, inverters, multipliers, and integrators.In the modular circuits, the model of operational amplifiers is TL082 and that of multipliers is AD633.According to Figure 6, the circuit equation of system (19) can be obtained: Let  = 10  ; one has Let again   = ; (23) can be changed as follows:   (19).

Conclusion
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  1,2 (±, ±, ) and satisfies  1  2 > 0 or  1  2 > 0 and |  | > |  | > 0 ( = 1, 2).In addition, because the circuit structure of implementing the multipiecewise linear chaotic system is simple and easy to implement, this chaos anticontrol method has more potential applications than other chaotic anticontrol methods of generating multiwings and multigrid wing chaotic attractors in engineering applications.

Figure 4 :
Figure 4: The diagram of LE versus parameter  of system (18).