How to Generate Chaos from Switching System: A Saddle Focus of Index 1 and Heteroclinic Loop-Based Approach

Abstract

There exist two different types of equilibrium points in 3-D autonomous systems, named as saddle foci of index 1 and index 2, which are crucial for chaos generation. Although saddle foci of index 2 have been usually applied for creating double-scroll or double-wing chaotic attractors, saddle foci of index 1 are further considered for chaos generation in this paper. A novel approach for constructing chaotic systems is investigated by applying the switching control strategy and yielding a heteroclinic loop which connects two saddle foci of index 1. A basic 3-D linear system with an arbitrary normal direction of the eigenplane, possessing a saddle focus of index 1 whose corresponding eigenvalues satisfy the Shil'nikov inequality, is first introduced. Then a heteroclinic loop connecting two saddle foci of index 1 will be formed by applying the switching control strategy to the basic 3-D linear system. The heteroclinic loop consists of an unstable manifold, a stable manifold, and a heteroclinic point. Under the necessary conditions for forming the heteroclinic loop, the intended two-segmented piecewise linear system which exhibits the chaotic behavior in the sense of the Smale horseshoe can be finally constructed. An illustrative example is given, confirming the effectiveness of the proposed method.

1. Introduction

It is well known that saddle foci of index 2 are crucial for chaos generation in 3-D autonomous systems, where each saddle focus of index 2 creates one corresponding scroll or wing. Typical examples include the double-scroll Chua system, the double-wing Lorenz system, and so forth [1–5]. Among the three equilibrium points possessed by the double scroll Chua system, two are the saddle foci of index 2 with eigenvalues 𝛾,𝜎±𝑗𝜔 satisfying 𝛾<0,𝜎>0 and|𝜎/𝛾|<1, and one is the saddle focus of index 1 with eigenvalues 𝛾,𝜎±𝑗𝜔 satisfying 𝛾>0,𝜎<0, and |𝜎/𝛾|<1. The two saddle foci of index 2 create two corresponding scrolls leading to the emergence of the double-scroll chaotic attractor [1]. Among the three equilibrium points possessed by the double-wing Lorenz system, two are the saddle foci of index 2 with eigenvalues 𝛾,𝜎±𝑗𝜔 satisfying 𝛾<0,𝜎>0, and |𝜎/𝛾|<1, and one is the saddle point with eigenvalues 𝛾₁, 𝛾₂, 𝛾₃ satisfying 𝛾₁<0, 𝛾₂>0, and𝛾₃<0. The two saddle foci of index 2 create two corresponding wings leading to the emergence of the double-wing chaotic attractor [2]. Similarly in many other double-scroll systems or the double-wing generalized Lorenz system family, two saddle foci of index 2 generate two corresponding wings or two scrolls [6]. Multiscroll or multiwing chaotic attractors can also be further generated by extending the number of saddle foci of index 2 [7–9].

The above-mentioned method is also applied for constructing switching systems, where only saddle foci of index 2 are considered for generating scrolls [10, 11]. For example, Li and Chen present an approach for constructing a piecewise linear chaotic system based on the heteroclinic Shil’nikov theorem, starting from a linear system with a saddle focus of index 2 [10]. A heteroclinic loop can be easily constituted by the proposed method; however it cannot be extended to the general case for the eigenplane of the linear system can only be located on the 𝑥-𝑦 plane. In [11], a piecewise linear chaotic system is constructed based on heteroclinic Shil’nikov theorem and the switching control strategy, by particularly selecting two linear systems with two saddle foci of index 2, the eigenplanes of which are supposed to be symmetric. The method presented in [11] cannot be applied into the general situation due to the orientation restriction on the linear systems.

The review above indicates that for both smooth continuous chaotic systems and piecewise linear chaotic ones, saddle foci of index 2 have been usually considered for creating chaotic attractors. One may ask whether or not there is a possible way to break such a limitation? This paper will provide a positive answer to the question.

Based on the heteroclinic Shil’nikov theorem and the switching control strategy, this paper proposes a novel approach for constructing piecewise linear chaotic systems by employing saddle foci of index 1. A basic 3-D linear system with an arbitrary normal direction of the eigenplane is first introduced. This 3-D linear system has a saddle focus of index 1 whose corresponding eigenvalues 𝛾,𝜎±𝑗𝜔 satisfy 𝛾>0,𝜎<0, and |𝜎/𝛾|<1. The necessary parameter conditions, which ensure the existence of the heteroclinic loop connecting two saddle foci of index 1 in the controlled switching system, are then deduced. In detail, the heteroclinic loop presented in the controlled switching chaotic system with saddle foci of index 1 consists of three geometric invariant subsets, which are a 1-D unstable manifold corresponding to real eigenvalues 𝛾 satisfying 𝛾>0, a 2-D stable manifold corresponding to complex conjugate eigenvalues 𝜎±𝑗𝜔 satisfying 𝜎<0 and |𝜎/𝛾|<1, and a heteroclinic point. Under the necessary parameter conditions for forming the heteroclinic loop, the intended 2-segmented piecewise linear system, which exhibits the chaotic behavior in the sense of the Smale horseshoe, can be finally constructed. The main differences between this presented method and [10, 11] lie in the following aspects: (i) two distinct types of saddle foci are considered for chaos generation, of which saddle foci of index 1 are considered in this paper while saddle foci of index  2 in [10, 11]; (ii) the normal direction of eigenplane proposed in this paper is arbitrarily orientated while the eigenplane of the selected linear system can only be located on the 𝑥-𝑦plane in [10] or the orientations of the two linear systems are also restricted since their eigenplanes are supposed to be symmetric in [11].

The rest of the paper is organized as follows. A basic 3-D linear system with a saddle focus of index 1 is introduced in Section 2. Based on the heteroclinic Shil’nikov theorem and the switching control strategy, the intended 2-segmented piecewise linear system which exhibits the chaotic behavior in the sense of the Smale horseshoe is constructed in Section 3. The basic dynamic behaviors of the piecewise-linear chaotic system are further analyzed in Section 4. Finally, concluding remarks are given in Section 5.

2. A Basic 3-D Linear Nominal System with a Saddle Focus of Index 1

In this section, the heteroclinic Shil’nikov theorem is first described. A basic 3-D linear system, possessing a saddle focus of index 1 whose eigenvalues satisfy the Shil’nikov inequality, is then introduced. After that, the mathematical expressions of the corresponding eigenvectors, the unstable, and stable manifolds are derived.

2.1. The Heteroclinic Shil’nikov Theorem

Consider a 3-D autonomous dynamical system [12]𝑥̇(𝑡)=𝜉(𝑥),𝑡∈𝐑,𝑥∈𝐑³, (2.1) where the vector field 𝜉∶𝐑³→𝐑³. Let 𝑃₁(𝑥₁,𝑦₁,𝑧₁)and 𝑃₂(𝑥₂,𝑦₂,𝑧₂) be two distinct equilibrium points for (2.1). Suppose the following:(i)both 𝑃₁ and 𝑃₂ are saddle foci that satisfy the Shil’nikov inequality |𝛾_𝑖|>|𝜎_𝑖|>0(𝑖=1,2) with the further constraint 𝜔_𝑖≠0, 𝜎₁𝜎₂>0 and 𝛾₁𝛾₂>0;(ii)there is a heteroclinic loop 𝐻_𝑙 joining 𝑃₁ to 𝑃₂ that is made up of two heteroclinic orbits 𝐻_𝑖(𝑖=1,2).

Then both the original system of (2.1) and its perturbed varieties exhibit the Smale horseshoe chaos.

2.2. Design of a Basic 3-D Linear Nominal System with a Saddle Focus of Index 1

The state equation of the basic 3-D linear nominal system to be designed is given by𝑋̇𝑌̇𝑍̇=𝑎₁₁𝑎₁₂𝑎₁₃𝑎₂₁𝑎₂₂𝑎₂₃𝑎₃₁𝑎₃₂𝑎₃₃𝑋𝑌𝑍𝑋𝑌𝑍=𝐽, (2.2) where 𝑎₁₁=0.2857,𝑎₁₂=10,𝑎₁₃=0,𝑎₂₁=0.1,𝑎₂₂=−0.5,𝑎₂₃=5,𝑎₃₁=0,𝑎₃₂=−15,and 𝑎₃₃=0. There exists a unique equilibrium point 𝑂(0,0,0) with the corresponding eigenvalues 𝛾=0.2896 and𝜎±𝑗𝜔=−0.2519±𝑗8.5988due to the full rank of the Jacobin matrix of (2.2). The system (2.2) therefore possesses a saddle focus of index 1 satisfying the Shil’nikov inequality |𝛾|>|𝜎|>0. The corresponding eigenvectors are described by𝜂=𝜉_𝑅±𝑗𝜉_𝐼=𝜉_1𝑅𝜉_2𝑅𝜉_3𝑅𝜉±𝑗𝜉^1𝐼_2𝐼𝜉_3𝐼=0,−0.49980.01260.7512±𝑗0.01660.4306𝜇=𝜉_𝛾=𝜉_1𝛾𝜉_2𝛾𝜉_3𝛾=.0.99980.0004−0.0199 (2.3)

The 1-D unstable manifold 𝐸^𝑈_𝛾(𝑂) corresponding to the real eigenvalue 𝛾=0.2896 and the 2-D stable manifold 𝐸^𝑆_{𝜎+𝑗𝜔}(𝑂) corresponding to the complex conjugate eigenvalues 𝜎±𝑗𝜔=−0.2519±𝑗8.5988 are given by𝐸^𝑈_𝛾(𝑂)∶𝑋𝑙=𝑌𝑚=𝑍𝑛,𝐸^𝑆_{𝜎+𝑗𝜔}(𝑂)∶𝐴𝑋+𝐵𝑌+𝐶𝑍=0, (2.4) where(𝑙,𝑚,𝑛)=(0.9998,0.0004,−0.0199), implying the direction of the 1-D unstable manifold, and (𝐴,𝐵,𝐶)=(0.3235,0.0125,0.2150), indicating the direction numbers of the 2-D stable manifold 𝐸^𝑆_{𝜎+𝑗𝜔}(𝑂).

3. Constructing a Switching Chaotic System with Saddle Foci of Index 1

In this section, starting from the basic 3-D linear nominal system (2.2), and based on the heteroclinic Shil’nikov theorem and the switching control strategy, the necessary parameter conditions, which ensure the existence of the heteroclinic loop connecting two saddle foci of index 1 in the controlled switching system, are deduced. The intended 2-segmented piecewise linear controlled system which exhibits the chaotic behavior in the sense of the Smale horseshoe is then constructed.

According to the heteroclinic Shil’nikov theorem and the switching control strategy, a switching controller 𝐅(𝑥,𝑦,𝑧) with 𝑆_𝑍={(𝑥,𝑦,𝑧)∣𝑧=0} being the switching plane is introduced into the system (2.2). A coordinate transformation at the equilibrium point 𝑂(0,0,0) is then performed by the switching controller𝐅(𝑥,𝑦,𝑧). As a result, one gets𝑥̇𝑦̇𝑧̇=𝑎₁₁𝑎₁₂𝑎₁₃𝑎₂₁𝑎₂₂𝑎₂₃𝑎₃₁𝑎₃₂𝑎₃₃𝑥𝑦𝑧−𝐅𝑥,𝑦,𝑧, (3.1) where𝑎₁₁=0.2857,𝑎₁₂=10,𝑎₁₃=0,𝑎₂₁=0.1,𝑎₂₂=−0.5,𝑎₂₃=5,𝑎₃₁=0,𝑎₃₂=−15, and𝑎₃₃=0.

Obviously, the system (3.1) possesses two equilibrium points 𝑃₁(𝑥₁,𝑦₁,𝑧₁)∈𝑉₁ and 𝑃₂(𝑥₂,𝑦₂,𝑧₂)∈𝑉₂ appearing to lie on the opposite sides of the switching plane 𝑆_𝑍. Note that the detailed mathematical expression of the switching controller 𝐅(𝑥,𝑦,𝑧)=[𝑓₁(𝑥,𝑦,𝑧),𝑓₂(𝑥,𝑦,𝑧),𝑓₃(𝑥,𝑦,𝑧)]^𝑇 in the system (3.1) will be determined by the necessary parameter conditions for forming a heteroclinic loop which connects 𝑃₁ and 𝑃₂, the two saddle foci of index 1.

Let 𝐸^𝑈_𝛾1(𝑃₁) denote the one-dimensional unstable manifold of the equilibrium point P₁ corresponding to the real eigenvalue 𝛾₁=0.2896, 𝐸^𝑈_𝛾2(𝑃₂) denote the one-dimensional unstable manifold of the equilibrium point 𝑃₂ corresponding to the real eigenvalue 𝛾₂=0.2896, 𝐸^𝑆_{𝜎1+𝑗𝜔1}(𝑃₁) denote the two-dimensional stable manifold of 𝑃₁ corresponding to the complex conjugate eigenvalues 𝜎₁±𝑗𝜔₁=−0.2519±𝑗8.5988, and𝐸^𝑆_{𝜎2+𝑗𝜔2}(𝑃₂)denote the two-dimensional stable manifold of 𝑃₂ corresponding to the complex conjugate eigenvalues 𝜎₂±𝑗𝜔₂=−0.2519±𝑗8.5988, respectively. Then, one gets𝐸^𝑈_𝛾𝑖𝑃_𝑖∶𝑥−𝑥_𝑖𝑙=𝑦−𝑦_𝑖𝑚=𝑧−𝑧_𝑖𝑛,𝐸^𝑆_{𝜎𝑖+𝑗𝜔𝑖}𝑃_𝑖∶𝐴𝑥−𝑥_𝑖+𝐵𝑦−𝑦_𝑖+𝐶𝑧−𝑧_𝑖=0, (3.2) where 𝑖=1,2, 𝑙=0.9998,𝑚=0.0004,𝑛=−0.0199,𝐴=0.3235,𝐵=0.0125, and 𝐶=0.2150. The eigen-subspace corresponding to𝑃₁ and 𝑃₂ with orbits on the stable planes spirally approaching the equilibrium points is shown in Figure 1.

Figure 1 

The eigen-subspace corresponding to 𝑃1and 𝑃2.

Let Q₁ be the cross point of 𝐸^𝑈_𝛾1(𝑃₁)and 𝑆_𝑍, 𝑄₂be the cross point of𝐸^𝑈_𝛾2(𝑃₂)and𝑆_𝑍, 𝐸^𝑆_{𝜎1+𝑗𝜔1}(𝑃₁) and𝑆_𝑍 intersect in the line𝐿₁, and 𝐸^𝑆_{𝜎2+𝑗𝜔2}(𝑃₂)and 𝑆_𝑍 intersect in the line𝐿₂, respectively. Then, one gets𝑄₁=𝐸^𝑈_𝛾1𝑃₁∩𝑆_𝑍∶𝑥₁−𝑙𝑛𝑧₁,𝑦₁−𝑚𝑛𝑧₁,𝑄,0₂=𝐸^𝑈_𝛾2𝑃₂∩𝑆_𝑍∶𝑥₂−𝑙𝑛𝑧₂,𝑦₂−𝑚𝑛𝑧₂,𝐿,0₁=𝐸^𝑆_{𝜎1+𝑗𝜔1}𝑃₁∩𝑆_𝑍∶𝐴𝑥−𝑥₁+𝐵𝑦−𝑦₁+𝐶𝑧−𝑧₁𝐿=0,𝑧=0,₂=𝐸^𝑆_{𝜎2+𝑗𝜔2}𝑃₂∩𝑆_𝑍∶𝐴𝑥−𝑥₂+𝐵𝑦−𝑦₂+𝐶𝑧−𝑧₂=0,𝑧=0. (3.3)

As indicated in Figure 1, there exists a heteroclinic orbit 𝐻₁=𝐸^𝑈_𝛾1(𝑃₁)∪𝑄₁∪𝐸^𝑆_{𝜎2+𝑗𝜔2}(𝑃₂) from 𝑃₁ to 𝑃₂ when 𝑄₁ appears to lie on 𝐿₂. There exists another heteroclinic orbit 𝐻₂=𝐸^𝑈_𝛾2(𝑃₂)∪𝑄₂∪𝐸^𝑆_{𝜎1+𝑗𝜔1}(𝑃₁) from 𝑃₂ to 𝑃₁ when 𝑄₂ appears to lie on 𝐿₁. Provided that 𝑄₁ appears to lie on 𝐿₂ and that 𝑄₂ simultaneously appears to lie on 𝐿₁, there must exist a heteroclinic loop 𝐻_𝑙 connecting 𝑃₁ and 𝑃₂ that is made up of the two heteroclinic orbits𝐻₁and 𝐻₂.

From (3.3), one gets that 𝑄₁ appears to lie on 𝐿₂, and 𝑄₂ simultaneously appears to lie on 𝐿₁when the coordinate parameters of the equilibrium points 𝑃₁(𝑥₁,𝑦₁,𝑧₁)∈𝑉₁ and 𝑃₂(𝑥₂,𝑦₂,𝑧₂)∈𝑉₂ satisfy the following conditions:𝐴𝑥₁−𝑙𝑛𝑧₁−𝑥₂+𝐵𝑦₁−𝑚𝑛𝑧₁−𝑦₂−𝐶𝑧₂𝐴𝑥=0,₂−𝑙𝑛𝑧₂−𝑥₁+𝐵𝑦₂−𝑚𝑛𝑧₂−𝑦₁−𝐶𝑧₁=0. (3.4) According to the theory of the Smale horseshoe and the heteroclinic Shil’nikov theorem, when the system (3.1) is chaotic, there exist infinite numbers of heteroclinic orbits, among which there must exist a heteroclinic loop 𝐻_𝑙 connecting 𝑃₁ and 𝑃₂ through 𝑄₁ and 𝑄₂. In this case, the system (3.1) exhibits the chaotic behavior in the sense of the Smale horseshoe.

In particular, when the switching plane is 𝑆_𝑍={(𝑥,𝑦,𝑧)∣𝑧=0}, and equilibrium points 𝑃₁(𝑥₁,𝑦₁,𝑧₁)∈𝑉₁ and 𝑃₂(𝑥₂,𝑦₂,𝑧₂)∈𝑉₂ are of certain symmetry, such as 𝑥₁=−𝑥₂=𝑥₀, 𝑦₁=𝑦₂=𝑦₀≥0, and 𝑧₁=−𝑧₂=𝑧₀>0, (3.4) can be simplified into a single constraint equation, given by𝑥₀=𝐴𝑙+𝐵𝑚−𝐶𝑛𝑧2𝐴𝑛⁰, (3.5) where 𝑥₀ exhibits dependence only on the value of 𝑧₀ and 𝑦₀ can be of arbitrary value. In brief, suppose that the equilibrium points 𝑃₁(𝑥₁,𝑦₁,𝑧₁)∈𝑉₁ and 𝑃₂(𝑥₂,𝑦₂,𝑧₂)∈𝑉₂ are appointed to satisfy the following coordinate parameter conditions:𝑥₁=−𝑥₂=𝑧₀(𝐴𝑙+𝐵𝑚−𝐶𝑛)(),𝑦2𝐴𝑛₁=𝑦₂,𝑧₁=−𝑧₂=𝑧₀, (3.6) There exists a heteroclinic loop connecting equilibrium points 𝑃₁ and𝑃₂ in the piecewise linear system (3.1).

For the sake of simplicity, let 𝑦₀=0 and 𝑧₀=1. From (3.6), one gets the coordinates of equilibrium points as: 𝑃₁(𝑥₁,𝑦₁,𝑧₁)=𝑃₁(−25.4089,0,1) and𝑃₂(𝑥₂,𝑦₂,𝑧₂)=𝑃₂(25.4089,0,−1), which are the necessary parameter conditions for forming a heteroclinic loop. Based on the equilibrium points with the above-mentioned coordinates, one can further obtain the mathematical expression of the switching controller 𝐅(𝑥,𝑦,𝑧) in the system (3.1), given by𝐅=𝑓𝑥,𝑦,𝑧𝑓¹𝑥,𝑦,𝑧₂𝑓𝑥,𝑦,𝑧₃=𝑥𝑥,𝑦,𝑧⁰(𝑧)𝑦sgn₀(𝑧)𝑧sgn₀sgn(𝑧), (3.7) where 𝑥₀=−25.4089,𝑦₀=0, and 𝑧₀=1.

The numerical simulation results of the chaotic attractor, obtained from the piecewise linear system with saddle foci of index 1 described by (3.1) with (3.7), can be finally demonstrated as an illustrative example shown in Figure 2.

(a) 𝑥-𝑦 plane

(b) 𝑥-𝑧 plane

(c) 𝑦-𝑧 plane

Figure 2 

Chaotic attractor obtained from the piecewise linear system with saddle foci of index 1.

4. Basic Dynamic Analysis of the Piecewise Linear Chaotic System

In this section, a basic dynamic analysis of the piecewise linear chaotic system is given, including the dissipation, the Lyapunov exponents, and the solution of the state equations.

4.1. Dissipation and Lyapunov Exponents

According to (3.1) and (3.7), one can obtain the dissipation of the piecewise linear chaotic system, given by∇𝑉=𝑎₁₁+𝑎₂₂+𝑎₃₃=0.2857−0.5+0<0. (4.1) Therefore, the piecewise linear system with saddle foci of index 1 described by (3.1) with (3.7) is dissipative.

The Lyapunov exponents (LE) are also obtained to further prove the existence of chaos in the piecewise linear system, which are as follows: LE₁=0.27, LE₂=0.00, and LE₃=−0.49. The existence of the positive LE implies the chaotic nature of the piecewise linear system.

4.2. The Solution of the State Equations

The solution𝑋(𝑡)=(𝑋(𝑡),𝑌(𝑡),𝑍(𝑡))^𝑇 of the linear system (2.2) with the initial value 𝑋(0)=(𝑋(0),𝑌(0),𝑍(0))^𝑇 is given by𝐗(𝑡)𝑒=𝑃^𝛾𝑡000𝑒^𝜎𝑡cos𝜔𝑡𝑒^𝜎𝑡sin𝜔𝑡0−𝑒^𝜎𝑡sin𝜔𝑡𝑒^𝜎𝑡𝑃cos𝜔𝑡⁻¹𝐗(0)=𝑏₁₁𝑏₁₂𝑏₁₃𝑏₂₁𝑏₂₂𝑏₂₃𝑏₃₁𝑏₃₂𝑏₃₃𝐗(0)=𝐵𝐗(0), (4.2) where𝜉𝑃=^1𝛾𝜉_1𝑅𝜉_1𝐼𝜉_2𝛾𝜉_2𝑅𝜉_2𝐼𝜉_3𝛾𝜉_3𝑅𝜉_3𝐼. (4.3)

From (2.3) and (4.3), the coefficients b_ij(0≤i,j≤3) in (4.2) are given by𝑏₁₁=1.0134𝑒^𝛾𝑡−0.0134𝑒^𝜎𝑡cos𝜔𝑡−0.0013𝑒^𝜎𝑡𝑏sin𝜔𝑡,₂₁=3.9007×10⁻⁴𝑒^𝛾𝑡−3.9007×10⁻⁴𝑒^𝜎𝑡cos𝜔𝑡+0.0116𝑒^𝜎𝑡𝑏sin𝜔𝑡,₃₁=−0.0202𝑒^𝛾𝑡+0.0202𝑒^𝜎𝑡cos𝜔𝑡+0.0013𝑒^𝜎𝑡𝑏sin𝜔𝑡,₁₂=0.0390𝑒^𝛾𝑡−0.0390𝑒^𝜎𝑡cos𝜔𝑡+1.1605𝑒^𝜎𝑡𝑏sin𝜔𝑡,₂₂=1.5014×10⁻⁵𝑒^𝛾𝑡+1.0000𝑒^𝜎𝑡cos𝜔𝑡−0.0289𝑒^𝜎𝑡𝑏sin𝜔𝑡,₃₂=−7.7778×10⁻⁴𝑒^𝛾𝑡+7.7778𝑒^𝜎𝑡cos𝜔𝑡−1.7444𝑒^𝜎𝑡𝑏sin𝜔𝑡,₁₃=0.6736𝑒^𝛾𝑡−0.6736𝑒^𝜎𝑡cos𝜔𝑡−0.0424𝑒^𝜎𝑡𝑏sin𝜔𝑡,₂₃=2.5926×10⁻⁴𝑒^𝛾𝑡−2.5926×10⁻⁴𝑒^𝜎𝑡cos𝜔𝑡+0.5815𝑒^𝜎𝑡𝑏sin𝜔𝑡,₃₃=−0.0134𝑒^𝛾𝑡+1.0134𝑒^𝜎𝑡cos𝜔𝑡+0.0301𝑒^𝜎𝑡sin𝜔𝑡, (4.4) where 𝛾=0.2896, 𝜎=−0.2519, and 𝜔=8.5988.

Since the piecewise linear chaotic system (3.1) is obtained from (2.2) by the coordinate transformation with the switching control strategy, the solution of (3.1) is obtained based on (4.2) as follows:𝐗₁(𝑡)=𝐗(𝑡)−𝐏₁𝐗if𝑥>0,₁(𝑡)=𝐗(𝑡)−𝐏₂𝐗̇if𝑥<0₁(𝑡)=𝐵𝐗(𝑡),, (4.5) where 𝐗₁(𝑡)=(𝑥(𝑡),𝑦(𝑡),𝑧(𝑡))^𝑇 is the solution of the system (3.1), 𝐗(𝑡)=(𝑋(𝑡),𝑌(𝑡),𝑍(𝑡))^𝑇is the solution of the system (2.2), 𝐗(0)=(𝑋(0),𝑌(0),𝑍(0))^𝑇 is the initial value, and 𝐏₁=(𝑥₁,𝑦₁,𝑧₁)^𝑇 and 𝐏₂=(𝑥₂,𝑦₂,𝑧₂)^𝑇 are the equilibrium points.

5. Conclusions

Both saddle foci of index 1 and saddle foci of index 2 play a crucial role in chaos generation in 3-D autonomous systems, whenever they are smooth continuous chaotic systems or piecewise linear chaotic ones. Saddle foci of index 2 have been usually considered for creating double-scroll or double-wing chaotic attractors in systems, such as the double-scroll Chua system, the double-wing Lorenz system, and switching chaotic systems. Few detailed reports have addressed the issue of applying saddle foci of index 1 for chaos generation. To break such a limitation, this paper has proposed the application of saddle foci of index 1 for constructing chaotic systems, by using the switching control strategy and yielding a heteroclinic loop which connects two saddle foci of index 1. On comparison with the existing methods, the approach presented in this paper is characterized mainly by selecting saddle foci of index 1 for chaos generation such that the intended 2-segmented piecewise linear system exhibits the chaotic behavior in the sense of the Smale horseshoe. Furthermore, this approach is more of generalization for constructing switching chaotic systems since the normal direction of uncontrolled basic 3-D linear system is arbitrarily orientated and is not supposed to be fixed on a special position.

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grants 60572073, 60871025 and 61172023, the Natural Science Foundation of Guangdong Province under Grants 8151009001000060, 8351009001000002, 9151009001000030, and S2011010001018, and the Science and Technology Program of Guangdong Province under Grant 2009B010800037.