Hopf Bifurcation Analysis in a New Chaotic System with Chaos Entanglement Function

Abstract

A new approach to generate chaotic phenomenon, called chaos entanglement, is introduced in this paper. The basic principle is to entangle two or multiple stable linear subsystems by entanglement functions to form an artificial chaotic system such that each of them evolves in a chaotic manner. The Hopf bifurcation of a new chaotic system with chaos entanglement function is studied. More precisely, we study the stability and bifurcations of equilibrium in the new chaotic system. Besides, we controlled the system to any fixed point to eliminate the chaotic vibration by means of sliding mode method. And the numerical simulations were presented to confirm the effectiveness of the controller.

1. Introduction

The discovery of the eminent Lorenz system [1] has led to an extensive study of chaotic behaviors in nonlinear systems due to many possible applications in science and technology. In the last three decades, many new three-dimension chaotic systems have been proposed, such as Rossler system [2], Chen and Ueta system [3], Lü and Chen system [4], Liu et al. system [5], and generalized Lorenz system family [6]. Later, many Lorenz-like or Lorenz-based chaotic systems were proposed and investigated. Some classical 3D autonomous chaotic systems have three particular fixed points: one saddle and two unstable saddle-foci [7]. The other 3D chaotic systems have two unstable saddle-foci [8, 9]. Yang and Chen [10] found another 3D chaotic system with three fixed points: one saddle and two stable equilibriums. Recently, Qi et al. [11] constructed a new three-dimensional chaotic system which has complex dynamics with some interesting characteristics. Yang and Chen [12] analyzed the complex dynamics of the unified Lorenz-type system with six parameters, which contain common chaotic systems as its particular cases.

As a representative of the nonlinear robust control theory, variable structure control theory has been widely researched throughout the world and also had an increasing number of industrial applications. Chen et al. [13] studied the nonlinear dynamics behavior of hydroturbine governing system. In order to eliminate the chaotic vibration, the author used sliding mode method and controlled the system to any fixed point and any periodic orbit. The results show that using sliding mode method can make the system track target orbit strictly and smoothly with short transition time. Chen et al. [14] proposed a no-chattering sliding mode control strategy for a class of fractional-order chaotic systems. The designed control scheme guarantees the asymptotical stability of an uncertain fractional-order chaotic system. Lee et al. [15] present a sliding mode controller with integral compensation for a magnetic suspension balance beam system. The control scheme comprises an integral controller which is designed for achieving zero steady-state error under step disturbances and a sliding mode controller which is designed for enhancing robustness under plant uncertainties. Wang et al. [16] present two methods to design a single-input/single-output integral variable structure system. To ensure the robustness of the system control, Chen et al. [17] stabilized the chaotic orbits to arbitrary chosen fixed points and periodic orbits by means of sliding mode method, and Matlab simulations were presented to confirm the validity of the controller.

Over the past one decade, as we have seen, researchers have paid great attention to generate chaos and analyze its dynamics, because more and more chaotic phenomena have been found in many research fields, and chaos has powerful applications in secure communication, information processing, nonlinear circuits, biological systems, chemical reactions, and so forth. Thus, the requirement for creating new chaotic systems is dramatically increasing. A new approach to generate chaotic phenomenon, called chaos entanglement, is introduced in this paper. The basic principle is to entangle two or multiple stable linear subsystems by entanglement functions to form an artificial chaotic system such that each of them evolves in a chaotic manner. In this paper, we mainly consider a new chaotic system constructed by Zhang et al. [18]. Zhang proposed a new approach to generate chaotic phenomenon, called chaos entanglement. And he constructed a three-dimensional continuous autonomous chaotic system. However, the basic dynamical properties of the system and the relationship between the Hopf bifurcation and the system parameters have not been clarified yet.

This paper is organized as follows. We first described the new chaotic system in Section 2. In Section 3 we present the linear analysis of equilibriums of system (1). A brief review of the methods used to study codimension one Hopf bifurcations is presented in Section 4. In Section 5, numerical simulation is given to illustrate the theoretical analysis. We controlled the system to any fixed point to eliminate the chaotic vibration by means of sliding mode method in Section 6. Finally, conclusions are given in Section 7.

2. New Chaotic Attractor

Chaos Entanglement [18]. If two or more linear subsystems can behave in chaotic manners when they are entangled by some nonlinear functions, we say it is chaos entanglement. These nonlinear functions are called entanglement functions.

Consider two linear subsystems; one is 2-dimensional and the other is 1-dimensional, ̇̇𝑥=𝑎𝑥+𝑏𝑦,𝑦=−𝑏𝑥+𝑎𝑦,𝑧̇=𝑐𝑧, (1)(2) where (𝑥,𝑦,𝑧)∈𝑅³ is the state variable. When 𝑎<0 and 𝑐<0, both subsystems are stable. We entangle these two subsystems in the following way: ̇̇̇𝑥=𝑎𝑥+𝑏𝑦,𝑦=−𝑏𝑥+𝑎𝑦+𝑑sin𝑧,𝑧=𝑐𝑧+𝑒sin𝑥, (3) where 𝑎<0, 𝑐<0, (𝑑,𝑒)∈𝑅² is the entanglement coefficient, and (sin𝑥,sin𝑧) is the entanglement function.

Zhang et al. [18] have proved that the system (3) is dissipative and bounded and possesses one positive Lyapunov exponent. So, the system (3) is chaotic. Despite the simplicity, system (3) has a rich dynamical behavior, ranging from stable equilibrium points to periodic and even chaotic oscillations, depending on the parameter values. When the real constants 𝑎=−1, 𝑏=1, 𝑐=−1, 𝑑=36, and 𝑒=18, the system (3) has a chaotic attractor as shown in Figures 1(a)–1(d).

(a)

(b)

(c)

(d)

Figure 1 

(a) Chaotic attractor given by (1) with 𝑎=−1, 𝑏=1, 𝑐=−1, 𝑑=36, and 𝑒=18, and projection of the chaotic attractor on the (b) 𝑥−𝑦 plane, (c) 𝑦−𝑧 plane, and (d) 𝑥−𝑧 plane.

Moreover, the dynamics of the system (3) can be characterized with its Lyapunov exponents which are computed numerically by Wolf et al. algorithm proposed in [19], where the Lyapunov exponents 𝜆₁=0.596592, 𝜆₂=0,  and  𝜆₃=−3.596124 as shown in Figure 2(a), and the Lyapunov dimension 𝐷_𝐾𝑌=2.16589. The time history, the frequency spectrum, and Poincaré map in 𝑦−𝑧{𝑥=0} plane of the chaotic attractor shown in Figures 2(b), 2(c) and 2(d) respectively.

(a)

(b)

(c)

(d)

Figure 2 

(a) Time history, (b) frequency spectrum, (c) Poincaré mapin 𝑦−𝑧{𝑥=0} plane, and (d) Lyapunov-exponent spectrum of chaotic system (3).

3. Linear Analysis of the New Chaotic System

This paper studies the stability and bifurcations with the respective qualitative changes in the dynamics of the system (3). System (3) has a rich dynamical behavior, ranging from stable equilibrium points to periodic and even chaotic oscillations, depending on the parameter values.

In a vectorial notation which will be useful in the calculations, system (3) can be written as 𝑥^′=(𝑥,𝜁), where 𝑓(𝑥,𝜁)=𝑎𝑥+𝑏𝑦,−𝑏𝑥+𝑎𝑦+𝑑sin𝑧,𝑐𝑧+𝑒sin𝑥, (4)𝑥=(𝑥,𝑦,𝑧)∈𝑅³, 𝜁=(𝑎,𝑏,𝑐,𝑑,𝑒)∈𝑅².

The equilibria of system (3) can be found by solving the following equations simultaneously: 𝑎𝑥+𝑏𝑦=0,−𝑏𝑥+𝑎𝑦+𝑑sin𝑧=0,𝑐𝑧+𝑒sin𝑥=0. (5) Usually, it is not easy to find out its accurate solutions. For simplification, we only consider 𝐸₀=(0,0,0).

For the sake of completeness we state the following lemma (Routh-Hurwitz stability criterion) whose proof can be found in [6].

Lemma 1 (see [9]). The polynomial 𝑝(𝜆)=𝜆³+𝑝₁𝜆²+𝑝₂𝜆+𝑝₃ with real coefficients has all roots with negative real parts if and only if the numbers 𝑝₁,𝑝₂,𝑝₃ are positive and the inequality 𝑝₁𝑝₂>𝑝₃ is satisfied.

We have the following proposition.

Proposition 2. Define the following subsets of W: 𝑤₁=(𝑎,𝑏,𝑐,𝑑,𝑒)∶𝑎<0,𝑏>0,𝑐<0,𝑑>2𝑎³+2𝑎𝑏²+4𝑎²𝑐+2𝑎𝑐²,,𝑤𝑏𝑒𝑒>0₂=(𝑎,𝑏,𝑐,𝑑,𝑒)∶𝑎<0,𝑏>0,𝑐<0,𝑑<2𝑎³+2𝑎𝑏²+4𝑎²𝑐+2𝑎𝑐²,,𝑤𝑏𝑒𝑒>0₃=(𝑎,𝑏,𝑐,𝑑,𝑒)∶𝑎<0,𝑏>0,𝑐<0,𝑑=2𝑎³+2𝑎𝑏²+4𝑎²𝑐+2𝑎𝑐².𝑏𝑒,𝑒>0 (6) The following statements hold.(1)If (𝑎,𝑏,𝑐,𝑑,𝑒)∈𝑤₁ then the equilibrium 𝐸₀ is locally asymptotically stable.(2)If (𝑎,𝑏,𝑐,𝑑,𝑒)∈𝑤₂ then the equilibrium 𝐸₀ is unstable.(3)If (𝑎,𝑏,𝑐,𝑑,𝑒)∈𝑤₃ then the equilibrium 𝐸₀ is unstable.

Proof. At the fixed point 𝐸₀=(0,0,0), the Jacobian matrix is defined as 𝐴=𝑎𝑏0−𝑏𝑎𝑑cos𝑧𝑒cos𝑥0𝑐_(0,0,0)=.𝑎𝑏0−𝑏𝑎𝑑𝑒0𝑐 (7) The characteristic polynomial of the Jacobian matrix of system (3) at 𝐸₀ has the form 𝑝(𝜆)=𝜆³−(2𝑎+𝑐)𝜆²+𝑎²+𝑏²𝜆−𝑎+2𝑎𝑐₂+𝑏²𝑐−𝑏𝑑𝑒. (8) From Lemma 1 the equilibrium 𝐸₀=(0,0,0) is locally asymptotically stable if the coefficients of the characteristic polynomial satisfy −𝑎²+𝑏²𝑐−𝑏𝑑𝑒>0,𝑎²+𝑏²()−()𝑎+2𝑎𝑐>0,−2𝑎+𝑐>0,2𝑎+𝑐²+𝑏²+2𝑎𝑐>−𝑎²+𝑏²𝑐−𝑏𝑑𝑒. (9) The last inequality in (9) can be written as 𝑑>(2𝑎³+2𝑎𝑏²+4𝑎²𝑐+2𝑎𝑐²)/𝑏𝑒. So if (𝑎,𝑏,𝑐,𝑑,𝑒)∈𝑤₁ then the equilibrium 𝐸₀ is locally asymptotically stable and if (𝑎,𝑏,𝑐,𝑑,𝑒)∈𝑤₂ then the equilibrium 𝐸₀ is unstable. This proves items 1 and 2 of the proposition. Item 3 of the proposition will be proved in the next section (see Theorem 3) with the calculation of the first Lyapunov coefficient at the equilibrium 𝐸₀ for parameters in 𝑤₃. In fact, when (𝑎,𝑏,𝑐,𝑑,𝑒)∈𝑤₃ the equilibrium 𝐸₀ is a Hopf point; that is, the characteristic polynomial of the Jacobian matrix of system (3) at 𝐸₀ has eigenvalues of the form 𝜆₁=2𝑎+𝑐,𝜆_2,3=±𝑖𝜔₀=±𝑖𝑎²+𝑏²+2𝑎𝑐. (10) Define 𝑑₀=2𝑎³+2𝑎𝑏²+4𝑎²𝑐+2𝑎𝑐².𝑏𝑒 (11) If (𝑎,𝑏,𝑐,𝑑,𝑒)∈𝑤₃ then the Jacobian matrix of (4) at 𝐸₀ has one negative real eigenvalue 𝜆₁ and a pair of purely imaginary eigenvalues 𝜆_2,3. From the center manifold theorem, at a Hopf point a two-dimensional center manifold is well-defined; it is invariant under the flow generated by (3) and can be continued with arbitrary high class of differentiability to nearby parameter values. This center manifold is attracting since 𝜆₀. So it is enough to study the stability of 𝐸₀ for the flow restricted to the family of parameter-dependent continuations of the center manifold.

4. Hopf Bifurcation Analysis in System (4)

This section is a review of the projection method described in [20, 21] for the calculation of the first Lyapunov coefficients associated with Hopf bifurcation, denoted by 𝑙₁.

Consider the differential equation 𝑥^′=𝑓(𝑥,𝜁), (12) where 𝑥∈𝑅³, 𝜁∈𝑅⁵ are, respectively, vectors representing phase variables and control parameters. Assume that 𝑓 is of class 𝐶^∞ in 𝑅³×𝑅⁵. Suppose that (12) has an equilibrium point 𝑥=𝑥₀ at 𝜁=𝜁₀ and, denoting the variable 𝑥−𝑥₀ also by 𝑥, write 𝐹(𝑥)=𝑓𝑥,𝜁₀, (13) as 𝐹(𝑥)1=𝐴𝑥+2𝐵()+1𝑥,𝑥6𝐶(𝑥,𝑥,𝑥)+𝑂‖𝑥‖⁴, (14) where 𝐴=𝑓_𝑥(0,𝜁₀), and for 𝑖=1,2,3, 𝐵_𝑖𝑥,𝑦=³_𝑗,𝑘=1𝜕²𝐹_𝑖(𝜉)𝜕𝜉_𝑖𝜕𝜉_𝑘𝜉=0𝑥_𝑖𝑦_𝑘,𝐶_𝑖𝑥,𝑦,𝑧=³_{𝑗,𝑘,𝑙=1}𝜕³𝐹_𝑖(𝜉)𝜕𝜉_𝑖𝜕𝜉_𝑘𝜕𝜉_𝑙𝜉=0𝑥_𝑖𝑦_𝑘𝑧_𝑙. (15)

Suppose that (𝑥₀,𝜁₀) is an equilibrium point of (12) where the Jacobian matrix 𝐴 has a pair of purely imaginary eigenvalues 𝜆_2,3=±𝑖𝜔₀,(𝜔₀>0), and admits no other eigenvalue with zero real part. Let 𝑇^𝑐 be the generalized eigenspace of 𝐴 corresponding to 𝜆_2,3. By this it is meant the largest subspace invariant by 𝐴 on the largest subspace invariant the eigenvalues are 𝜆_2,3.

Let 𝑝,𝑞∈𝐶³ be vectors such that 𝐴𝑞=𝑖𝜔₀𝑞,𝐴^𝑇𝑝=−𝑖𝜔₀𝑝,𝑝,𝑞=³_𝑖=1𝑝_𝑖𝑞_𝑖=1, (16) where 𝐴^𝑇 is the transpose of the matrix 𝐴 and 𝑞 is the conjugate complex of the matrix 𝑞. Any vector 𝑦∈𝑇^𝑐 can be represented as 𝑦=𝜔𝑞+𝜔𝑞, where 𝜔=⟨𝑝,𝑦⟩∈𝐶 and 𝜔 is the conjugate complex of the matrix 𝜔. The two-dimensional center manifold associated with the eigenvalues 𝜆_2,3=±𝑖𝜔₀ can be parameterized by the variables 𝜔 and 𝜔 by means of an immersion of the form 𝑥=𝐻(𝜔,𝜔), where 𝐻∶𝐶²→𝑅³ has a Taylor expansion of the form 𝐻(𝜔,𝜔)=𝜔𝑞+𝜔𝑞+_{2≤𝑗+𝑘≤7}1ℎ𝑗!𝑘!^𝑗𝑘𝜔^𝑗𝜔^𝑘+𝑂|𝜔|⁸, (17) with ℎ_𝑗𝑘∈𝐶³, and ℎ_𝑗𝑘=ℎ_𝑘𝑗. Substituting this expression into (12) we obtain the following differential equation: 𝐻_𝜔𝜔^′+𝐻_𝜔𝜔^′=𝐹(𝐻(𝜔,𝜔)), (18) where 𝐹 is given by (13). The complex vectors ℎ_𝑗𝑘 are obtained solving the system of linear equations defined by the coefficients of (18), taking into account the coefficients of 𝐹, so that system (18), on the chart 𝜔 for a central manifold, writes as follows: 𝜔^′=𝑖𝜔₀1𝜔+2𝐺₂₁𝜔|𝜔|²+1𝐺12³²𝜔|𝜔|⁴+1𝐺144⁴³𝜔|𝜔|⁶+𝑂|𝜔|⁸, (19) with 𝐺_𝑗𝑘∈𝐶.

The first Lyapunov coefficient 𝑙₁ is defined by 𝑙₁=12Re𝐺₂₁, (20) where 𝐺₂₁=⟨𝑝,𝐻₂₁⟩, 𝐻₂₁=𝐶(𝑞,𝑞,𝑞)+𝐵(𝑞,ℎ₂₀)+2𝐵(𝑞,ℎ₁₁), ℎ₂₀=(2𝑖𝜔₀𝐼₃−𝐴)⁻¹𝐵(𝑞,𝑞), ℎ₁₁=−𝐴⁻¹𝐵(𝑞,𝑞), and 𝐼₃ is the unit 3×3 matrix.

A Hopf point (𝑥₀,𝜁₀) of system (12) is an equilibrium point where the Jacobian matrix 𝐴 has a pair of purely imaginary eigenvalues 𝜆_2,3=±𝑖𝜔₀,(𝜔₀>0) and the other eigenvalue 𝜆₁≠0. From the center manifold theorem, at a Hopf point a two-dimensional center manifold is well-defined; it is invariant under the flow generated by (12) and can be continued with arbitrary high class of differentiability to nearby parameter values.

A Hopf point is called transversal if the parameter dependent complex eigenvalues cross the imaginary axis with nonzero derivative. In a neighborhood of a transversal Hopf point with 𝑙₁≠0 the dynamic behavior of the system (12), reduced to the family of parameter-dependent continuations of the center manifold, is orbitally topologically equivalent to the following complex normal form 𝜔^′=(𝜂+𝑖𝜔)𝜔+𝑙₁𝜔|𝜔|², where 𝜔∈𝐶. 𝜂,𝜔, and 𝑙₁ are real functions having derivatives of arbitrary higher order, which are continuations of 0, 𝜔₀, and the first Lyapunov coefficient at the Hopf point. When 𝑙₁<0(𝑙₁>0) one family of stable (unstable) periodic orbits can be found on this family of manifolds, shrinking to an equilibrium point at the Hopf point.

Next we study the stability of 𝐸₀ for parameters in 𝐻. We have the following theorem.

Theorem 3. Consider the system (3); the first Lyapunov coefficient at 𝐸₀ for parameter values in 𝑤₃ is given by 𝑙₁𝑎,𝑏,𝑐,𝑑₀,𝑒=𝑒𝑎𝜔²𝑘+𝑒²−𝑎²𝑘−𝑒²2𝑒³2𝑎²+𝑏²,+2𝑎𝑐 (21) where 𝑘=𝑎²+𝑏²+𝑐²+2𝑎𝑐,𝜔=𝑎²+𝑏²+2𝑎𝑐. (22)
As 𝑏>𝑒>0 then 𝑙₁(𝑎,𝑏,𝑐,𝑑₀,𝑒)<0 and system (3) has a transversal Hopf point at 𝐸₀ for 𝑑=𝑑₀ and 𝑏>𝑒>0 (see Figure 4). More precisely, Hopf point at 𝐸₀ is stable and for each 𝑑>𝑑₀, but close to 𝑑₀, there exists a stable periodic orbit near the asymptotically stable equilibrium point 𝐸₀.

Proof. For parameters on the Hopf surface 𝑤₃ one has 𝜆₁=2𝑎+𝑐,𝜆_2,3=±𝑖𝜔₀=±𝑖𝑎²+𝑏²𝑖+2𝑎𝑐,𝑞=𝑎²+𝑏²+2𝑎𝑐−𝑐𝑒,𝑖𝑎²+𝑏²+2𝑎𝑐−𝑎𝑖𝑎²+𝑏²+2𝑎𝑐−𝑐,𝑝𝑏𝑒,1𝑝=₁,𝑝₂,𝑝₃,𝐶𝑞,𝑞,𝑞=𝑖0,−𝑑,𝑎²+𝑏²+2𝑎𝑐−𝑐²𝑖𝑎²+𝑏²+2𝑎𝑐+𝑐𝑒², (23) where 𝑝₁=𝑒𝑖𝑎²+𝑏²+2𝑎𝑐+𝑎𝑖𝑎²+𝑏²+2𝑎𝑐+𝑐−2𝑎²+𝑏²+𝑐²+2𝑎𝑐𝑎²+𝑏²,𝑝+2𝑎𝑐𝑖−𝑏𝑑𝑒₂=𝑏𝑒𝑖𝑎²+𝑏²+2𝑎𝑐+𝑐−2𝑎²+𝑏²+𝑐²+2𝑎𝑐𝑎²+𝑏²,𝑝+2𝑎𝑐𝑖−𝑏𝑑𝑒₃=𝑏𝑑𝑒−2𝑎²+𝑏²+𝑐²+2𝑎𝑐𝑎²+𝑏².+2𝑎𝑐𝑖−𝑏𝑑𝑒 (24) We can get 𝐻₂₁=𝑖0,−𝑑,𝑎²+𝑏²+2𝑎𝑐−𝑐²𝑖𝑎²+𝑏²+2𝑎𝑐+𝑐𝑒²,𝐺₂₁=𝑝,𝐻₂₁=−2𝑏𝑑𝑒𝜔²𝑘𝑘+𝑒²+𝑏²𝑐𝑑²𝑒𝑘−𝑒²−2𝑏𝑐𝑑𝑒𝑘𝑘−𝑒²+𝑏𝑑𝑒²𝑏𝑑𝑘+𝑒²×−()𝜔𝑖2𝑒𝑘𝜔²−𝑏𝑑𝑒²²⁻¹, (25) where 𝑘=𝑎²+𝑏²+𝑐²+2𝑎𝑐,𝜔=𝑎²+𝑏²+2𝑎𝑐. (26)
Now it remains only to verify the transversality condition of the Hopf bifurcation, so consider the family of differential equation (3) regarded as dependent on the parameter 𝑑. The real part 𝜉 of the pair of complex eigenvalues at the critical parameter 𝑑=𝑑₀ verifies 𝜉^′𝑑₀=Re𝑝,𝑑𝐴𝑑𝑑_𝑑=𝑑0𝑞=−𝑏𝑒𝜔²+𝑎𝑏𝑐𝑒2𝑘𝜔+2𝑎²𝑘<0. (27) Since 𝜉^′(𝑑₀)≠0, the transversality condition at the Hopf point holds.

5. Numerical Example

Next, we will give a numerical example of system (3). Let 𝑎=−1, 𝑏=1, 𝑐=−1, and 𝑒=18; we can compute the Hopf bifurcation value 𝑑₀=−0.555556. The equilibrium is stable when 𝑑=−0.2>𝑑₀ and unstable when 𝑑=−0.8<𝑑₀, as shown in Figure 3 and Figure 5, respectively. Thus, the periodic solutions bifurcating from the equilibrium point 𝐸₀ are supercritical and stable.

(a)

(b)

Figure 3 

Time history and phase diagram of system (3) with 𝑎=−1, 𝑏=1, 𝑐=−1, 𝑒=18, and 𝑑=−0.2.

(a)

(b)

Figure 4 

Time history and phase diagram of system (3) with 𝑎=−1, 𝑏=1, 𝑐=−1, 𝑒=18, and 𝑑=−0.555556.

(a)

(b)

Figure 5 

Time history and phase diagram of system (3) with 𝑎=−1, 𝑏=1, 𝑐=−1, 𝑒=18, and 𝑑=−0.8.

The characteristic polynomial of the Jacobian matrix of system (3) at 𝐸₀ has the form 𝑝(𝜆)=𝜆³+𝐴𝜆²+𝐵𝜆+𝐶, (28) where 𝐴=−2𝑎−𝑐,𝐵=𝑎²+𝑏²𝑎+2𝑎𝑐,𝐶=−𝑐²+𝑏²−𝑏𝑑𝑒. (29) If 𝐴>0, 𝐵>0, 𝐶>0, and 𝐴𝐵−𝐶>0, then the equilibrium 𝐸₀ is asymptotically stable. If 𝐴>0, 𝐵>0, 𝐶>0, and 𝐴𝐵−𝐶=0, then the characteristic polynomial of the Jacobian matrix of system(1) at 𝐸₀ with one pair of conjugate nonzero purely imaginary eigenvalues and the real part of other characteristic root are less than zero.

When the parameters 𝑐=−1, 𝑑=36, and 𝑒=18 are fixed while parameters 𝑎,𝑏 are varied, some different dynamical behaviors of system (3) are obtained. Let 𝐴=0, 𝐵=0, 𝐶=0, and 𝐴𝐵−𝐶=0, and use MATLAB to draw the stability region on the parameter plane 𝑎−𝑏, as shown in Figure 6. In this figure, the symbol 𝐿_𝑖, 𝑖=1,2,3,4, represents 𝐴=0, 𝐵=0, 𝐶=0, and 𝐴𝐵−𝐶=0, respectively. And in region (I), 𝐴>0, 𝐵<0, 𝐶<0, 𝐴𝐵−𝐶>0, in region (II),𝐴>0, 𝐵<0, 𝐶>0, 𝐴𝐵−𝐶<0, in region (III), 𝐴<0, 𝐵<0, 𝐶<0, 𝐴𝐵−𝐶>0, in region (IV): 𝐴<0, 𝐵<0, 𝐶>0, 𝐴𝐵−𝐶<0, in the region (V): 𝐴<0,𝐵>0,𝐶>0,𝐴𝐵−𝐶<0, in the region (VI), 𝐴<0,𝐵>0, 𝐶<0, 𝐴𝐵−𝐶>0, in region (VII), 𝐴<0, 𝐵>0, 𝐶<0, 𝐴𝐵−𝐶<0, in region (VIII), 𝐴>0, 𝐵>0, 𝐶<0, 𝐴𝐵−𝐶>0, in region (IX), 𝐴>0, 𝐵>0, 𝐶>0, 𝐴𝐵−𝐶<0, and in region (X), 𝐴>0, 𝐵>0, 𝐶>0, 𝐴𝐵−𝐶>0. All of the points are stable in region (X) and in the other regions are unstable.

Figure 6 

The stable region on the parameter plane (𝑎,𝑏).

According to the above analysis, we know the 𝐿₄ is the Hopf bifurcation boundaries of the system. And the system (3) will generate the Hopf bifurcation when the bifurcation parameter passes through 𝐿₄ from region (X) to region (IX). Next, we combined with Figure 6 and fixed 𝑎=−6; then we can compute the Hopf bifurcation value 𝑏₀=−0.9232. The system (3) has a transversal Hopf point at 𝐸₀ when the bifurcation parameter 𝑏 passes through 𝑏₀, as shown in Figure 7(b). In addition, we choose (𝑎,𝑏)=(−6,−0.8), (𝑎,𝑏)=(−6,−1.2), and (𝑎,𝑏)=(−1,1); the points are located in region (X), region (IX), and region (VIII), respectively, as shown in Figures 7(a) and 7(c) and Figure 7(d).

(a) 𝑎=−6,𝑏=−0.8

(b) 𝑎=−6,𝑏=−0.9232

(c) 𝑎=−6,𝑏=−1.2

(d) 𝑎=−1,𝑏=1

Figure 7 

Phase diagram of system (3) with different parameters.

When the parameters 𝑎=−1, 𝑑=36, and 𝑒=18 are fixed while parameters 𝑏,𝑐 are varied, some different dynamical behaviors of system (3) are obtained. Let 𝐴=0, 𝐵=0, and 𝐶=0, 𝐴𝐵−𝐶=0, and use MATLAB to draw the stability region on the parameter plane 𝑏−𝑐, as shown in Figure 8. In this figure, the symbol 𝐿_𝑖, 𝑖=1,2,3,4, represents 𝐴=0, 𝐵=0, 𝐶=0, and 𝐴𝐵−𝐶=0, respectively. And in region (I), 𝐴>0, 𝐵<0, 𝐶>0, 𝐴𝐵−𝐶<0, in region (II), 𝐴>0, 𝐵<0, 𝐶<0, 𝐴𝐵−𝐶>0, in region (III), 𝐴<0, 𝐵<0, 𝐶<0, 𝐴𝐵−𝐶>0, in region (IV), 𝐴<0, 𝐵<0, 𝐶>0, 𝐴𝐵−𝐶>0, in region (V), 𝐴>0, 𝐵>0, 𝐶>0, 𝐴𝐵−𝐶>0, in region (VI), 𝐴>0, 𝐵>0, 𝐶<0, 𝐴𝐵−𝐶>0, in region (VII), 𝐴<0, 𝐵>0, 𝐶<0, 𝐴𝐵−𝐶>0, in region (VIII), 𝐴<0, 𝐵<0, 𝐶>0, 𝐴𝐵−𝐶>0, in region (IX), 𝐴<0, 𝐵>0, 𝐶>0, 𝐴𝐵−𝐶<0, and in region (X), 𝐴>0, 𝐵>0, 𝐶>0, 𝐴𝐵−𝐶<0. All of the points are stable in region (V) and in the other regions are unstable.

Figure 8 

The stable region on the parameter plane (𝑏,𝑐).

According to the above analysis, we know the 𝐿₄ is the Hopf bifurcation boundaries of the system. And the system (3) will generate the Hopf bifurcation when the bifurcation parameter passes through 𝐿₄ from region (V) to region (X). Next, we combined with Figure 8 and fixed 𝑏=−0.5; then we can compute the Hopf bifurcation value 𝑐₀=−11.7181. The system (3) has a transversal Hopf point at 𝐸₀ when the bifurcation parameter 𝑐 passes through 𝑐₀, as shown in Figure 9(b). In addition, we choose (𝑏,𝑐)=(−0.5,−10), (𝑏,𝑐)=(−0.5,−14), and (𝑏,𝑐)=(6,−5); the points are located in region (V), region (X), and region (VI), respectively, as shown in Figures 9(a), 9(c), and 9(d).

(a) 𝑏=−0.5,𝑐=−14

(b) 𝑏=−0.5,𝑐=−11.7181

(c) 𝑏=−0.5,𝑐=−10

(d) 𝑏=6,𝑐=−5

Figure 9 

Phase diagram of system (3) with different parameters.

6. Sliding Mode Control of Chaotic Vibrations

6.1. The Design of the Controller

We designed a sliding surface with good nature and made the system possess the desired properties when it has limitations on the sliding surface. Then to facilitate control, make the system reach the sliding surface and keep sliding. After joining the controller, the system (3) has the following form: 𝑥̇=𝑎𝑥+𝑏𝑦+𝑑₁+𝑢₁,𝑦̇=−𝑏𝑥+𝑎𝑦+𝑑sin𝑧+𝑑₂+𝑢₂,𝑧̇=𝑐𝑧+𝑒sin𝑥+𝑑₃+𝑢₃, (30) where 𝑢₁, 𝑢₂, and 𝑢₃ are control inputs. If we join the reasonable controller, we can control the chaos system to the required range or the fixed point.

Define the following matrix: ,𝑑𝐀=−110−1−10004,𝐁=100010001𝐝=𝑑¹₂𝑑₃0,𝐠=36sin𝑧,18sin𝑥 (31) where 𝐀 is the linear matrix of the system, 𝐁 is the bounded perturbation matrix, 𝐝 is the control matrix, and 𝐠 is the nonlinear matrix of the system. The purpose of control is to let the system state 𝐱=[𝑥₁,𝑥₂,𝑥₃]^𝑇 to track a time-varying state 𝐱_𝑑=[𝑥_𝑑1,𝑥_𝑑2,𝑥_𝑑3]^𝑇. So, we can define the tracking error 𝐞=𝐱−𝐱_𝑑. (32)

The error dynamic system can be written as 𝐞̇=𝐱̇−𝐱̇_𝑑=𝐀𝐱+𝐁𝐠+𝐁𝐮+𝐝−𝐱̇_𝑑. (33)

Define the time-varying proportional integral sliding mode surface 𝐒=𝐊𝐞−^𝑡₀𝐊(𝐀−𝐁𝐋)𝐞(𝜏)𝑑𝜏, (34) where 𝐊∈𝐑^3×3,det(𝐊𝐁)≠0. For the convenience of calculation, we get 𝐊=diag(1,1,1). The additional matrix 𝐋∈𝐑^3×3 and 𝐀−𝐁𝐋 is negative definite matrix. The equation 𝐒=𝐒̇=0 must be satisfied under the sliding mode, where 𝐒̇=𝐊𝐁𝐠+𝐊𝐁𝐋𝐞+𝐊𝐁𝐮+𝐊𝐝+𝐊𝐀𝐱_𝑑−𝐊𝐱̇_𝑑. (35)

In order to satisfy the sliding conditions, the following controller is designed: 𝐮=−[𝐠+𝐋𝐞]−(𝐊𝐁)⁻¹𝐊𝐀𝐱_𝑑−𝐊𝐱̇_𝑑−(𝐊𝐁)⁻¹[𝜀+‖𝐊𝐁𝐠‖]sign(𝐒), (36) where sign(𝐒) is symbolic function.

Proposition 4. The controller (36) can make the system (28) reach the sliding mode 𝐒=0 in a limited time if the constant 𝜀 satisfied the inequality 𝜀>𝛿₁+𝛿₂+1, where 𝛿₁,𝛿₂ are an arbitrary small positive numbers. The state variables and the selected reference state 𝐱_𝑑 are identical.

Proof. Construct the Lyapunov function 𝑉=𝐒^𝑇𝐒=∑³_𝑖=1𝐒²_𝑖; according to (34), (35), and (36) one has 𝐒^𝑇𝐒̇=𝐒^𝑇𝐊𝐁𝐠+𝐊𝐁𝐋𝐞+𝐊𝐁𝐮+𝐊𝐝+𝐊𝐀𝐱_𝑑−𝐊𝐱̇_𝑑=𝐒^𝑇𝐊𝐝−(𝜀+‖𝐊𝐁𝐠‖)sign(𝐒)≤𝐒^𝑇(𝐒≤𝐝−𝜀sign)_3𝑖=1𝐒_𝑖𝛿₁+³_𝑖=1𝐒_𝑖𝛿₂−³_𝑖=1𝐒_𝑖𝜀=𝛿₁+𝛿₂−𝜀³_𝑖=1𝐒_𝑖<−³_𝑖=1𝐒_𝑖. (37) By the same token, we get 𝐒̇^𝑇𝐒<−³_𝑖=1𝐒_𝑖,𝑉̇=𝐒̇^𝑇𝐒+𝐒^𝑇𝐒̇<−2³_𝑖=1𝐒_𝑖. (38) So the proposition follows.

6.2. The Numerical Simulation

In the case of 𝑢₁=𝑢₂=𝑢₃=0, the time-domain charts of the state variables of system (28) are shown in Figure 10. Figure 10 illustrates that the system (28) has a periodic motion state before control.

(a) Time domain chart of 𝑥 before control

(b) Time domain chart of 𝑦 before control

(c) Time domain chart of 𝑧 before control

Figure 10 

Time domain charts of state variables before control.

In order to control the system (28) to the target state, we select the eigenvalue of 𝐀−𝐁𝐋 to be 𝐏=[−5,−5,−5]. The pole-placement method is adopted to get the following matrix: 𝐋=.410−140004 (39)

Select proportional integral sliding mode surface as follows: 𝑆₁=𝑒₁+^𝑡₀5𝑒₁(𝜏)𝑆𝑑𝜏,₂=𝑒₂+^𝑡₀5𝑒₂(𝜏)𝑆𝑑𝜏,₃=𝑒₃+^𝑡₀5𝑒₃(𝜏)𝑑𝜏. (40)

Set the initial value [𝑥₁(0),𝑥₂(0),𝑥₃(0)]=[0.1,0.1,0.1], and the reference state 𝑥_𝑑1=𝑥_𝑑2=𝑥_𝑑3=𝑥_𝑑. The control signal is as follows: 𝑢₁=−4𝑒₁−𝑒₂+𝑥̇_𝑑−𝜀sign𝑆₁,𝑢₂=−36sin𝑧+𝑒₁−4𝑒₂+2𝑥_𝑑+𝑥̇_𝑑−(𝜀+|36sin𝑧|)sign𝑆₂,𝑢₃=−18sin𝑥−4𝑒₃+𝑥_𝑑+𝑥̇_𝑑−(𝜀+|18sin𝑥|)sign𝑆₃. (41)

6.3. Control to the Fixed Point

We can stabilize the system (28) to any point by this method. In this paper, we select the fixed point [0,0,0], reference state 𝐱_𝑑=0, small parameter 𝜀=7, and the initial value of the sliding mode surface [𝑆₁(0),𝑆₂(0),𝑆₃(0)]=[0.1,0.1,0.1]. The controller 𝐮(𝑡) is activated at 𝑡=1𝑠, the time domain charts of state variables, and sliding surfaces as shown in Figure 11 and Figure 12, respectively.

(a) Time domain chart of 𝑥 after control

(b) Time domain chart of 𝑦 after control

(c) Time domain chart of 𝑧 after control

Figure 11 

Time domain charts of state variables after control.

(a) Time domain chart of 𝑆1 after control

(b) Time domain chart of 𝑆2 after control

(c) Time domain chart of 𝑆3 after control

Figure 12 

Time domain charts of sliding surfaces after control.

Figures 11 and 12 show that, after joining the controller, the system (28) tracks to reference state [0,0,0] ultimately, and the sliding mode surface 𝐒 becomes 0. It is proven that the system (28) reached the sliding mode.

7. Conclusion

In this paper, a new approach to generate artificial chaos by entangling two or multiple stable linear subsystems has been introduced. Meanwhile, review the linear stability analysis which accounts for the characterization, in the space of parameters, of the Lyapunov stability of the equilibria of the new system. It continues with the extension of the analysis to the first-order, codimension one Hopf bifurcation, based on the calculation of the first Lyapunov coefficient. At last, we controlled the system to any fixed point to eliminate the chaotic vibration by means of sliding mode method. And the numerical simulations were presented to confirm the effectiveness of the controller. Apparently there are more interesting problems about this chaotic system in terms of complexity, control, and synchronization, which deserve further investigation. Therefore, further research into the system is still important and insightful.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors gratefully acknowledge the support from the National Natural Science Foundation (nos. 11161027 and 61364001), the Key Project of Chinese Ministry of Education (no. 212180), and the Fundamental Research Funds for the Universities of Gansu Province (no. 620023) Government of China.