# Controlling Hopf Bifurcation of a New Modified Hyperchaotic Lü System

Authors: Ping Cai; Jia-Shi Tang; Zhen-Bo Li
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

Controlling Hopf bifurcation of a new modified hyperchaotic Lü system is investigated in this paper. A hybrid control strategy using both state feedback and parameter control is proposed. The control strategy realizes the delay of Hopf bifurcation. Furthermore, by applying the normal form theory, the stability of the bifurcation is determined. Numerical simulation results are given to support the theoretical analysis.


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## Body

## 1. Introduction

Chaos and bifurcation are of great importance in many physical, chemical, and biological nonlinear systems. Chaos theory is one of the most significant achievements of nonlinear science. Nowadays complex nonlinear systems are being used in many fields of science and engineering. Most of the recent works have focused on solving complex chaos control, synchronization, and so on [1–4]. Bifurcation control has been a rapidly growing interest by many research works in recent years [5–7]. Hopf bifurcation is a kind of important dynamic bifurcation. Contributions in Hopf bifurcation control mainly focused on amplitude control of limit cycle [8, 9], changing the critical points of an existing bifurcation [10], delaying the onset of an inherent bifurcation, and stabilizing an existing bifurcation [11–13], creating a desired bifurcation at a desired location, which is called anticontrol of bifurcation [14–17]. Among these researches, 3D chaotic systems play a leading role, such as Lorenz system [18], Chen system [19], and Lü system [20]. The study of 4D hyperchaotic systems has recently become a hot topic [21–24]. 4D hyperchaotic system has more complicated dynamical behaviors compared to 3D chaotic system. So the analysis and calculation are also more difficult.The idea of the work is to design a control law to control Hopf bifurcation in nonlinear system. Consider the following general nonlinear system:(1)x˙=fx,μ,x∈Rn,μ∈R,where the dot denotes differentiation with respect to time t and x is an n-dimensional state vector, while μ is bifurcation parameter. Let xi*(i=1,2,…,k) be k equilibria of system (1); that is, f(xi*,μ)≡0(i=1,2,…,k) for any value of μ. A hybrid control strategy is added to model (1), and then we obtain the following controlled system: (2)x˙=αfx,μ+1-αux,μ,α∈Rn,where α is a control parameter and u is state feedback controller. In order for the controlled system (2) to keep all the equilibria unchanged under the control, the following conditions should be satisfied:(3)uxi*,μ=0,i=1,2,…,k.A general formula satisfying condition (3) can be constructed as follows:(4)ux,x1*,x2*,…,xk*,μ=∏i=1kx-xi*+∑i=1k(x-xi*)∏j=1k(x-xj*)+∑i=1kx-xi*2∏j=1kx-xj*+⋯.Obviously, we use nonlinear feedback with polynomial function in the controller u. Generally speaking, linear part of a control strategy is used to shift the bifurcation value, in order to eliminate or delay an existing bifurcation. The nonlinear part, on the other hand, can be designed to change the stability of bifurcation solutions. Controller u involves higher-order terms, which may not be necessary for stability control. It is preferable to have the simplest possible design for engineering applications. In most cases, using fewer components or just one component may be enough to satisfy the predesigned control objectives. So, it is not necessary to take all the components in the controller u for practical system. This greatly simplifies the control formula. For example, if system (1) has two equilibria x1*, x2*, then the general controller can be taken as the following simple form:(5)ux,x1*,x2*,μ=x-x1*x-x2*+x-x1*2x-x2*+x-x1*x-x2*2.If system (1) has only one equilibrium x*, the controller can be taken as (6)ux,x*,μ=x-x*2+x-x*3.We also omit the linear term (x-x*) in (6), because parameter α has the same control effect. In fact, it even can be more simple as(7)ux,x*,μ=x-x*3.For Hopf bifurcation control, as a result of the calculation formula of stability index [25], terms up to the third-order term are enough and the second-order term (x-x*)2 might not be necessary due to the presence of the third-order term (x-x*)3 for the simplicity of the calculation. The following are the conditions of system (2) undergoing Hopf bifurcation at the equilibrium.LetJ(μ) be the Jacobian matrix of system (2) evaluated at x*. By the Hopf theory [25], J(μ) contains a complex conjugate pair of eigenvalues λ1,2(μ)=φ(μ)±ω(μ) satisfying(8)φ(μ~)=0,dφ(μ~)dμ≠0and the remaining eigenvalues of J(μ) have negative real parts at the critical point μ=μ~. That is to say, when μ is varied, the pair of the complex conjugates moves to cross the imaginary axis at μ=μ~. The second condition of (8) is usually called the transversality condition, implying that the crossing of the complex conjugate pair at the imaginary axis is not tangent to the imaginary axis. Without loss of generality, assume that when μ is varied from μ<μ~ to μ>μ~, λ1,2(μ) moves from the left-half of complex plane to the right. Thus, a family of limit cycles will bifurcate from the equilibrium x* at the critical point μ~.Next, it will be shown that the parameterα can change the bifurcation critical value, and the nonlinear state feedback can ensure the stability of bifurcation solutions.


## 2. Hybrid Control of Hopf Bifurcation

In this paper, the hybrid control is applied to Hopf bifurcation control of a new modified 4D hyperchaotic Lü system of the form [24](9)x˙1=ax2-x1+x2x3,x˙2=-x1x3+bx2+x4,x˙3=x1x2-cx3,x˙4=-kx1,where x1,x2,x3,x4 are state variables and a,b,c,k are real parameters. System (9) has a hyperchaotic attractor when a=35, b=14, c=3, and k=5 [24]. Obviously, system (9) has only one isolated equilibrium O(0,0,0,0) when k≠0. The controlled system is (10)x˙1=α1ax2-x1+x2x3+1-α1x13,x˙2=α2-x1x3+bx2+x4+1-α2x23,x˙3=α3x1x2-cx3+1-α3x33,x˙4=α4-kx1+1-α4x43.O(0,0,0,0) is also the equilibrium of system (10). The Jacobian matrix of system (10) at O(0,0,0,0) is (11)A=-aα1aα1000bα20α200-cα30-kα4000.The characteristic equation of A is (12)λ4+aα1-bα2+cα3λ3+-abα1α2+acα1α3-bcα2α3λ2+-abcα1α2α3+akα1α2α4λ+ackα1α2α3α4=0.Taking k as the Hopf bifurcation parameter and supposing that (12) has a pair of pure imaginary roots λ1,2=±iω0(ω0>0), which leads to (13)k0=α2b2-α1abα4,(14)ω0=-abα1α2.The other two roots are (15)λ3=α2b-α1a,λ4=-cα3.Thus, the necessary conditions for system (10) to exhibit Hopf bifurcation at O(0,0,0,0) are aα1>0, bα2<0, cα3>0, and k=k0. Under these conditions, the transversality condition (16)Reλ′k0λ=iω0=aα1α2aα1bα2-cα3α4-1-cα3cα3α4-bα2α4-1·2a2α12-3abα1α2+b2α22abα1α2-c2α32-1≠0is also satisfied. Therefore, system (10) undergoes Hopf bifurcation at the equilibrium O(0,0,0,0) based on Hopf bifurcation theory [25].Remark 1.
In particular, ifα1=α2=α3=α4=1, system (10) is reverted to the original system (9). The Hopf bifurcation value of the original system is k~0=b2-ab. So, parameter α can change the Hopf bifurcation value.Remark 2.
By formula (13), we notice that α3 does not affect the bifurcation critical value, so we can set α3=1 in system (10). And for simplicity, we also set α4=1 in the following discussion; that is to say, we may only choose the first two equations of system (9) under control. In this case, k0=α2b2-α1ab, ω0=-abα1α2 are obtained.


## 3. Analysis of Stability of Hopf Bifurcation

In this section, we apply the normal form theory [25] to study the stability of the Hopf bifurcation for system (10).By the linear transform(x1,x2,x3,x4)T=P(y1,y2,y3,y4)T, where(17)P=ω0b(α1a-α2b)0-1b0-ω0α2α22b2+ω02-α22bα22b2+ω02-α2α1a000010110,then system (10) has the following normal form:(18)y˙1=-ω0y2+F1y1,y2,y3,y4,y˙2=ω0y1+F2y1,y2,y3,y4,y˙3=α2b-α1ay3+F3y1,y2,y3,y4,y˙4=-cy4+F4y1,y2,y3,y4,where(19)F1=-α22a2bα12α22aα1-bα2k1+a3α13aα1-bα2k2ω05k3,F2=aα1k1+k2b2k3,F3=-F2,F4=0,(20)k1=α1-1α1ay3-α2by3-y1ω03,k2=α2-1α22b2y3+abα1α2y2-y3+α1ay1ω03a3α13,k3=α1a-α2b2α12a2-3abα1α2+α22b2.Then the stability condition of the bifurcated limit circle can be derived [25]:(21)β2=3aα1α2α12a2α2-1+α1α2ab5-2α1-3α2+α22b2-3+α1+2α2·4bα1a-α2b2α12a2-3α1α2ab+α22b2-1.If β2<0, the bifurcated periodic solution is orbitally asymptotically stable, and if β2>0, it is unstable. The following three special cases are considered.Case 1.
Ifα1=α2=α, then β2=3a(a2-5ab+3b2)(α-1)/4b(a-b)2(a2-3ab+b2). For a>0, b<0, we choose α>1; then the bifurcation critical value of system (10) satisfies k0=α(b2-ab)>k~0=b2-ab. Moreover, we have β2<0.Case 2.
Ifα2=1, that is to say, only the first equation of system (9) is under control, then β2=-3aα1(α1-1)(2aα1-b)/4(b-aα1)2(b2-3abα1+a2α12). For a>0, b<0, we choose α1>1; then the bifurcation critical value of system (10) satisfies k0=b2-α1ab>k~0=b2-ab. Moreover, we have β2<0.Case 3.
Ifα1=1, that is to say, only the second equation of system (9) is under control, then β2=3aα2(α2-1)(a-2bα2)/4b(a3-4a2bα2+4ab2α22-b3α23). For a>0, b<0, we choose α2>1; then the bifurcation critical value of system (10) satisfies k0=α2b2-ab>k~0=b2-ab. Moreover, we have β2<0.So, theoretical analyses show that the control strategy not only delays Hopf bifurcation but also achieves the stability control of the bifurcation.


## 4. Numerical Simulations

In this section, numerical simulations are given to illustrate the above theoretical analyses. We choosea=2, b=-1, and c=2, and the original system (9) undergoes Hopf bifurcation at k~0=3. The bifurcation figure of the original system (9) is shown in Figure 1. If we set α1=α2=2, the Hopf bifurcation critical value of the controlled system (10) is k0=6, and β2=-0.257576. The bifurcation figure of the controlled system (10) is shown in Figure 2. Time displacement curves and phase space trajectories are shown in Figures 3 and 4, respectively.Figure 1
Bifurcation figure of original system witha=2, b=-1, and c=2.Figure 2
Bifurcation figure of controlled system witha=2, b=-1, c=2, and α1=α2=2.Figure 3
Time displacement curves and phase space trajectories witha=2, b=-1, c=2, α1=α2=2, and k=5; (a) time displacement curves and (b) phase space.
(a)
(b)Figure 4
Time displacement curves and phase space trajectories witha=2, b=-1, c=2, α1=α2=2, and k=6.5; (a) time displacement curves and (b) phase space.
(a)
(b)Therefore, the numerical simulation results are consistent with the theory analyses.


## 5. Conclusions

In this paper, a hybrid control strategy is applied to control the Hopf bifurcation in a new modified hyperchaotic Lü system for the first time. This method keeps the equilibrium construction of the original system and does not increase the dimension of the system. By choosing an appropriate control parameter, the control strategy can effectively delay the Hopf bifurcation, so the stable range of the system is extended. By using the normal form theory, the stability of bifurcating solutions is analyzed. Numerical simulations show the effectiveness of the method. Bifurcation control of high dimensional nonlinear systems is much more difficult than low dimensional systems. This control strategy is simple and convenient, so it is meaningful for the study of bifurcation control of high dimensional nonlinear systems.



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