Hopf Bifurcation Analysis and Anticontrol of Hopf Circles of the Rössler-Like System

Abstract

A new Rössler-like system is constructed by the linear feedback control scheme in this paper. As well, it exhibits complex dynamical behaviors, such as bifurcation, chaos, and strange attractor. By virtue of the normal form theory, its Hopf bifurcation and stability are investigated in detail. Consequently, the stable periodic orbits are bifurcated. Furthermore, the anticontrol of Hopf circles is achieved between the new Rössler-like system and the original Rössler one via a modified projective synchronization scheme. As a result, a stable Hopf circle is created in the controlled Rössler system. The corresponding numerical simulations are presented, which agree with the theoretical analysis.

1. Introduction

In the last three decades, chaos has been studied extensively and attracted increasing interests from mathematicians, physicists, engineers, and so on. Since chaotic systems not only admit abundant complex and interesting dynamical behaviors, such as bifurcations, chaos, and strange attractors, but also have many potential practical applications, great efforts have been devoted to investigating chaotic systems, for example, Lorenz system [1] and Rössler system [2, 3], and there has been an increasing effort to construct different types of chaotic systems. During the last few years, some new Lorenz-like chaotic systems [4], including Chen system [5], Lü system [6], Liu system [7], and T system [8], were proposed and studied.

Research on bifurcation, such as Hopf bifurcation, homoclinic bifurcation, and period doubling bifurcation, is one of the most hot topics in the field of nonlinear science [9]. It has been found that bifurcation will frequently lead to chaos in nonlinear systems. So it is necessary to explore the bifurcation of dynamical systems so as to understand the complex dynamical behaviors. Recently, Hopf bifurcation of some famous chaotic systems has been investigated and it has been becoming one of the most active topics in the field of chaotic systems.

In this paper, a new Rössler-like system is constructed by adding a linear feedback control term to the original Rössler system [2]. It is found that the new system admits complex dynamical behaviors with varying parameters, such as bifurcation, chaos, and strange attractors. Now its basic dynamical behaviors are analyzed, especially its Hopf bifurcation is investigated in detail via theoretical as well as numerical analysis. The bifurcated periodic solutions will be created under some parameter conditions.

As is well known, Hopf bifurcation gives rise to limit circles, which are typical oscillatory behaviors of many nonlinear systems in physical, social, economic, biological, and chemical fields. These oscillatory behaviors [10–12] can be beneficial in practical applications, such as in mixing, monitoring, and fault diagnosis in electromechanical systems. Also, the properties of limit cycles are very useful in modern control engineering, such as autotuning of PID controller [13] and process identification [14]. Early efforts in Hopf bifurcation control focused on delaying the onset of this bifurcation [15] or stabilizing an existing bifurcation [16]. In this paper, we propose a control method based on modified projective synchronization (MPS) [17] to generate a Hopf limit circle in the original Rössler system. The proposed MPS involves less calculations. Furthermore, the shape and size of the created Hopf circle can be adjusted by choosing different scaling factors.

The paper is organized as follows. In Section 2, a new Rössler-like system is constructed by the linear feedback control. In Section 3, its dissipation and fractal dimension are analyzed. The local stability and existence of Hopf bifurcation, as well as the direction and stability of bifurcating periodic solutions, are discussed in detail by the normal form theory in Section 4. In Section 5, the anticontrol of Hopf circles is achieved between the new Rössler-like system and the original Rössler system via the modified projective synchronization scheme, and a stable Hopf limit circle is created in the original Rössler system successfully. Finally, some conclusions are drawn in Section 6.

2. The New Modified Rössler System

The Rössler system [2] is described by ̇𝑥̇=−𝑦−𝑧,𝑦̇=𝑥,𝑧=𝑎𝑦−𝑦²−𝑏𝑧, (2.1) which is chaotic when 𝑎=𝑏=0.5, and its strange attractor is shown in Figure 1.

Figure 1 

Chaotic attractor of system (2.1).

Based on this Rössler system, by adding a linear term to the second equation and changing the third equation of system (2.1), a new Rössler-like system is obtained and given by ̇𝑥̇=−𝑦−𝑧,𝑦̇=𝑥+𝑐𝑧,𝑧=𝑎𝑦−𝑏𝑧−𝑦², (2.2) where 𝑥,𝑦,𝑧 are state variables, and 𝑎,𝑏,𝑐 are parameters. In order to ensure that system (2.2) is a dissipative system, assume that the parameter 𝑏 is positive in the following discussions. When 𝑎=0.5,𝑏=1,𝑐=1.2, system (2.2) is chaotic, which is shown in Figure 2. The bifurcation diagram of state variable 𝑦 versus parameter 𝑐 is shown in Figure 3. Figure 4 is the state trajectory of 𝑦.

Figure 2 

Chaotic attractor of system (2.2).

Figure 3 

Bifurcation diagram of system (2.2).

Figure 4 

State trajectory of 𝑦.

3. Analysis of Basic Dynamic Behaviors

3.1. Dissipation and Existence of Attractor

The divergence of system (2.2) is defined by ∇𝑉=𝜕𝑥̇+𝜕𝑥𝜕𝑦̇+𝜕𝑧̇𝜕𝑦𝜕𝑧=−𝑏=−1<0. (3.1) Since ∇𝑉<0, system (2.2) is a dissipative system and converges with an index rate of 𝑒^−𝑡. Volume element 𝑉₀ shrinks to 𝑉₀𝑒^−𝑡 at the time 𝑡. When 𝑡→∞, volume element 𝑉₀ shrinks to 0. Therefore, all trajectories of system (2.2) will be confined to a congregation, whose volume is 0. Its gradual movement behaviors are fixed in an attractor.

3.2. The Lyapunov Dimension

As we know, the Lyapunov exponents measure the exponential rates of divergence or convergence of nearby trajectories in phase space. Using Matlab software, the three Lyapunov exponents of system (2.2) are, respectively, 𝜆₁=0.0701,𝜆₂=0,𝜆₃=−1.0796 when 𝑎=0.5,𝑏=1,𝑐=1.2. The Lyapunov dimension of chaotic attractor of this new Rössler-like system is fractional, which is described as 𝐷_𝐿1=𝑗+𝜆_𝑗+1^𝑗_𝑖=1𝜆_𝑖1=2+||()−1.07960.0701+0=2.0649. (3.2)

3.3. Local Stability

By simple computation, it is easy to obtain that system (2.2) has two equilibria 𝑂(0,0,0) and 𝐸(𝑐𝑦₀,𝑦₀,−𝑦₀), where 𝑦₀=𝑎+𝑏. The Jacobian matrix for system (2.2) at the equilibrium 𝑂(0,0,0) and the equilibrium 𝐸(𝑐𝑦₀,𝑦₀,−𝑦₀) are, respectively, given by 𝐽=,̃0−1−110𝑐0𝑎−𝑏𝐽=0−1−110𝑐0−𝑎−2𝑏−𝑏, (3.3) and their corresponding characteristic equation are, respectively, 𝑓(𝜆)=𝜆³+𝑏𝜆²+()𝑓(𝜆)1−𝑎𝑐𝜆+𝑎+𝑏=0,=𝜆³+𝑏𝜆²+(1+2𝑏𝑐+𝑎𝑐)𝜆−𝑎−𝑏=0. (3.4)(3.5) According to Routh-Hurwitz’s criterion, the real parts of the roots of (3.4) are all negative if and only if 𝑏>0,𝑎+𝑎𝑏𝑐<0,𝑎+𝑏>0, (3.6) and the real parts of the roots of (3.5) are all negative if and only if 𝑏>0,(𝑏𝑐+1)(𝑎+2𝑏)>0,𝑎+𝑏<0. (3.7) The above analysis is summarized as follows.

Theorem 3.1. For the two equilibria 𝑂 and 𝐸 of the new Rössler-like system (2.2),(1)when 𝑏>0,𝑎+𝑎𝑏𝑐<0,𝑎+𝑏>0, the equilibrium 𝑂 is asymptotically stable.(2)when 𝑏>0,(𝑏𝑐+1)(𝑎+2𝑏)>0,𝑎+𝑏<0, the equilibrium 𝐸 is asymptotically stable.

4. Hopf Bifurcation Analysis of the New System

4.1. Hopf Bifurcation Analysis about 𝑂(0,0,0)

Let us assume that the solutions to the new Rössler-like system (2.2) undergo a Hopf bifurcation on some submanifold in parameter space corresponding to fixed 𝑐=𝑐₀. Then the characteristic equation (3.4) has roots 𝜆₀∈𝑅 and 𝜆_±=±𝑖𝜔₀, where 𝜔₀∈𝑅⁺. Then, one can obtain that 𝑓(𝜆)=𝜆−𝜆₀𝜆²+𝜔²₀=𝜆³+𝑏𝜆²+(1−𝑎𝑐)𝜆+𝑎+𝑏. (4.1) Clearly, 𝜆₀=−𝑏,𝜔²₀=1−𝑎𝑐,−𝜆₀𝜔²₀=𝑎+𝑏. (4.2) From (4.2), it is easy to get that (3.4) has a pair of purely imaginary conjugate roots 𝜆_±=±𝑖1+𝑎/𝑏≡±𝑖𝜔₀ and a real root 𝜆₀=−𝑏 if and only if 𝑐=𝑐₀=−1𝑏𝑎,1+𝑏>0. (4.3) According to (3.4), one has 𝜆^′(𝑐)=𝑎𝜆3𝜆².+2𝑏𝜆−𝑎𝑐+1 (4.4) Therefore, Re𝜆^′𝑐₀=𝑎𝑏³𝜔²₀2(𝑎+𝑏)²+2𝑏⁴𝜔²₀. (4.5) By Hopf bifurcation theorem [18] and the above analysis, we have the following

Theorem 4.1 (Existence of Hopf bifurcation). When 𝑎≠0,𝑏>0,𝑎+𝑏>0 and 𝑐 passes through the critical value 𝑐₀=−1/𝑏, system (2.2) undergoes a Hopf bifurcation at the equilibrium point 𝑂(0,0,0).

4.2. Direction and Stability of Bifurcating Periodic Orbits

In this section, we apply the normal form theory [18] to study the direction, stability, and period of bifurcating periodic solutions for system (2.2).

The eigenvectors 𝜐₁, 𝜐₀ associated with 𝜆₊=𝑖𝜔₀,𝜆₀=−𝑏 are, respectively, 𝜐₁=−1𝑎−√𝑏²+𝑎𝑏𝑎𝑖𝑏𝑎−√𝑏²+𝑎𝑏𝑖1𝑎𝑏,𝜐₀=1𝑏01. (4.6) Define 𝑃=Re𝜐₁,Im𝜐₁,𝜐₀=−1𝑎−√𝑏²+𝑎𝑏𝑎1𝑏𝑏𝑎−√𝑏²+𝑎𝑏0,𝑎𝑏101𝑥,𝑦,𝑧^𝑇=𝑥₁,𝑦₁,𝑧₁^𝑇. (4.7) Then system (2.2) can be written into 𝑥̇₁=−𝜔₀𝑦₁+𝐹₁𝑥₁,𝑦₁,𝑧₁,𝑦₁̇=𝜔₀𝑥₁+𝐹₂𝑥₁,𝑦₁,𝑧₁,𝑧₁̇=−𝑏𝑧₁+𝐹₃𝑥₁,𝑦₁,𝑧₁, (4.8) where 𝐹₁𝑥₁,𝑦₁,𝑧₁=𝑘−𝑏²𝑥²₁+√𝑏²+𝑎𝑏2𝑥₁𝑦₁−𝑎+𝑏𝑏𝑦²₁,𝐹₂𝑥₁,𝑦₁,𝑧₁=𝑏²√𝑏²𝐹+𝑎𝑏¹𝑥₁,𝑦₁,𝑧₁,𝐹₃𝑥₁,𝑦₁,𝑧₁=𝑏³+𝑏𝑎𝐹₁𝑥₁,𝑦₁,𝑧₁, (4.9) in which 1𝑘=𝑎𝑏³.+𝑏+𝑎 (4.10) In the following, we will follow the procedures proposed by Hassard et al. [18] to figure out the necessary quantities. One can get 𝑔₁₁=14𝜕²𝐹₁𝜕𝑥²₁+𝜕²𝐹₁𝜕𝑦²₁𝜕+𝑖²𝐹₂𝜕𝑥²₁+𝜕²𝐹₂𝜕𝑦²₁=𝑘𝑎+𝑏+𝑏³𝑘2𝑏+𝑖𝑎𝑏+𝑏²−𝑏⁴2√𝑏²,𝑔+𝑎𝑏₀₂=14𝜕²𝐹₁𝜕𝑥²₁−𝜕²𝐹₁𝜕𝑦²₁𝜕−2²𝐹₂𝜕𝑥₁𝜕𝑦₁𝜕+𝑖²𝐹₁𝜕𝑥²₁−𝜕²𝐹₁𝜕𝑦²₁𝜕+2²𝐹₂𝜕𝑥₁𝜕𝑦₁𝑘𝑏=−³+2𝑎+2𝑏𝑘4𝑏−𝑖2𝑏⁴+3𝑏²+3𝑎𝑏4√𝑏²,𝑔+𝑎𝑏₂₀=14𝜕²𝐹₁𝜕𝑥²₁−𝜕²𝐹₁𝜕𝑦²₁𝜕+2²𝐹₂𝜕𝑥₁𝜕𝑦₁𝜕+𝑖²𝐹₁𝜕𝑥²₁−𝜕²𝐹₁𝜕𝑦²₁𝜕−2²𝐹₂𝜕𝑥₁𝜕𝑦₁𝑘=−3𝑏³+2𝑎+2𝑏𝑘4𝑏−𝑖2𝑏⁴+𝑏²+𝑎𝑏4√𝑏²,𝐺+𝑎𝑏₂₁=18𝜕³𝐹₁𝜕𝑥³₁+𝜕³𝐹₁𝜕𝑥₁𝜕𝑦²₁+𝜕³𝐹₂𝜕𝑥²₁𝜕𝑦₁+𝜕³𝐹₂𝜕𝑦³₁𝜕+𝑖³𝐹₂𝜕𝑥³₁+𝜕³𝐹₂𝜕𝑥₁𝜕𝑦²₁−𝜕³𝐹₁𝜕𝑥²₁𝜕𝑦₁−𝜕³𝐹₁𝜕𝑦³₁𝐺=0,₁₁₀=12𝜕²𝐹₁𝜕𝑥₁𝜕𝑧₁+𝜕²𝐹₂𝜕𝑦₁𝜕𝑧₁𝜕+𝑖²𝐹₂𝜕𝑥₁𝜕𝑧₁−𝜕²𝐹₁𝜕𝑦₁𝜕𝑧₁𝐺=0,₁₀₁=12𝜕²𝐹₁𝜕𝑥₁𝜕𝑧₁−𝜕²𝐹₂𝜕𝑦₁𝜕𝑧₁𝜕+𝑖²𝐹₂𝜕𝑥₁𝜕𝑧₁+𝜕²𝐹₁𝜕𝑦₁𝜕𝑧₁=0. (4.11) Then we have 𝑔₂₁=𝐺₂₁+2𝐺₁₁₀𝑤₁₁+𝐺₁₀₁𝑤₂₀=0, (4.12) where 𝑤₁₁,𝑤₂₀ can be obtained by solving the following equations: 𝜆₃𝑤₁₁=−ℎ₁₁,𝜆₃−2𝑖𝜔₀𝑤₂₀=−ℎ₂₀, (4.13) in which ℎ₁₁=14𝜕²𝐹₃𝜕𝑥²₁+𝜕²𝐹₃𝜕𝑦²₁,ℎ₂₀=14𝜕²𝐹₃𝜕𝑥²₁−𝜕²𝐹₃𝜕𝑦²₁𝜕−2𝑖²𝐹₃𝜕𝑥₁𝜕𝑦₁, (4.14) here, it is not necessary to calculate 𝑤₁₁,𝑤₂₀,ℎ₁₁,ℎ₂₀ for obtaining the value of 𝑔₂₁.

From the above analysis, one can compute the following quantities: 𝐶₁(0)=𝑖2𝜔₀𝑔₂₀𝑔₁₁−2𝑔₁₁²−13𝑔₀₂²+12𝑔₂₁=11𝑏⁶−35𝑏⁴−35𝑎𝑏³+39𝑏²−78𝑎𝑏+39𝑎²48𝜔₀𝑎²√𝑏²+𝑎𝑏+𝑖16𝑏⁹−37(𝑏−𝑎)𝑏⁶+(77𝑏+106𝑎−48)𝑏⁴96𝜔₀𝑎²𝑏²(𝑎+𝑏)𝑏³++𝑏+𝑎72𝑎²𝑏−36𝑏−120𝑎𝑏²−105𝑎²+12𝑏−40𝑎³96𝜔₀𝑎²𝑏²(𝑎+𝑏)𝑏³,𝛽+𝑏+𝑎₂=2Re𝐶₁(0)=11𝑏⁶−35𝑏⁴−35𝑎𝑏³+39𝑏²−78𝑎𝑏+39𝑎²24𝜔₀𝑎²√𝑏²,𝜇+𝑎𝑏₂=−Re𝐶₁(0)𝛼^′(0)=−(𝑎+𝑏)𝑏³+𝑎+𝑏11𝑏⁶−35𝑏⁴−35𝑏³+39𝑏²−78𝑎𝑏+39𝑎²24𝑎³𝑏⁴𝜔⁴₀,𝜏₂=−Im𝐶₁(0)+𝜇₂𝜔^′(0)𝜔₀=2(𝑎+𝑏)𝑏³+𝑏+𝑎11𝑏⁶−35𝑏⁴−35𝑎𝑏³+39𝑏²−78𝑎𝑏+39𝑎²96𝜔²₀𝑎²𝑏²()−𝑎+𝑏16𝑏⁹−37(𝑏−𝑎)𝑏⁶+(77𝑏+106𝑎−48)𝑏⁴96𝜔²₀𝑎²𝑏²(𝑎+𝑏)𝑏³−+𝑏+𝑎72𝑎²𝑏−36𝑏−120𝑎𝑏²−105𝑎²+12𝑏−40𝑎³96𝜔²₀𝑎²𝑏²(𝑎+𝑏)𝑏³.+𝑏+𝑎 (4.15) Now we can get the following theorem.

Theorem 4.2. System (2.2) exhibits a Hopf bifurcation at the equilibrium 𝑂(0,0,0) as 𝑐 passes through 𝑐₀, with the following properties:(a)if 𝜇₂>0(<0), the Hopf bifurcation is supercritical (subcritical) and the bifurcating periodic solutions exist for 𝑐>𝑐₀(<𝑐₀);(b)if 𝛽₂<0(>0), the bifurcating periodic solutions are orbitally stable (unstable);(c)if 𝜏₂>0(<0), the period of bifurcating periodic solutions increases (decreases).

Remark 4.3. Here we only study the Hopf bifurcation about the equilibrium 𝑂(0,0,0). For the Hopf bifurcation of system (2.2) at the equilibrium 𝐸(𝑐𝑦₀,𝑦₀,−𝑦₀), We first introduce the transformation 𝑥=𝑥+𝑐𝑦₀,𝑦=𝑦+𝑦₀, and z = 𝑧−𝑦₀. Then, system (2.2) becomes 𝑥̇‌=−𝑦−̇𝑧,𝑦=𝑥+𝑐̇𝑧,𝑧=−(𝑎+2𝑏)𝑦−𝑏𝑧−𝑦²; (4.16) hence, the discussion about the Hopf bifurcation of system (2.2) at the equilibrium 𝐸(𝑐𝑦₀,𝑦₀,−𝑦₀) is equivalent to the case of system (4.16) at (0,0,0). So the Hopf bifurcation of the equilibrium point 𝐸(𝑐𝑦₀,𝑦₀,−𝑦₀) can be treated similarly.

4.3. Numerical Simulations

When 𝑎=0.5,𝑏=1, we can calculate 𝑐=−1 according to Theorem 4.1. It follows from the results in Section 4.2 that 𝜇₂=21.5000,𝛽₂=−3.5833,𝜏₂=−0.1052. In the light of Theorem 4.2, since 𝜇₂>0, the Hopf bifurcation is supercritical, which means that the equilibrium 𝑂(0,0,0) of system (2.2) is stable when 𝑐<𝑐₀ as shown in Figure 5. A Hopf bifurcation occurs when 𝑐 increases past 𝑐₀, that is, a family of periodic solutions bifurcate from the equilibrium, as shown in Figure 6. Since 𝛽₂<0, each individual periodic solution is stable. Since 𝜏₂<0, periods of bifurcating periodic solutions increase with increasing 𝑐.

Figure 5 

Phase diagram for system (2.2) with 𝑐=−1.2.

Figure 6 

Phase diagram for system (2.2) with 𝑐=−0.5.

5. Anticontrol of Hopf Limit Circles Based on MPS

Next, we will design a stable Hopf limit circle into the Rössler system (2.1) via modified projective synchronization (MPS) proposed by [17]. To this end, the closed-loop control system based on MPS is formulated as follows: (𝑥)𝑦,𝑥̇=𝐴𝑥+𝑓+𝑢,𝑦̇=𝐵𝑦+𝑔 (5.1)(5.2) where (5.1) denotes the response system and (5.2) stands for the drive system with a stable Hopf limit circle. 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛)^𝑇 and 𝑦=(𝑦₁,𝑦₂,…,𝑦_𝑛)^𝑇 are state variables of the response system and the drive system, respectively. 𝐴,𝐵 are matrices, whereas 𝑓,𝑔 are nonlinear functions. The vector 𝑢=(𝑢₁,𝑢₂,…,𝑢_𝑛)^𝑇 represents the controller to be designed, with which the two systems can achieve synchronization by MPS. The symbols 𝑒_𝑖=𝑥_𝑖−𝛼_𝑖𝑦_𝑖,𝑖=1,2,…,𝑛 (5.3) stand for the errors between the state variables of the response system and those of the drive one. 𝛼=diag(𝛼₁,𝛼₂,…,𝛼_𝑛) denotes the scaling factor matrix of the MPS, which can change the shape and size of the created Hopf limit circle. In order to achieve the modified projective synchronization, the controller 𝑢 for the active control [19] is chosen as 𝑢=−𝐴𝛼𝑦−𝑓(𝑥)+𝛼𝐵𝑦+𝑔𝑦−𝑃𝐾𝑒, (5.4) where 𝑃∈𝑅^𝑛×𝑚,𝐾∈𝑅^𝑚×𝑛, and 𝐾 is the feedback gain.

Then we obtain the error system 𝑒̇=(𝐴−𝑃𝐾)𝑒, (5.5) when the choice of 𝐾 makes the real parts of all eigenvalues of (𝐴−𝑃𝐾) negative by Routh-Hurwitz criterion, the errors 𝑒_𝑖(𝑖=1,2,…,𝑛) exponentially converge to zero as time 𝑡→∞ and the MPS between the response system (5.1) and the drive system (5.2) occurs.

Remark 5.1. For a feasible control, the feedback gain 𝐾 can also be selected by the pole placement technique [20] such that the real parts of all eigenvalues of (𝐴−𝑃𝐾) are negative, when the controllability matrix is of full rank. In addition, 𝑃 can also be adjusted feasibly.

Remark 5.2. Notice that the values of the scaling factor components 𝛼_𝑖≠0,𝑖=1,2,…,𝑛 have no effect on the controllability of the error system (5.5), because the eigenvalues of (𝐴−𝑃𝐾) are independent on the scaling factor matrix 𝛼. This implies that we can arbitrarily adjust the scaling factor 𝛼_𝑖≠0,𝑖=1,2,…,𝑛 to change the shape of the created Hopf limit circle without worrying about the robustness during control.

Now take the Rössler system (2.1) as the response system in which we want to create a stable Hopf limit circle. First, rewrite system (2.1) as follows: 𝑥̇₁=−𝑥₂−𝑥₃,𝑥̇₂=𝑥₁,𝑥̇₃=𝑎𝑥₂−𝑥²₂−𝑏𝑥₃. (5.6) Note that the system is chaotic with parameters (𝑎,𝑏)=(0.5,0.5) and the chaotic attractor is shown in Figure 1.

Next, choose the new Rössler-like system (2.2) with parameters (𝑎,𝑏,𝑐)=(0.5,1,−0.5) as the drive system, and rewrite it as follows: 𝑦₁̇=−𝑦₂−𝑦₃,𝑦₂̇=𝑦₁−0.5𝑦₃,𝑦₃̇=0.5𝑦₂−𝑦₃−𝑦²₂. (5.7) According to dynamical analysis in Section 4, its stable Hopf limit circle surrounding the equilibrium 𝑂(0,0,0) will appear, as shown in Figure 6. By the MPS scheme proposed above, the signal of the stable Hopf circle is employed to drive the Rössler system (5.6) to generate a stable Hopf circle surrounding the equilibrium 𝑂(0,0,0).

In the controller 𝑢 as stated in (5.4), we have (𝑥)=0𝐴=00−1−110000.5−0.5,𝑓−0.5𝑥²,𝐵𝑦+𝑔𝑦=−𝑦₂−𝑦₃𝑦₁−0.5𝑦₃0.5𝑦₂−𝑦₃−𝑦²₂001,𝑃=. (5.8) Let 𝐾=(𝑘₁,𝑘₂,𝑘₃), then 𝐴−𝑃𝐾=0−1−1100−𝑘₁0.5−𝑘₂−0.5−𝑘₃, (5.9) and the corresponding characteristic equation is 𝑓(𝜆)=𝜆³+𝑘₃+12𝜆²+1−𝑘₁𝜆+𝑘₃−𝑘₂+1=0. (5.10) According to Routh-Hurwitz criterion, the real parts of the roots of (5.10) are all negative if and only if 𝑘₃+12>0,𝑘₃−𝑘₂+1>0,𝑘₂−𝑘₁12−𝑘₃−12>0. (5.11) As long as the feedback gain 𝐾=(𝑘₁,𝑘₂,𝑘₃) satisfying condition (5.11) is chosen, the MPS between the response system (5.6) and the drive system (5.7) will be achieved according to the analysis above, that is, we can successfully generate a stable Hopf limit circle surrounding 𝑂(0,0,0) in Rössler system (5.6) and the scaling factor 𝛼_𝑖≠0,𝑖=1,2,3, can be arbitrarily adjusted to change the shape of the created Hopf circle.

Case 1. Choose 𝛼= diag(1,2,3) and 𝐾=(0,1,1). It is easy to verify that the feedback gain 𝐾=(𝑘₁,𝑘₂,𝑘₃) satisfies condition (5.11). The time evolution of the errors is shown in Figure 7. A stable Hopf limit circle surrounding 𝑂(0,0,0) is created as shown in Figure 8.

Figure 7 

Synchronization errors in Case 1.

Figure 8 

The created Hopf circle in Case 1.

Case 2. Choose 𝛼= diag(−1,−1,−1) and 𝐾=(−1,2,3). It is easy to verify that the feedback gain 𝐾=(𝑘₁,𝑘₂,𝑘₃) satisfies condition (5.11). The time evolution of the errors is shown in Figure 9. Another stable Hopf limit circle surrounding 𝑂(0,0,0) is created as shown in Figure 10.

Figure 9 

Synchronization errors in Case 2.

Figure 10 

The created Hopf circle in Case 2.

The above numerical simulation results are presented to illustrate the effectiveness of the MPS-based anticontrol of Hopf circles. Furthermore, the feedback gain 𝐾=(𝑘₁,𝑘₂,𝑘₃) can change the speed of convergence of the error system, and the scaling factor 𝛼_𝑖≠0,𝑖=1,2,3, can be adjusted to change the shape and size of the created Hopf limit circle.

6. Conclusions

In this paper, a new Rössler-like system has been proposed and its dynamical behaviors are analyzed. By choosing an appropriate bifurcation parameter, a Hopf bifurcation occurs at the equilibrium point in this system when the bifurcation parameter exceeds a critical value. The direction of the Hopf bifurcation and the stability of bifurcating periodic solutions are analyzed in detail. Further, a stable Hopf limit circle is generated surrounding the equilibrium point in the response system (the original Rössler system) via a modified projective synchronization. Numerical simulations illustrated the effectiveness of the anticontrol of Hopf limit circles. There are still some unknown dynamical behaviors such as heteroclinic (homoclinic) orbits about this system, as well as chaotic control to it, which deserve to be further investigated.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (No. 610741 023), the Specialized Research Fund for the Doctoral Program of Higher Education of China (No. 20093401120001), the Natural Science Foundation of Anhui Province (No. 11040606M12) and the Natural Science Foundation of Anhui Education Bureau (No. KJ2010A035), and the 211 Project of Anhui University (No. KJJQ1102).