#### Abstract

In order to further understand a Lorenz-like system, we study the stability of the equilibrium points and the existence of Hopf bifurcation by center manifold theorem and normal form theory. More precisely, we designed a washout controller such that the equilibrium undergoes a controllable Hopf bifurcation, and by adjusting the controller parameters, we delayed Hopf bifurcation phenomenon of the equilibrium . Besides, numerical simulation is given to illustrate the theoretical analysis. Finally, two possible electronic circuits are given to realize the uncontrolled and the controlled systems.

#### 1. Introduction

Over the past decades, as we have seen, researchers have paid a great attention to the control of nonlinear dynamical systems exhibiting Hopf bifurcation phenomena, because the presence of bifurcation is very important in many physical, biological, and chemical nonlinear systems [1–3]. Chen et al. [4, 5] created a certain bifurcation at a desired location with preferred properties by appropriate control. And they developed a washout-filter-aided dynamic feedback control laws for the creation of Hopf bifurcations. Wei and Yang [6] proposed nonlinear control scheme to the system and the controlled system can exhibit codimensions one, two, and three Hopf bifurcations in a much larger parameter regain. An et al. [7] based on washout filter designed a state feedback controller for Hopf bifurcation of the nonlinear systems. Ma et al. [8] designed a bifurcation controller using the method of washout filter so as to control the dynamic bifurcation in power system. Sotomayor et al. [9] use the projection method described in [10] to calculate the first and second Lyapunov coefficients associated with Hopf bifurcations of the Watt governor system, and it was extended to the calculation of the third and fourth Lyapunov coefficients.

Recently, Dias et al. [11] studied the existence of singularly degenerate heteroclinic cycles for a suitable choice of the parameters in the following Lorenz-like system: where are the state variables and are real constants. The system (1) has a chaotic attractor when the real constants , , and , as shown in Figures 1(a)–1(d).

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Moreover, the dynamics of the system (1) can be characterized with its Lyapunov exponents which are computed numerically by Wolf algorithm proposed in [12], where the Lyapunov exponents , , and as shown in Figure 2(d) and the Lyapunov dimension . The time history, frequency spectrum, and Poincaré map in plane of the chaotic attractor are shown in Figures 2(a), 2(b), and 2(c), respectively.

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The system possesses three equilibria and if . However, the Hopf bifurcation at the equilibrium does not occur. Dias et al. [11] studied the existence of singularly degenerated heteroclinic cycles for a suitable choice of the parameters at the equilibrium . The system (1) has a transversal Hopf point at for , , . And the equilibrium is locally asymptotically stable in the parameter space , .

In this work, we will design a control laws such that our feedback system undergoes a controllable Hopf bifurcation. To accomplish the control of Hopf bifurcation in the system (1), we design the controller in the following structure: where , are the control gain vectors and is the washout-filter time constant, which satisfied and .

This paper is organized as follows. In Section 2, a brief review of the methods used to study codimensions one and two Hopf bifurcations is presented. Through a linear analysis of system (16), we obtain the Hopf conditions for , and the main results of this paper in Sections 3 and 4, respectively. In Section 5, two possible electronic circuits are given to realize the uncontrolled system and the controlled system. Finally, in Section 6 conclusion is given.

#### 2. Outline of the Hopf Bifurcation Methods

This section is a review of the projection method described in [9–11, 13] for the calculation of the first Lyapunov coefficient and second Lyapunov coefficient associated with Hopf bifurcation, denoted by and , respectively.

Consider the differential equation where , are, respectively, vectors representing phase variables and control parameters. Assume that is of class in . Suppose that (3) has an equilibrium point at , and denoting the variable also by , write as where , and for , and so on for and .

Suppose that is an equilibrium point of (3), where the Jacobian matrix has a pair of purely imaginary eigenvalues , and admits no other eigenvalues with zero real part. Let be the generalized eigenspace of corresponding to . By this, the largest subspace invariant by on which the eigenvalues are is meant.

Let be vectors such that where is the transpose of the matrix . Any vector can be represented as , where . The two-dimensional center manifold associated with the eigenvalues can be parameterized by the variables and by means of an immersion of the form , where has a Taylor expansion of the form with and . Substituting this expression into (3), we obtain the following differential equation: where is given by (4). The complex vectors are obtained solving the system of linear equations defined by the coefficients of (9), taking into account the coefficients of , so that system (9), on the chart for a central manifold, writes as follows: with .

The first Lyapunov coefficient is defined by where , , , , and are unit matrices.

The second Lyapunov coefficient is defined by where , The complex vector can be found by solving the nonsingular -dimensional system with the condition ,

A Hopf point of system (3) is an equilibrium point where, the Jacobian matrix has a pair of purely imaginary eigenvalues , , and the other eigenvalue . 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 the 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 , the dynamic behavior of the system (3), reduced to the family of parameter-dependent continuations of the center manifold, is orbitally topologically equivalent to the following complex normal form: , where . , , and are real functions having derivatives of arbitrary higher order, which are continuations of 0, and the first Lyapunov coefficient at the Hopf point. When , one family of stable (unstable) periodic orbits can be found on this family of manifolds, shrinking to an equilibrium point at the Hopf point.

A Hopf point of codimension 2 is a Hopf point, where vanishes. It is called transversal if and have transversal intersections, where is the real part of the critical eigenvalues. In a neighborhood of a transversal Hopf point of codimension 2 with , the dynamic behavior of the system (3), reduced to the family of parameter-dependent continuations of the center manifold, is orbitally topologically equivalent to , where and are unfolding parameters.

#### 3. Hopf Bifurcations at

In this section, we will study the stability of in the following controlled system: where , , are the control gain vectors, and is the washout-filter time constant. Obviously, the controller keeps the equilibrium structure and does not change the divergence of the dynamic system (1).

The Jacobian matrix of the system (16) at is given by The characteristic polynomial of the Jacobian matrix of system (16) at has the form If , , , and then the equilibrium is asymptotically stable. If , the equilibrium is unstable.

The equation in (19) gives the equation of the Hopf hypersurfaces in the parameter space . This equation will be used in this Section in the study of Hopf bifurcations which occur at the equilibria of system (16).

Then, using the notion of the previous section, the multilinear symmetric functions corresponding to can be written as

The eigenvalues of are And from (7), one has where where One also has where

Now, consider the family of differential equation (16) regarded as dependent on the parameter . The real part of the pair of complex eigenvalues at the critical parameter verifies If , the transversality condition at the Hopf point holds.

Using these calculations, we prove the next theorem.

Theorem 1. *Consider the six-parameter family of differential equations (16). The first Lyapunov coefficient associated with the equilibrium is given by
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If is different from zero and the transversality condition at the Hopf point holds, then system (16) has a transversal Hopf point at .*

Let the damping coefficient . The washout filter is designed as If , then the system (16) has a transversal Hopf point at , when , . More specifically, if , there exists a stable periodic orbit near the unstable equilibrium point ; if , there exists an unstable periodic orbit near the asymptotically stable equilibrium point .

When the parameters and are fixed, while parameter is varied in the interval , some different dynamical behaviors of system (1) are obtained. The bifurcation diagram of system (1) in terms of the parameter is depicted in Figure 3(a). Figure 3(b) shows the corresponding Lyapunov exponent spectrum versus the increasing . As increases, system (1) is undergoing the following dynamical routes:(1)if , the system is stable,(2)if , the system is chaotic. But there are two periodic windows in the chaotic band,(3)if , there is a reverse period-three bifurcation route with a flip bifurcation,(4)if , the system is chaotic. But there are several periodic windows in the chaotic band,(5)if , there is a very long reverse period-doubling bifurcation window.

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With the analysis performed here, one can find that the Hopf bifurcation at the equilibrium does not occur when . We design a control laws such that our feedback system (16) undergoes a controllable Hopf bifurcation when the parameter , as shown in Figures 4–7.

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According to Routh-Hurwitz criterion, if and only if (30) is satisfied, then the equilibrium is asymptotically stable where If , , , then the characteristic polynomial of the Jacobian matrix of system (16) at with one pair of conjugate nonzero purely imaginary eigenvalues and the real part of other characteristic roots are less than .

Let , let , let , and let , using MATLAB to draw the stability region on the parameter plane , as shown in Figure 4. In the figure, the symbol , , represents , . And in the region (I): , , , and , in the region (II): , , , and , in the region (III): , , , and , in the region (IV): , , , and , in the region (V): , , , and , in the region (VI): , , , and , in the region (VII): , , , and , in the region (VIII): , , , and , in the region (IX): , , , and , and in the region (X): , , , and . All the points are stable in the region (I).

Next, we combined with Figure 4 and let , , , and ; then, we can compute the Hopf bifurcation value . The equilibrium is stable when and unstable when , as shown in Figures 5 and 7, respectively. The system (16) has a transversal Hopf point at when as shown in Figure 6. From the formulas in previous sections, we have . Thus, the periodic solution bifurcating from is supercritical and stable.

#### 4. Hopf Bifurcations at

In this section, we will study the stability of in the controlled system (16). The Jacobian matrix of the system (16) at is given by The characteristic polynomial of the Jacobian matrix of system (16) at has the form where If , , , , and then system (16) has a transversal Hopf point at .

According to Dias et al. [11], the equilibrium is locally asymptotically stable when . In this section, the washout filter is designed as to realize Hopf bifurcation at in the parameter space . We fixed , , and and then study the Hopf bifurcation at in the controlled system (16). Assuming that the controlled system (16) undergoes a Hopf bifurcation at the equilibrium , then (35) is satisfied. By solving (35), we can get .

Then, using the notion of the previous section, the multilinear symmetric functions corresponding to can be written as

The eigenvalues of are And from (7), one has

Now, consider the family of differential equation (16) regarded as dependent on the parameter . The real part of the pair of complex eigenvalues at the critical parameter verifies So, the transversality condition at the Hopf point holds.

Using these calculations, we prove the next theorem.

Theorem 2. *Consider the six-parameter family of differential equations (16). The first Lyapunov coefficient associated with the equilibrium is given by
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If , then and system (16) has a transversal Hopf point at for , , and . More specifically, if , there exists a stable periodic orbit near the unstable equilibrium point ; if , there exists a unstable periodic orbit near the asymptotically stable equilibrium point .*

The sign of the first Lyapunov coefficient is determined by . Observe that the first Lyapunov coefficient vanishes when . In the following theorem, we study the sign of the second Lyapunov coefficient where the first coefficient vanishes.

Define Considering the system (16) for the parameters , one has

Using these calculations, we prove the next theorem.

Theorem 3. *If , consider the six-parameter family of differential equations (16). The second Lyapunov coefficient associated with the equilibrium is given by
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As , then system (16) has a transversal Hopf point of codimension 2 at for parameters in .*

Next, we shall give a numerical example of system (16). Let the damping coefficient , , and ; the washout filter is designed as The equilibrium is locally asymptotically stable in the uncontrolled system (1) if we fixed , , and . Compared with the uncontrolled system (1), the controlled system (16) has a transversal Hopf point at under this parameter region, as shown in Figures 8 and 9, respectively.

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When the parameters and are fixed, while parameter is varied in the interval , some different dynamical behaviors of system (1) and system (16) are obtained. The bifurcation diagram of system (1) and system (16) in terms of the parameter is depicted in Figures 10(a) and 11(a), respectively. As shown in Figure 10(a), while increases, the system (1) is undergoing the following dynamical routes. When , there is a period-doubling bifurcation window. And the system is chaotic when . Obviously, the uncontrolled system (1) undergoes a Hopf bifurcation at the equilibrium when . When , the system is period-one orbits. We design a control laws such that our controlled system (16) undergoes a Hopf bifurcation when the parameter , as shown in Figure 11. In Figure 11(a), the system is chaotic when . When , the system is period-one orbits. The system is chaotic when . And when , the system is period-one orbits. Figures 10(b) and 11(b) show the corresponding Lyapunov exponent spectrum versus the increasing , respectively. With the analysis performed here one can find that we delayed the Hopf bifurcation.

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Letting , , , and , we can compute the Hopf bifurcation value . The equilibrium is stable when and unstable when , as shown in Figures 12 and 13, respectively. From the formulas in the previous section, we have . Thus, the periodic solution bifurcating from is supercritical and stable.

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#### 5. Circuit Design for the Chaotic Attractor

In this section, two possible electronic circuits are given to realize (1) and (16), as shown in Figures 14 and 16, respectively. The first circuit includes three layers, each of which implements one equation of (1). The operational amplifiers and associated circuitry perform the basic operations of addition, subtraction, and integration. The occurrence of the chaotic attractor can be clearly seen from Figures 15(a)–15(c). By comparing them with Figures 1(b)–1(d), it can be concluded that there is a good qualitative agreement between the numerical simulation and the experimental realization. And the second circuit includes four layers, each of which implements one equation of (16). The occurrence of the phase diagram of system (16) can be clearly seen from Figures 17, 18, and 19. By comparing them with Figures 5–7, it can be concluded that there is a good qualitative agreement between the numerical simulation and the experimental realization.

(a) Projection on plane |

(b) Projection on plane |

(c) Projection on plane |

**(a) The time history**

**(b) The phase diagram**

**(a) The time history**

**(b) The phase diagram**

**(a) The time history**

**(b) The phase diagram**

#### 6. Concluding Remarks

In this paper, we consider the problem of anticontrol of Hopf bifurcations; that is, an anticontroller for a Lorenz-like system is designed with desired location and properties by appropriate controls. By the numerical analysis, we prove that Hopf bifurcation occurs when the bifurcation parameter passes through the critical value. In particular, we designed a washout controller such that the equilibrium undergoes a controllable Hopf bifurcation, and by adjusting the controller parameters, we delayed Hopf bifurcation phenomenon of the equilibrium . The proposed anticontrol scheme is effective and easy to manipulate with the aid of symbolic computation. The calculation of the first and second Lyapunov coefficients, which makes the determination of the Lyapunov stability at the equilibriums possible, can make the controlled system exhibit Hopf bifurcation in a much larger parameter region. Besides, we proposed two possible electronic circuits to realize the uncontrolled and the controlled systems. Apparently, there are more interesting problems about this chaotic system in terms of complexity, control, and synchronization, which deserve further investigation.

#### Acknowledgments

The authors express their gratitude to the referee for their valuable comments on the first version of the paper. The authors also gratefully acknowledge the support from the National Natural Science Foundation (nos. 11161027 and 61364001), the Key Project of Chinese Ministry of Education (no. 212180), the Fundamental Research Funds for the Universities of Gansu Province (no. 620023), and the Natural Science Foundation of Gansu Province (no. 1010RJZA067), Government of China.