Control of Hopf Bifurcation in Autonomous System Based on Washout Filter

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 E 0 undergoes a controllable Hopf bifurcation, and by adjusting the controller parameters, we delayed Hopf bifurcation phenomenon of the equilibriumE + . 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.


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][2][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.
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  1 = 0.0295,  2 = 0, and  3 = −1.1092as shown in Figure 2(d) and the Lyapunov dimension   = 2.02664.The time history, frequency spectrum, and Poincaré map in  − { = 1.3} plane of the chaotic attractor are shown in Figures 2(a), 2(b), and 2(c), respectively.
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  1 ,  2 are the control gain vectors and  is the washoutfilter time constant, which satisfied  > 0 and (0) = 0.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  0 ,  + , 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.

Outline of the Hopf Bifurcation Methods
This section is a review of the projection method described in [9][10][11]13] for the calculation of the first Lyapunov coefficient and second Lyapunov coefficient associated with Hopf bifurcation, denoted by  1 and  2 , respectively.
Consider the differential equation where  ∈ R 4 ,  ∈ R  are, respectively, vectors representing phase variables and control parameters.Assume that  is of class C ∞ in R 4 × R  .Suppose that (3) has an equilibrium point  =  0 at  =  0 , and denoting the variable  −  0 also by , write as  where A =   (0,  0 ), and for  = 1, 2, 3, 4, and so on for   and   .Suppose that ( 0 ,  0 ) is an equilibrium point of (3), where the Jacobian matrix A has a pair of purely imaginary eigenvalues  2,3 = ± 0 , ( 0 > 0), and admits no other eigenvalues with zero real part.Let T  be the generalized eigenspace of A corresponding to  2,3 .By this, the largest subspace invariant by A on which the eigenvalues are  2,3 is meant.
Let ,  ∈ C 4 be vectors such that where A  is the transpose of the matrix A. Any vector  ∈ T  can be represented as  =  +  , where  = ⟨, ⟩ ∈ C. The two-dimensional center manifold associated with the eigenvalues  2,3 = ± 0 can be parameterized by the variables  and  by means of an immersion of the form  = (, ), where  : C 2 → R 4 has a Taylor expansion of the form with ℎ  ∈ C 4 and ℎ  = ℎ  .Substituting this expression into (3), we obtain the following differential equation: where  is given by (4).The complex vectors ℎ  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   ∈ C.
The first Lyapunov coefficient  1 is defined by where The complex vector ℎ 21 can be found by solving the nonsingular  + 1-dimensional system with the condition ⟨, ℎ 21 ⟩ = 0, A Hopf point ( 0 ,  0 ) of system (3) is an equilibrium point where, the Jacobian matrix A has a pair of purely imaginary eigenvalues  2,3 = ± 0 , ( 0 > 0), and the other eigenvalue  1 ̸ = 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 (3) and can be continued with arbitrary high class of differentiability to the nearby parameter values.
A Hopf point is called transversal if the parameterdependent complex eigenvalues cross the imaginary axis with nonzero derivative.In a neighborhood of a transversal Hopf point with  1 ̸ = 0, 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  ∈ C. , , and  1 are real functions having derivatives of arbitrary higher order, which are continuations of 0,  0 and the first Lyapunov coefficient at the Hopf point.When  1 < 0 ( 1 > 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.
A Hopf point of codimension 2 is a Hopf point, where  1 vanishes.It is called transversal if  1 = 0 and  = 0 have transversal intersections, where  = () is the real part of the critical eigenvalues.In a neighborhood of a transversal Hopf point of codimension 2 with  2 ̸ = 0, the dynamic behavior of the system (3), reduced to the family of parameter-dependent continuations of the center manifold, is orbitally topologically equivalent to   = ( +  0 ) + || 2 +  1 || 4 , where  and  are unfolding parameters.

Hopf Bifurcations at 𝐸 0
In this section, we will study the stability of  0 in the following controlled system: where  =  1 (−V)+ 2 ( − V) 3 ,  1 ,  2 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  0 is given by The characteristic polynomial of the Jacobian matrix of system (16) at  0 has the form If  > 0,  +  −  1 > 0,  2  < 0, and then the equilibrium  0 is asymptotically stable.If  >  0 , the equilibrium  0 is unstable.
The equation  =  0 in (19) gives the equation of the Hopf hypersurfaces in the parameter space (, ,  1 ).This equation will be used in this Section in the study of Hopf bifurcations which occur at the equilibria  0 of system (16).
Then, using the notion of the previous section, the multilinear symmetric functions corresponding to  can be written as The eigenvalues of A are And from (7), one has where , where (24) 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  =  0 verifies If   ( 0 ) ̸ = 0, 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  0 is given by If  1 is different from zero and the transversality condition at the Hopf point holds, then system (16) has a transversal Hopf point at  0 .Let the damping coefficient  = 0.5.The washout filter is designed as If 0 <  1 < 1, then the system (16) has a transversal Hopf point at  0 , when  = 1,  = 0.08.More specifically, if  2 > 0, there exists a stable periodic orbit near the unstable equilibrium point  0 ; if  2 < 0, there exists an unstable periodic orbit near the asymptotically stable equilibrium point  0 .When the parameters  = 1 and  = 0.08 are fixed, while parameter  is varied in the interval [−0.5, 70], 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 −0.5 <  ≤ 0.95, the system is stable, (2) if 0.95 <  ≤ 31, the system is chaotic.But there are two periodic windows in the chaotic band, (3) if 31 <  ≤ 33.4,there is a reverse period-three bifurcation route with a flip bifurcation, (4) if 33.4 <  ≤ 59.2, the system is chaotic.But there are several periodic windows in the chaotic band, (5) if 59.2 <  ≤ 70, there is a very long reverse perioddoubling bifurcation window.
With the analysis performed here, one can find that the Hopf bifurcation at the equilibrium  0 does not occur when  ∈ [−0.5, 70].We design a control laws such that our feedback system (16) undergoes a controllable Hopf bifurcation when the parameter  = −0.175,as shown in Figures 4-7.
According to Routh-Hurwitz criterion, if and only if (30) is satisfied, then the equilibrium  0 is asymptotically stable where If Δ 3 = 0, Δ  > 0,  = 1, 2, 4, then the characteristic polynomial of the Jacobian matrix of system (16) at  0 with one pair of conjugate nonzero purely imaginary eigenvalues and the real part of other characteristic roots are less than 0.

Hopf Bifurcations at 𝐸 +
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 If  > 0,  > 0,  > 0,  > 0, 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  = 1,  = 2, and  = 0.6998 > (−1)/(+1) 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  1 = 0.2.
Then, using the notion of the previous section, the multilinear symmetric functions corresponding to  can be written as The eigenvalues of A are And from (7) Using these calculations, we prove the next theorem.
Next, we shall give a numerical example of system (16).Let the damping coefficient  = 0.5,  1 = 0.2, and  2 = 0.5; the washout filter is designed as The equilibrium  + is locally asymptotically stable in the uncontrolled system (1) if we fixed  = 1,  = 2, and  = 0.6998.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.
When the parameters  = 1 and  = 2 are fixed, while parameter  is varied in the interval [0, 1], 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  ∈ [0, 0.0625), there is a perioddoubling bifurcation window.And the system is chaotic when  ∈ [0.0625, 0.3334).Obviously, the uncontrolled system (1) undergoes a Hopf bifurcation at the equilibrium  + when  = 0.3334.When  ∈ [0.3334, 1), the system is period-one orbits.We design a control laws such that our controlled system (16) undergoes a Hopf bifurcation when the parameter  = 0.6998, as shown in Figure 11.In Figure 11(a), the system is chaotic when  ∈ [0, 0.575).When  ∈ [0.575, 0.68), the system is period-one orbits.The system is chaotic when  ∈ [0.68, 0.6998).And when  ∈ [0.6998, 1), 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.

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.

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  0 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.

Figure 1 :
Figure 1: Phase trajectory in 3-D space and various projections of the chaotic attractor.

Figure 14 :
Figure 14: Circuit diagram for realizing the chaotic attractor of system.
Figure 15: Experimental observations of the chaotic attractor in different planes.

Figure 16 :
Figure 16: Circuit diagram for realizing the chaotic attractor of system (inside the black-dotted box is the controller).