# Control of Hopf Bifurcation in Autonomous System Based on Washout Filter

Authors: Wenju Du; Yandong Chu; Jiangang Zhang; Yingxiang Chang; Jianning Yu; Xinlei An
Journal: Journal of Applied Mathematics
 (2013)

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## 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 equilibriumE0 undergoes a controllable Hopf bifurcation, and by adjusting the controller parameters, we delayed Hopf bifurcation phenomenon of the equilibrium E+. 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.


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## Body

## 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: (1)x˙=a(y-x),y˙=a(bx-xz),z˙=-cz+xy,
where (x,y,z)∈R3 are the state variables and (a,b,c)∈R3 are real constants. The system (1) has a chaotic attractor when the real constants a=1, b=1, and c=0.08, as shown in Figures 1(a)–1(d).Phase trajectory in 3-D space and various projections of the chaotic attractor.
(a)
(b)
(c)
(d)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 λ1=0.0295, λ2=0, and λ3=-1.1092 as shown in Figure 2(d) and the Lyapunov dimension DKY=2.02664. The time history, frequency spectrum, and Poincaré map in x-y{z=1.3} plane of the chaotic attractor are shown in Figures 2(a), 2(b), and 2(c), respectively.(a) Time history, (b) frequency spectrum, (c) Poincaré map inx-y{z=1.3} plane, and (d) Lyapunov exponent spectrum of chaotic system (1).
(a)
(b)
(c)
(d)The system possesses three equilibriaE0=(0,0,0) and E±=(±bc,±bc,b) if bc>0. However, the Hopf bifurcation at the equilibrium E0 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 E+. The system (1) has a transversal Hopf point at E+ for c=a(b-1)/(b+1)>0, a>0, b>0. And the equilibrium E+ is locally asymptotically stable in the parameter space (a,b,c)∈υ, υ={(a,b,c):a>0,b>0,c>a(b-1)/(b+1)}.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:
(2)u=k1(y-mv)+k2(y-mv)3,
where k1, k2 are the control gain vectors and m is the washout-filter time constant, which satisfied m>0 and u(0)=0.This paper is organized as follows. In Section2, 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 E0,E+, 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 l1 and l2, respectively.Consider the differential equation(3)x′=f(x,ζ),
where x∈R4, ζ∈Rm are, respectively, vectors representing phase variables and control parameters. Assume that f is of class C∞ in R4×Rm. Suppose that (3) has an equilibrium point x=x0 at ζ=ζ0, and denoting the variable x-x0 also by x, write
(4)F(x)=f(x,ζ0),
as
(5)F(x)=Ax+12B(x,x)+16C(x,x,x)+124D(x,x,x,x)+1120E(x,x,x,x,x)+O(∥x∥6),
where A=fx(0,ζ0), and for i=1,2,3,4,
(6)Bi(x,y)=∑j,k=14∂2Fi(ξ)∂ξj∂ξk|ξ=0xjyk,Ci(x,y,z)=∑j,k,l=14∂3Fi(ξ)∂ξj∂ξk∂ξl|ξ=0xjykzl,
and so on for Di and Ei.Suppose that(x0,ζ0) is an equilibrium point of (3), where the Jacobian matrix A has a pair of purely imaginary eigenvalues λ2,3=±iω0,(ω0>0), and admits no other eigenvalues with zero real part. Let Tc be the generalized eigenspace of A corresponding to λ2,3. By this, the largest subspace invariant by A on which the eigenvalues are λ2,3 is meant.Letp,q∈C4 be vectors such that
(7)Aq=iω0q,ATp=-iω0p,〈p,q〉=∑i=14p-iqi=1,
where AT is the transpose of the matrix A. Any vector y∈Tc can be represented as y=ωq+ω-q-, where ω=〈p,y〉∈C. The two-dimensional center manifold associated with the eigenvalues λ2,3=±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→R4 has a Taylor expansion of the form
(8)H(ω,ω-)=ωq+ω-q-+∑2≤j+k≤51j!k!hjkωjω-k+O(|ω|6),
with hjk∈C4 and hjk=h-kj. Substituting this expression into (3), we obtain the following differential equation:
(9)Hωω′+Hω-ω-′=F(H(ω,ω-)),
where F is given by (4). The complex vectors hjk are obtained solving the system of linear equations defined by the coefficients of (9), taking into account the coefficients of F, so that system (9), on the chart ω for a central manifold, writes as follows:
(10)ω′=iω0ω+12G21ω|ω|2+112G32ω|ω|4+O(|ω|6),
with Gjk∈C.The first Lyapunov coefficientl1 is defined by
(11)l1=12ReG21,
where G21=〈p,H21〉, H21=C(q,q,q-)+B(q-,h20)+2B(q,h11), h20=(2iω0I4-A)-1B(q,q), h11=-A-1B(q,q-), and I4 are unit 4×4 matrices.The second Lyapunov coefficient is defined by(12)l2=112ReG32,
where G32=〈p,H32〉,
(13)H32=6B(h11,h21)+B(h-20,h30)+3B(h-21,h-20)+3B(q,h22)+2B(q-,h31)+6C(q,h11,h11)+3C(q,h-20,h20)+3C(q,q,h-21)+6C(q,q-,h21)+6C(q-,h20,h11)+C(q-,q-,h30)+D(q,q,q,h-20)+6D(q,q,q-,h11)+3D(q,q-,q-,h20)+E(q,q,q,q-,q-)-6G21h21-3G-21h21.
The complex vector h21 can be found by solving the nonsingular n+1-dimensional system
(14)(iω0I4-Aqp-0)(h21s)=(H21-G21q0),
with the condition 〈p,h21〉=0,
(15)h30=(3iω0I4-A)-1[3B(q,h20)+C(q,q,q)],h31=(2iω0I4-A)-1[3B(q,h21)+B(q-,h30)+3B(h20,h11)+3C(q,q,h11)+3C(q,q-,h20)+D(q,q,q,q-)-3G21h20],h22=-A-1[(h-20,h20)D(q,q,q,q-)+4C(q,q-,h11)+C(q-,q-,h20)+C(q,q,h-20)+2B(h11,h11)+2B(q,h-21)+2B(q-,h21)+B(h-20,h20)].A Hopf point(x0,ζ0) of system (3) is an equilibrium point where, the Jacobian matrix A has a pair of purely imaginary eigenvalues λ2,3=±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 (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 withl1≠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: ω′=(η+iω0)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.A Hopf point of codimension 2 is a Hopf point, wherel1 vanishes. It is called transversal if l1=0 and η=0 have transversal intersections, where η=η(μ) is the real part of the critical eigenvalues. In a neighborhood of a transversal Hopf point of codimension 2 with l2≠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 ω′=(η+iω0)ω+τω|ω|2+l1ω|ω|4, where η and τ are unfolding parameters.


## 3. Hopf Bifurcations atE0

In this section, we will study the stability ofE0 in the following controlled system:
(16)x˙=a(y-x),y˙=a(bx-xz)+u,z˙=-cz+xy,v˙=y-mv,
where u=k1(y-mv)+k2(y-mv)3, k1, k2 are the control gain vectors, and m is the washout-filter time constant. Obviously, the controller u keeps the equilibrium structure and does not change the divergence of the dynamic system (1).The Jacobian matrix of the system (16) at E0 is given by
(17)A=(-aa00abk10-mk100-c0010-m).
The characteristic polynomial of the Jacobian matrix of system (16) at E0 has the form
(18)p(λ)=(λ+c)[λ3+(a+m-k1)λ2+(am-ak1-a2b)λ+a2bm].
If c>0, a+m-k1>0, a2bm<0, and
(19)b<b0=m2+k12-2mk1+am-ak1a2+2am-ak1,
then the equilibrium E0 is asymptotically stable. If b>b0, the equilibrium E0 is unstable.The equationb=b0 in (19) gives the equation of the Hopf hypersurfaces in the parameter space (a,m,k1). This equation will be used in this Section in the study of Hopf bifurcations which occur at the equilibria E0 of system (16).Then, using the notion of the previous section, the multilinear symmetric functions corresponding tof can be written as
(20)B(x,y)=(0,-a(x1y3+x3y1),x1y2+x2y1,0)T,C(x,y,z)=(0,6k2(x2y2z2-m3x4y4z4+m2x2y4z4+m2x4y2z4+m2x4y4z2-mx2y2z4-mx2y4z2-mx4y2z2m3x4y4z4),0,0)T.The eigenvalues ofA are
(21)λ1,2=±iω0=±iam-ak1-a2b,λ3=-c,λ4=-(a+m-k1).
And from (7), one has
(22)q=(a(am+ω02)+aω0(a-m)ia2+ω02,m+ω0i,0,1)T,p=(p1+p2i,p3+p4i,0,p5i)T,
where
(23)δ1=(a2+ω02)(m2-ω02+mk1)-(2k1m+ω02)(am+ω02)-ω02(m-k1)(a-m),δ2=2mω0(a2+ω02)-ω0(m-k1)(am+ω02)-ω0(a-m)(2k1m+ω02),p1=(a2+ω02)[δ2ω0(m-k1)-δ1(2k1m-ω02)]a(δ12+δ22),p3=(a2+ω02)(δ1m+δ2ω0)a(δ12+δ22),p2=(a2+ω02)[-δ2(2k1m-ω02)-δ1ω0(m-k1)]a(δ12+δ22),p4=(a2+ω02)(δ1ω0-δ2m)a(δ12+δ22),p5=mk1(a2+ω02)(δ1+δ2)a(δ12+δ22),B(q,q)=(0,2a[(aω02-am2-2mω02)+(m2ω0+2amω0-ω03)i]a2+ω02,0,02a[(aω02-am2-2mω02)+(m2ω0+2amω0-ω03)i]a2+ω02)T,B(q,q-)=(0,2a(m2+ω02)a2+ω02,0,0)T,h11=(-2a(m2+ω02)b(a2+ω02),-2a(m2+ω02)b(a2+ω02),0,-2a(m2+ω02)bm(a2+ω02))T,h20=(2iω0I3-A)-1B(q,q)=(h1,h2,0,h3)T,
where
(24)δ3=(ci-2ω0)(2amω0-8ω03-2ak1ω0-4k1ω02i-2a2bω0+4mω02i+4aω02i+a2bmi),δ4=a(-2ω0+mi)(8ω03-4k1ω02i+2ak1ω0i+2ak1ω0-2amω0+2a2bω0+4mω02i+4aω02i+a2bmi),δ5=a2b2m2+4a2b2ω02+8a2bk1ω02-8a2bk1mω02+8a2bm2ω02+32a2bω04+4a2k12ω02-8a2k1mω02+4a2m2ω02+16a2ω02+16k12ω04-32k1mω04+16m2ω04+64ω06,h1=(2aδ4[(aω02-am2-2mω02)+(m2ω0+2amω0-ω03)i])×(δ5(a2+ω02))-1,h2=([(aω02-am2-2mω02)+(m2ω0+2amω0-ω03)i]2(a+ω0)(c+2ω0i)(m+2ω0i)×[(aω02-am2-2mω02)+(m2ω0+2amω0-ω03)i])×(δ3(a2+ω02))-1,h3=(δ3(a2+ω02))-1([(aω02-am2-2mω02)+(m2ω0+2amω0-ω03)i]2(c+2ω0i)(a+2ω0i)×[(aω02-am2-2mω02)+(m2ω0+2amω0-ω03)i])×(δ3(a2+ω02))-1.
One also has
(25)B(q,h11)=((b(a2+ω02)2)-1)0,0,(×(b(a2+ω02)2)-1(-2a(m2+ω02)×[aω02+m2a2+mω02+ω0(2a2+ω02-am)i])×(b(a2+ω02)2)-1),0)T,B(q-,h20)=(0,0,(×(a2+ω02)-1(ah2[(am+ω02)-ω0(a-m)i]+h1(a2+ω02)(m-ω0i)ah2[(am+ω02)-ω0(a-m)i])×(a2+ω02)-1(ah2[(am+ω02)-ω0(a-m)i]),0)T,C(q,q,q-)=(0,h4+h5i,0,0)T,H21=(0,h4+h5i,h6+h7i,0)T,G21=(h4p3-h5p4)+(h5p3+h4p4)i,
where
(26)h4=6mk2[(m+1)(ω02+3m2)-(3m2+2ω02)],h5=6k2ω03,h6=ab(am+ω02)+h1bm(a2+ω02)-4a(m2+ω02)[a(am+ω02)+m(a2+ω02)],h7=-abω0(a-m)-h1bω0(a2+ω02)-4aω0(m2+ω02)[a(am+ω02)+m(a2+ω02)].Now, consider the family of differential equation (16) regarded as dependent on the parameter b. The real part ξ of the pair of complex eigenvalues at the critical parameter b=b0 verifies
(27)ξ′(b0)=Re〈p,dAdb|b=b0q〉=p3(am+ω03)-p4ω0(a-m).
If ξ′(b0)≠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 E0 is given by
(28)l1=12(h4p3-h5p4).
If l1 is different from zero and the transversality condition at the Hopf point holds, then system (16) has a transversal Hopf point at E0.Let the damping coefficientm=0.5. The washout filter is designed as
(29)u=k1(y-0.5v)+k2(y-0.5v)3.
If 0<k1<1, then the system (16) has a transversal Hopf point at E0, when a=1, c=0.08. More specifically, if k2>0, there exists a stable periodic orbit near the unstable equilibrium point E0; if k2<0, there exists an unstable periodic orbit near the asymptotically stable equilibrium point E0.When the parametersa=1 and c=0.08 are fixed, while parameter b 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 b is depicted in Figure 3(a). Figure 3(b) shows the corresponding Lyapunov exponent spectrum versus the increasing b. As b increases, system (1) is undergoing the following dynamical routes:(1)
if-0.5<b≤0.95, the system is stable,(2)
if0.95<b≤31, the system is chaotic. But there are two periodic windows in the chaotic band,(3)
if31<b≤33.4, there is a reverse period-three bifurcation route with a flip bifurcation,(4)
if33.4<b≤59.2, the system is chaotic. But there are several periodic windows in the chaotic band,(5)
if59.2<b≤70, there is a very long reverse period-doubling bifurcation window.Nonlinear dynamics of system (1) for specific values a=1, c=0.08 versus the control parameter b. (a) Bifurcation diagram of y. (b) Lyapunov exponent spectrum.
(a)
(b)With the analysis performed here, one can find that the Hopf bifurcation at the equilibriumE0 does not occur when b∈[-0.5,70]. We design a control laws such that our feedback system (16) undergoes a controllable Hopf bifurcation when the parameter b=-0.175, as shown in Figures 4–7.Figure 4
The stable domain on the parameter plane(a,b) with c=0.08, and k1=0.8, k2=0.5.Time history and phase diagram of system (16) with a=1, c=0.08, m=0.5, k1=0.8, k2=0.5, and b=-0.2.
(a)
(b)Time history and phase diagram of system (16) with a=1, c=0.08, m=0.5, k1=0.8, k2=0.5, and b=-0.175.
(a)
(b)Time history and phase diagram of system (16) with a=1, c=0.08, m=0.5, k1=0.8, k2=0.5, and b=-0.15.
(a)
(b)According to Routh-Hurwitz criterion, if and only if (30) is satisfied, then the equilibrium E0 is asymptotically stable
(30)Δ1=p0>0,Δ2=|p01p2p1|>0,Δ3=|p010p2p1p00p3p2|>0,Δ4=p3Δ3>0,
where (31)p0=a+m-k1,p1=am-ak1-a2b+ac+ma-ck1,p2=a2bm+amc-ack1-a2bc,p3=a2bmc.
If Δ3=0, Δi>0, i=1,2,4, then the characteristic polynomial of the Jacobian matrix of system (16) at E0 with one pair of conjugate nonzero purely imaginary eigenvalues and the real part of other characteristic roots are less than 0.LetΔ1=0, let Δ2=0, let Δ3=0, and let Δ4=0, using MATLAB to draw the stability region on the parameter plane a-b, as shown in Figure 4. In the figure, the symbol Li, i=1,2,3,4, represents Δi=0, i=1,2,3,4. And in the region (I): Δ1>0, Δ2>0, Δ3>0, and Δ4>0, in the region (II): Δ1>0, Δ2>0, Δ3>0, and Δ4<0, in the region (III): Δ1>0, Δ2<0, Δ3>0, and Δ4<0, in the region (IV): Δ1>0, Δ2<0, Δ3>0, and Δ4>0, in the region (V): Δ1<0, Δ2<0, Δ3<0, and Δ4<0, in the region (VI): Δ1<0, Δ2>0, Δ3>0, and Δ4>0, in the region (VII): Δ1>0, Δ2<0, Δ3<0, and Δ4<0, in the region (VIII): Δ1<0, Δ2>0, Δ3<0, and Δ4<0, in the region (IX): Δ1<0, Δ2>0, Δ3<0, and Δ4>0, and in the region (X): Δ1>0, Δ2>0, Δ3<0, and Δ4<0. All the points are stable in the region (I).Next, we combined with Figure4 and let a=1, c=0.08, k1=0.8, and k2=0.5; then, we can compute the Hopf bifurcation value b0=-0.175. The equilibrium is stable when b=-0.2<b0 and unstable when b=-0.15>b0, as shown in Figures 5 and 7, respectively. The system (16) has a transversal Hopf point at E0 when b=b0=-0.175 as shown in Figure 6. From the formulas in previous sections, we have l1=-2.8234<0,ξ′(b0)=-1.0719<0. Thus, the periodic solution bifurcating from E0 is supercritical and stable.


## 4. Hopf Bifurcations atE+

In this section, we will study the stability ofE+ in the controlled system (16). The Jacobian matrix of the system (16) at E+ is given by
(32)A=(-aa000k1-bc-mk1bcbc-c0010-m).
The characteristic polynomial of the Jacobian matrix of system (16) at E+ has the form
(33)p(λ)=λ4+Aλ3+Bλ2+Cλ+D,
where
(34)A=a+m+c-k1,B=abc+mc-ck1+am+ac-ak,C=a2bc-ack1+mac+mabc+abc,D=2a2bcm.
If A>0, B>0, C>0, D>0, and
(35)C(C-AB)+A2D=0,
then system (16) has a transversal Hopf point at E+.According to Dias et al. [11], the equilibrium E+ is locally asymptotically stable when (a,b,c)∈υ. In this section, the washout filter is designed as
(36)u=k1(y-0.5v)+k2(y-0.5v)3,
to realize Hopf bifurcation at E+ in the parameter space (a,b,c)∈υ. We fixed a=1, b=2, and c=0.6998>a(b-1)/(b+1) and then study the Hopf bifurcation at E+ in the controlled system (16). Assuming that the controlled system (16) undergoes a Hopf bifurcation at the equilibrium E+, then (35) is satisfied. By solving (35), we can get k1=0.2.Then, using the notion of the previous section, the multilinear symmetric functions corresponding tof can be written as
(37)B(x,y)=(0,-x1y3-x3y1,x1y2+x2y1,0)T,C(x,y,z)=(0,6k2(x2y2z2-m3x4y4z4+m2x2y4z4+m2x4y2z4+m2x4y4z2-mx2y2z4-mx2y4z2-mx4y2z42m3x4y4z4),0,0)T.The eigenvalues ofA are
(38)λ1,2=±iω0=±1.3587i,λ3=-0.5,λ4=-1.3296.
And from (7), one has
(39)q=(-0.8311-0.2364i,0.5+1.3587i,1.26+1.68i,1)T,p=(0.0689+0.2062i,-0.25+0.4503i,0.2987+0.54i,0.1192+0.0221i)T,B(q,q)=(0,10.6+3.5i,-0.1628-0.452i,0)T,B(q,q-)=(0,1.55,-0.82,0)T,h11=(0.7975,0.7975,1.55,1.595)T,h20=(-0.1567+0.5445i,-1.627+0.1213i,0.2712+0.7712i,-0.0644+0.5907i)T,B(q,h11)=(0,0.2-0.895i,-0.24+0.96i,0)T,B(q-,h20)=(0,-0.28-0.27i,-0.779+0.78i,0)T,C(q,q,q-)=(0,5.525k2+5.194k2i,0,0)T,H21=(0,(5.25k2+0.12)+(5.194k2-2.06)i,-1.259+2.7i,0(5.25k2+0.12))T,G21=-(3.721k2+0.936)+(1.2k2+0.699)i.Now, consider the family of differential equation (16) regarded as dependent on the parameter c. The real part ξ of the pair of complex eigenvalues at the critical parameter c=c0 verifies
(40)ξ′(c0)=Re〈p,dAdb|c=c0q〉=0.3945>0.
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 E+ is given by
(41)l1=-1.8605k2-0.468.
If k2≠-0.2515, then l1≠0 and system (16) has a transversal Hopf point at E+ for a=1, b=2, and c=0.6998. More specifically, if k2>-0.2515, there exists a stable periodic orbit near the unstable equilibrium point E+; if k2<-0.2515, there exists a unstable periodic orbit near the asymptotically stable equilibrium point E+.The sign of the first Lyapunov coefficient is determined byk2. Observe that the first Lyapunov coefficient vanishes when k2=-0.2515. In the following theorem, we study the sign of the second Lyapunov coefficient where the first coefficient vanishes.Define(42)Ω={(a,b,c,m,k1,k2)∣a=1,b=2,c=0.5,m=0.6998,k1=0.2,k2=-0.2515}.
Considering the system (16) for the parameters (a,b,c,m,k1,k2)∈Ω, one has
(43)C(q,q,q-)=(0,-1.3209-1.3063i,0,0)T,H21=(0,-1.2009-3.3663i,-1.259+2.7i,0)T,G21=1.0008i,h30=(-0.2201+0.1447i,-0.9726-0.9996i,-0.0057+0.8692i,-0.2083+0.167i)T,h21=(-4.4617+0.4572i,-4.8018-4.9892i,-4.6476+2.7212i,-3.613+2.1395i)T,h31=(-0.5359-0.1395i,-1.6454-1.16i,-1.1993+1.297i,-0.4584-0.1371i)T,h22=(2.6732+5.18i,2.6732+5.18i,-9.5872-28.1i,5.3464+10.36i)T,G32=-3.87607-7.1839i,l2=112ReG32=-0.323005834.Using these calculations, we prove the next theorem.Theorem 3.
If(a,b,c,m,k1,k2)∈Ω, consider the six-parameter family of differential equations (16). The second Lyapunov coefficient associated with the equilibrium E+ is given by
(44)l2=-0.323005834.
As l2≠0, then system (16) has a transversal Hopf point of codimension 2 at E+ for parameters in Ω.Next, we shall give a numerical example of system (16). Let the damping coefficient m=0.5, k1=0.2, and k2=0.5; the washout filter is designed as
(45)u=0.2(y-0.5v)+0.5(y-0.5v)3.
The equilibrium E+ is locally asymptotically stable in the uncontrolled system (1) if we fixed a=1, b=2, and c=0.6998. Compared with the uncontrolled system (1), the controlled system (16) has a transversal Hopf point at E+ under this parameter region, as shown in Figures 8 and 9, respectively.Time history and phase diagram of system (1) with a=1, b=2, and c=0.6998.
(a)
(b)Time history and phase diagram of system (16) with a=1, b=2, and c=0.6998.
(a)
(b)When the parametersa=1 and b=2 are fixed, while parameter c 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 c is depicted in Figures 10(a) and 11(a), respectively. As shown in Figure 10(a), while c increases, the system (1) is undergoing the following dynamical routes. When c∈[0,0.0625), there is a period-doubling bifurcation window. And the system is chaotic when c∈[0.0625,0.3334). Obviously, the uncontrolled system (1) undergoes a Hopf bifurcation at the equilibrium E+ when c=0.3334. When c∈[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 c=0.6998, as shown in Figure 11. In Figure 11(a), the system is chaotic when c∈[0,0.575). When c∈[0.575,0.68), the system is period-one orbits. The system is chaotic when c∈[0.68,0.6998). And when c∈[0.6998,1), the system is period-one orbits. Figures 10(b) and 11(b) show the corresponding Lyapunov exponent spectrum versus the increasing b, respectively. With the analysis performed here one can find that we delayed the Hopf bifurcation.Nonlinear dynamics of system (1) for specific values a=1, b=2 versus the control parameter c. (a) Bifurcation diagram of y. (b) Lyapunov exponent spectrum.
(a)
(b)Nonlinear dynamics of controlled system (16) for specific values a=1, b=2 versus the control parameter c. (a) Bifurcation diagram of y. (b) Lyapunov exponent spectrum.
(a)
(b)Lettinga=1, b=2, k1=0.2, and k2=0.5, we can compute the Hopf bifurcation value c0=0.6998. The equilibrium is stable when c=0.8>c0 and unstable when c=0.65<c0, as shown in Figures 12 and 13, respectively. From the formulas in the previous section, we have l1=-1.39825<0. Thus, the periodic solution bifurcating from E+ is supercritical and stable.Time history and phase diagram of system (16) with a=1, b=2, m=0.5, k1=0.2, k2=0.5, and c=0.8.
(a)
(b)Time history and phase diagram of system (16) with a=1, b=2, m=0.5, k1=0.2, k2=0.5, and c=0.65.
(a)
(b)


## 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.Figure 14
Circuit diagram for realizing the chaotic attractor of system.Experimental observations of the chaotic attractor in different planes.
(a)
Projection ony-z plane
(b)
Projection onx-z plane
(c)
Projection onx-y planeFigure 16
Circuit diagram for realizing the chaotic attractor of system (inside the black-dotted box is the controller).Experimental observations of the time history and phase diagram of system (16) with a=1, c=0.08, m=0.5, k1=0.8, k2=0.5, and b=-0.2.
(a)
The time history
(b)
The phase diagramExperimental observations of the time history and phase diagram of system (16) with a=1, c=0.08, m=0.5, k1=0.8, k2=0.5, and b=-0.175.
(a)
The time history
(b)
The phase diagramExperimental observations of the time history and phase diagram of system (16) with a=1, c=0.08, m=0.5, k1=0.8, k2=0.5, and b=-0.15.
(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 equilibriumE0 undergoes a controllable Hopf bifurcation, and by adjusting the controller parameters, we delayed Hopf bifurcation phenomenon of the equilibrium E+. 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.



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