HOPF BIFURCATION CONTROL USING NONLINEAR ......International Journal of Bifurcation and Chaos, Vol. 14, No. 5 (2004) 1683–1704 c World Scientiﬁc Publishing Company HOPF BIFURCATION

International Journal of Bifurcation and Chaos, Vol. 14, No. 5 (2004) 1683–1704c© World Scientific Publishing Company

HOPF BIFURCATION CONTROL USING NONLINEARFEEDBACK WITH POLYNOMIAL FUNCTIONS

PEI YU∗Department of Applied Mathematics, The University of Western Ontario,

London, Ontario, N6A 5B7, [email protected]

GUANRONG CHENDepartment of Electronic Engineering, City University of Hong Kong,

Kowloon, Hong Kong, P.R. China

Received October 31, 2002

A general explicit formula is derived for controlling bifurcations using nonlinear state feedback.This method does not increase the dimension of the system, and can be used to either delay(or eliminate) existing bifurcations or change the stability of bifurcation solutions. The methodis then employed for Hopf bifurcation control. The Lorenz equation and Rossler system areused to illustrate the application of the approach. It is shown that a simple control can beobtained to simultaneously stabilize two symmetrical equilibria of the Lorenz system, and keepthe symmetry of Hopf bifurcations from the equilibria. For the Rossler system, a control isalso obtained to simultaneously stabilize two nonsymmetric equilibria and meanwhile stabilizepossible Hopf bifurcations from the equilibria. Computer simulation results are presented toconfirm the analytical predictions.

Keywords : Control system; Hopf bifurcation control; limit cycle; normal form; Lorenz equation;Rossler system.

1. Introduction

In the past two decades, there has been rapidlygrowing interest in bifurcation dynamics of controlsystems, including controlling and anti-controllingof bifurcations and chaos. Such bifurcation andchaos control techniques have been widely appliedto solve physical and engineering problems (e.g. see[Abed & Fu, 1987; Berns et al., 2000; Chen et al.,2001; Chen et al., 2000; Chiang et al., 1994; Guet al., 1997; Kang & Krener, 2000; Laufenberg et al.,1997; Nayfeh et al., 1996; Ono et al., 1998; Wang& Abed, 1995; Wang et al., 1997; Yu & Huseyin,1988]). The general goal of bifurcation control isto design a controller such that the bifurcationcharacteristics of a nonlinear system undergoing

bifurcation can be modified to achieve certain desir-able dynamical behavior, such as changing a Hopfbifurcation from subcritical to supercritical, elimi-nating chaotic motions, etc. Anti-control of chaos,on the other hand, is to purposefully create chaoswhen it is beneficial. Many applications have beenfound, for example, in the areas of mechanical sys-tems, fluid dynamics, biological systems and securecommunications. In engineering applications, oneoften expects to design a system to be either chaoticor nonchaotic as one wishes.

In this paper, we consider bifurcation controlusing nonlinear state feedback. A general explicitformula is derived for the control strategy, given inthe form of simple homogeneous polynomials. The

∗Author for correspondence.

1683

1684 P. Yu & G. Chen

formula keeps the equilibria of the original sys-tem unchanged. The linear part of the formulacan be used to modify the system’s linear stability,in order to eliminate or delay an existing bifurca-tion. The nonlinear part, on the other hand, canchange the stability of bifurcation solutions, for ex-ample, converting a subcritical Hopf bifurcation tosupercritical.

To that end, we will apply the general controlformula to particularly study controlling Hopf bifur-cation. To be more specific, consider the followinggeneral nonlinear system:

x = f(x, µ) , x ∈ Rn , µ ∈ R ,

f : Rn+1 → Rn ,(1)

where the dot denotes differentiation with respectto time t, x is an n-dimensional state vector while µis a scalar parameter, called bifurcation parameter.(Note that in general, one may assume that µ is anm-dimensional vector for m ≥ 1.) The function f isassumed analytic with respect to both x and µ.

Equilibrium solutions of system (1) can befound by solving the nonlinear algebraic equationf(x, µ) = 0 for an arbitrary µ, which usually yieldsmultiple solutions. Let x∗ be an equilibrium (orfixed point) of the system, i.e. f(x∗, µ) ≡ 0 for anyvalue of µ. Further, suppose that the Jacobian of thesystem evaluated at the equilibrium x∗ has eigen-values, λ1(µ), λ2(µ), . . . , λn(µ), which may be realor complex. Assume that when µ is varied, one pairof the complex conjugates, denoted as λ1,2(µ) withλ1 = λ2 = α(µ)+iω(µ), where α(µ) and ω(µ) repre-sent the real and imaginary parts of λ1,2(µ), respec-tively, moves to cross the imaginary axis at µ = µ∗such that

α(µ∗) = 0 anddα(µ∗)

dµ�= 0 . (2)

The second condition of Eq. (2) is usually called thetransversality condition, implying that the crossingof the complex conjugate pair at the imaginary axisis not tangent to the imaginary axis. Without lossof generality, one may assume that when µ is variedfrom µ < µ∗ to µ > µ∗, the λ1,2(µ) moves from theleft-half of complex plane to the right, and the re-maining eigenvalues have negative real parts in thevicinity of the critical point µ = µ∗. According tothe Hopf theory, a family of limit cycles will bifur-cate from the equilibrium x∗ at the critical pointµ∗, where the equilibrium x∗ changes its stability.

The goal of Hopf bifurcation control here is todesign a controller, given by

u = u(x;µ) , (3)

such that the original equilibrium point x∗ isunchanged, but the Hopf bifurcation point (x∗, µ∗)is moved to a new position, (x, µ) �= (x∗, µ∗).Therefore, a necessary condition for the controlleris obtained as

u(x∗;µ) = 0 , (4)

for all values of µ ∈ R, in order not to change theoriginal equilibrium x∗.

A general control formula for this task is de-rived in the next section. Section 3 outlines thegeneral strategy of Hopf bifurcation control. Thewell-known Lorenz and Rossler systems are thenconsidered in Secs. 4 and 5, respectively, to illus-trate the applicability of the new control approach.Finally, concluding remarks are given in Sec. 6.

2. A General and Explicit ControlFormula

Before discussing how to use a state feedback tocontrol Hopf bifurcation, a general formula for thecontroller that satisfies the necessary condition (4)is given. The formula is not restricted to Hopf bi-furcation; it can be applied to study the controlof other singularities or bifurcations such as dou-ble Hopf and double zero bifurcations. Thus, inthis general discussion we consider a more generalsystem, given by

x = f(x,µ) , x ∈ Rn ,

µ ∈ Rm , f : Rn+m → Rn ,(5)

where x is an n-dimensional state vector while µ isan m-dimensional parameter vector.

Suppose system (5) has k equilibria, determinedfrom the equation f(x,µ) = 0, given by

x∗i (µ)=(x∗

1i, x∗2i, . . . , x

∗ni) , i = 1, 2, . . . , k , (6)

satisfying f(x∗i ,µ) = 0 for i = 1, 2, . . . , k. A gen-

eral nonlinear state feedback control is applied sothat system (5) becomes

x = f(x, µ) + u(x,µ) . (7)

In order for the controlled system (7) to keep allthe original k equilibria unchanged under the con-trol u, it requires that the following conditions besatisfied:

u(x∗i ,µ) ≡ (u1, u2, . . . , un)T = 0 (8)

for i = 1, 2, . . . , k.

Hopf Bifurcation Control Using Nonlinear Feedback with Polynomial Functions 1685

A general formula satisfying condition (8) canbe constructed as follows:

uq(x,x∗1,x

∗2, . . . ,x

∗k,µ)

=n∑

i=1

Aqi

k∏j=1

(xi − x∗ij)

+n∑

i=1

k∑j=1

Bqij(xi − x∗ij)

k∏p=1

(xi − x∗ip)

+n∑

i=1

k∑j=1

Cqij(xi − x∗ij)

2k∏

p=1

(xi − x∗ip)

+n∑

i=1

k∑j=1

Dqij(xi − x∗ij)

2k∏

p=1

(xi − x∗ip)

2

+ · · · (q = 1, 2, . . . , n) . (9)

It is easy to verify that uq(x∗i ,x

∗1,x

∗2, . . . ,x

∗k,µ) = 0

for i = 1, 2, . . . , k.Usually terms given in Eq. (9) up to Dqij are

enough for controlling a bifurcation if the singu-larity of the system is not highly degenerate. Thecoefficients Aqi, Bqij, Cqij and Dqij , which may befunctions of µ, are determined from the stabilitiesof an equilibrium under consideration and that ofthe associated bifurcation solutions. More precisely,linear terms are determined by the requirement ofshifting an existing bifurcation (e.g. delaying an ex-isting Hopf bifurcation). The nonlinear terms, onthe other hand, can be used to change the stabilityof an existing bifurcation or create a new bifurcation(e.g. changing an existing subcritical Hopf bifurca-tion to supercritical). Note that not just Aqi termsmay involve linear terms; Bqij terms, etc. may alsocontain linear terms.

Remarks

(1) By no means the control formula given inEq. (9) is a unique control law. There are manyother feasible controllers (e.g. nonpolynomialtypes) which satisfy the necessary condition (8).

(2) It is not necessary to take all the compo-nents uq, i = 1, 2, . . . , n, in the control. Inmost cases, using fewer components or just onecomponent may be enough to satisfy the pre-designed control objectives. It is preferable tohave a simplest possible design for engineeringapplications.

(3) If x∗i1 = x∗

i2 = · · · = x∗ik for some i, then one

only needs to use these terms and omit theremaining terms in the control law. Moreover,lower-order terms related to these equilibriumcomponents can be added. This greatly simpli-fies the control formula. For example, if i = 1,then the general controller can be taken as

uq =k−1∑i=1

aqi(x1 − x∗11)

i + Aq1(x1 − x∗11)

k

+ Bq11(x1 − x∗11)

k+1

+ Cq11(x1 − x∗11)

k+2 , (10)

where aqi’s denote the added lower-order terms.(4) uq involves higher-order terms, which may not

be necessary for determining the stability of bi-furcation solutions, but necessary for disablingthe control at all the k equilibria so that theoriginal equilibria are not changed under thecontrol. Since all different order terms are in-volved in the terms associated with the coeffi-cients Aqi, Bqij , Cqij , Dqij , etc., all these termsmay be used to control Hopf bifurcation. For ex-ample, later in considering the Rossler system,the Dqij coefficients are indeed used to controlHopf bifurcation.

(5) In general, it may be difficult to compute theexplicit expressions of the equilibrium solutions,in particular, for higher dimensional practicalsystems. Also, it is not easy to choose an opti-mal (best) control law from the general controlformula, in particular, when a large number ofparameters are included. This is related to theinverse problem in optimizing the parameterdesign. Therefore, for practical, higher dimen-sional systems, efficient computational method-ologies need to be developed by combiningnumerical and symbolic approaches, with theaid of parameter optimization. This technicalproblem is not only pertaining to this paper butexists in general.

3. Hopf Bifurcation Control

Return to the problem of controlling Hopf bifur-cation. Adding the controller given by Eq. (3) tosystem (1) yields

x = f(x, µ) + u(x;µ) ≡ F(x, µ) . (11)

Assume that the original system without controlhas an equilibrium x∗ and Hopf bifurcation occurs


at the critical point (x∗, µ∗). The goals of controlare:

(i) to move the critical point (x∗, µ∗) to a designateposition (x, µ);

(ii) to stabilize all possible Hopf bifurcations.

Goal (i) only requires linear analysis, while goal (ii)must apply nonlinear systems theory. In general, ifthe purpose of the control is to avoid bifurcations,one should employ linear analysis to maximize thestable interval for the equilibrium. The best is tocompletely eliminate possible bifurcations using afeedback control. If this is not feasible, then one mayhave to consider stabilizing the bifurcating limit cy-cles using a nonlinear state feedback [Chen et al.,2000]. In certain circumstances one wants to createa Hopf bifurcation, which can be easily achievedusing the above two steps in a reverse way [Chenet al., 2001].

At the designed position, x,

f(x, µ) = 0 (12)

for all µ ∈ R. To achieve objective (i), calculate theJacobian of system (11) at x to obtain

J(µ) =[∂F∂x

]x=x

=[∂f∂x

+∂u∂x

]x=x

. (13)

Thus, by the Hopf theory, J(µ) contains a complexconjugate pair of eigenvalues λ1,2(µ) = α(µ)+iω(µ)satisfying

α(µ) = 0 anddα(µ)dµ

�= 0 , (14)

and the remaining eigenvalues of J(µ) have negativereal parts at the critical point µ = µ.

Substituting the control law (9) (or (10)) intoEqs. (12) and (14) yields the equations for deter-mining the linear coefficients in the controller u.It should be noted that the solution for the lin-ear coefficients is not unique since there are manymore variables in the control law than the numberof predesigned conditions. This is why many coeffi-cients in the controller can be chosen zero and theactual controller can be quite simple. This will bedemonstrated in the next two sections. Therefore,in general, no fixed formula can be given for deter-mining the linear part of the control.

Once the first step discussed above is done,one may decide if it is necessary to continue to-ward the next step. If the aim of the control isto eliminate an existing Hopf bifurcation but thelinear analysis does not reach the goal, then one

must use the nonlinear part of the control to sta-bilize the Hopf bifurcation. This can be achievedusing normal form theory. The main task in apply-ing normal form theory is to compute the leadingnonzero coefficient in the normal form, which deter-mines whether the Hopf bifurcation is supercriticalor subcritical. For Hopf bifurcation, this coefficientis also called the first Liapunov coefficient or thefirst-order focus value. This coefficient can be ex-plicitly expressed in terms of the second- and third-order derivatives of the vector field of (11) evaluatedat the critical point. For example, if the originalsystem (1) is described on a two-dimensional cen-ter manifold, i.e. x = (x1, x2)T and F = (F1, F2)T ,then the first Liapunov coefficient is given by [Yu &Huseyin, 1988]

v3 =34

[F1111 + F1122 + F2211 + F2222

+1ω2

(F111F211 + F211F221

+ F221F222 − F111F112

− F112F122 − F122F222)], (15)

where ω = ω(µ), Fijk’s and Fijkl’s are the second-and third-order derivatives of Fi evaluated at(x1, x2, µ), for example, F2211 = ∂3F2(x1, x2, µ)/∂2x1∂x2, etc. If v3 < 0 (>0), the Hopf bifurcationis supercritical (subcritical). The nonlinear controlcoefficients are involved in the above ten deriva-tive terms, and are not difficult to be chosen suchthat v3 < 0 under which the bifurcating limit cy-cles are stable. However, if Eq. (1) is a generaln-dimensional system, then the expression of v3

is much more complicated since those noncriticalcomponents (associated with those eigenvalues thathave negative real parts) also have contributions.The symbolic computation with the aid of computeralgebra systems will be used, as shown in the nexttwo sections where the Lorenz equation and Rosslersystem are studied in detail.

4. Lorenz Equation

It is well known that the Lorenz equation can ex-hibit complex dynamics, including equilibria, limitcycles and chaos. In [Wang & Abed, 1995; Chenet al., 2000], the Lorenz equation is considered forHopf bifurcation control, where an approach wasproposed by introducing a feedback control whichincreases the stability interval for one of the nonzeroequilibria in terms of the control parameter.


-14

-10

-6

-2

2

6

z

-14

-10

-6

-2

2

6

z

-14

-10

-6

-2

2

6

-12 -8 -4 0 4 8 12

z

x

Fig. 1. Trajectories of the uncontrolled Lorenz system (16)for p = 4 and r = 13.20 with initial conditions (x0, y0, z0):(a) (0.0, 5.0, 5.7), chaos; (b) (10.0,−10.0, 5.7), stable C−; and(c) (−10.0, 10.0, 5.7), stable C+.

The convection equation, considered in [Wang& Abed, 1995; Chen et al., 2000] is

x = −p(x− y) ,

y = −xz − y ,

z = xy − z − r ,

(16)

where p and r are positive constants, which areconsidered as control parameters. Note that system(16) is a special case of the Lorenz equation (e.g. see[Guckenheimer & Holmes, 1993]):

x = −σ(x− y) ,

y = ρx− xz − y ,

z = xy − βz ,

(17)

which has one more parameter than system (16)does. One can easily show that system (16) is aspecial case of system (17) by first setting β = 1 inEq. (17), and then letting p = σ and r = ρ, andfinally using a constant shift z = z − r in system(16), which together will transform Eq. (16) to theLorenz equation (17) with β = 1.

It is easy to show that system (16) has threeequilibrium solutions, C0, C+ and C−, given below:

C0 : x0e = y0

e = 0 , z0e = −r ,

C± : x±e = y±e = ±√

r − 1 , z±e = −1 .(18)

Suppose the parameters p and r are positive. ThenC0 is stable for 0 ≤ r < 1, and pitchfork bifurcationoccurs at r = 1, where the equilibrium C0 loses itsstability and bifurcates into either C+ or C−. Thetwo equilibria C+ and C− are stable for 1 < r < rH ,where

rH =p(p + 4)p− 2

(p > 2) , (19)

and at this critical point C+ and C− lose their sta-bilities, giving rise to Hopf bifurcation. In orderto compare with the results obtained in [Wang &Abed, 1995; Chen et al., 2000], where p = 4 wasused, p = 4 is used here and thus rH = 16.

It is known that the Lorenz system (16) exhibitschaotic motions when r > 16. In fact, one can use anumerical integration scheme to show that the sys-tem can have coexistence of locally stable equilibriaC± and (global) chaotic motions for limited valuesof r. Figure 1 depicts the coexistence of the equi-libria and chaos when r = 15.20 for different initialconditions.

The next two subsections give a comparisonbetween the cases with and without controls.

4.1. Without control

When no control is applied, the critical pointat which Hopf bifurcation occurs is defined byEq. (19). At the critical point, defined as p = 4,rH = 16, the Jacobian of system (16) evaluated atC+ and C− has a real eigenvalue −6 and a purelyimaginary pair ±2

√5. Using the shift, given by

x = ±√r − 1 + x , y = ±√

r − 1 + y,

z = −1 + z ,(20)

to move C± to the origin and then applying an ap-propriate linear transformation to system (16), we


obtain the following system:

˙x = 2√

5y +184

(x + 4√

5y − 6z)µ−√

1521

(x− 2√

5y)(x− 2z) + · · ·

˙y = −2√

5x−√

52100

(155x − 10√

5y − 6z)µ−√

3105

(55x − 5√

5y + 42z)(x− 2z) + · · ·

˙z = −6z +1

168(x + 4

√5y − 6z)µ−

√15

42(x− 2

√5y)(x− 2z) + · · ·

(21)

where µ = r − 16 is a bifurcation parameter.Employing the Maple programs developed in

[Yu, 1998] for computing the normal forms of Hopfand generalized Hopf bifurcations yields the follow-ing normal form:

ρ = ρ

(156

µ +31

3248ρ2

)+ · · · ,

θ = 2√

5(1 +

17560

µ− 85148720

ρ2

)+ · · · ,

(22)

where ρ and θ represent the amplitude and phase ofthe motion, respectively. The first equation of (22)can be used for bifurcation and stability analysis. Itis easy to see that the Hopf bifurcation is subcriti-cal since the coefficient of ρ3 in the first equation of(22) is 31/3248 > 0.

Next, we apply feedback controls to stabilizesystem (16).

4.2. With feedback control

We first consider controlling the Hopf bifurcation ofLorenz equation (16) by a washout filter [Wang &Abed, 1995; Chen et al., 2000], which increases thesystem’s dimension by one. Then, we will use for-mula (9) or (10) to design a control law without thewashout filter, which turns out to be better thanthe one with the washout filter.

4.2.1. Control using washout filter

A feedback control u, utilizing a washout filter, wasfirst proposed by Wang and Abed [1995] and laterreconsidered by Chen et al. [2000] to obtain the con-trolled Lorenz system

x = −p(x− y) ,

y = −xz − y ,

z = xy − z − r − u ,

v = y − cv ,

(23)

where v is the state of the washout filter used ascontrol:

u = −kc(y − cv) − kn(y − cv)3 , (24)

with constant gains kc and kn to be determined,and c is a constant chosen for the filter. Note inEq. (24) that we use negative sign for u while posi-tive sign is used in [Wang & Abed, 1995; Chen et al.,2000] where the numerical results showed that withc = 0.5, kc = 2.5 and kn = 0.009, the critical pointrH at which limit cycles bifurcating from C+ is in-creased from rH = 16 to rH ≈ 36.

One purpose of introducing the washout fil-ter is to keep the equilibria of the original systemunchanged, which can be seen from the last twoequations of (24): Setting v = 0 results in y−cv = 0and thus u = 0 at the equilibria.

In the following, we first apply an analyticalapproach to study the controlled system (23), andthen in the next subsection present a better controllaw using the newly derived formula (9). It is easyto show that system (23) still has three equilibriumsolutions:

C0 : xe = ye = 0 , ze = −r , ve = 0 ,

C± : x±e = y±e = ±√

r − 1 , z±e = −1 ,

v±e = ±1c

√r − 1 .

(25)

Comparing Eqs. (25) with (18) clearly indicatesthat the controlled system (23) keeps the equilibriaof the original system (16) unchanged. By a linearanalysis with the aid of the Hurwitz criterion, wefound that when c = 0.5 and kc = 2.5 (kn does notaffect linear stability), the equilibrium C+ is stablefor 1 < r < rH ≈ 36.0043 while C− is stable for1 < r < rH ≈ 1.0854. By noting that both the twoequilibria C± of the original system are (locally) sta-ble for 1 < r < rH = 16, this suggests that the feed-back control (24) is beneficial for the stability of C+

since it receives 125% increase of the stable intervalover the original one. However, it dramatically de-creases the stability range of C−, with almost zero


stable interval, indicating that this control strategymay be no good if one wants to treat the two equi-libria C+ and C− more or less equally.

Let us consider C+ first. We shift C+ to theorigin by using the transformation (20) and v =v±e + v. Then the Jacobian of the new system, evalu-ated at the critical point r = rH ≈ 36.0043, has foureigenvalues: One purely imaginary pair: ±7.4338iand two negative real eigenvalues: −0.4165 and−6.0835. A linear transformation is then applied tochange system (23) into the following form:

˙x = 7.4338y + (0.0114x + 0.1072y − 0.0076z+ 0.0038v)µ + gx2 + kngx3 + · · ·

˙y = −7.4338x − (0.0684x − 0.0049y + 0.0105z+ 0.0353v)µ + gy2 + kngy3 + · · ·

˙z = −0.4165z − (0.0042x − 0.0045y + 0.0010z+ 0.0023v)µ + gz2 + kngz3 + · · ·

˙v = −6.0835v − (0.0299x + 0.0031y + 0.0042z+ 0.0153v)µ + gv2 − kngv3 + · · ·

(26)

where gα2 and gα3 (α = x, y, z, v) denote the secondand third degree homogeneous polynomials of x, y,z and v, respectively. Now, applying the Maple pro-gram [Yu, 1998] yields the following normal form:

ρ+ = ρ+[0.008178µ+ − (0.043699kn

− 0.001437)ρ2+ ] + · · ·

θ+ = 7.433823 + (0.172467kn − 0.009716)ρ2+ + · · ·

(27)

where µ+ = r − 36.0043, and the subscript and su-perscript + denote the equilibrium C+. A similaranalysis on the equilibrium point C− leads to thefollowing normal form:

ρ− = ρ−[0.818325µ− − (1.782058

− 0.042870kn)ρ2−] + · · ·θ− = 0.216173 + (2.961924 − 0.005917kn)ρ2− + · · ·

(28)

where µ− = r−1.0854, and the subscript − and thesuperscript − indicate the equilibrium C−. It fol-lows from the first equations of (27) and (28) thatthe Hopf bifurcation from C+ is supercritical when

0.043699kn − 0.001437 > 0 , i.e. kn > 0.0329 ,(29)

while that from C− is supercritical if

1.782058 − 0.042870kn > 0 , i.e. kn < 41.5690 .(30)

Thus, in order for both Hopf bifurcations emergingfrom C+ and C− to be supercritical, it requires that

0.0329 < kn < 41.5690 . (31)

In other words, when the control parameters c,kc and kn are taken as c = 0.5, kc = 2.5 andkn ∈ (0.0329, 41.5690), all the limit cycles bifur-cating from either C+ or C− become stable. Whenthe values of c and kc are varied, the interval for kn

is changed, and the new values of kn can be foundusing the above procedure. This becomes quite easyby using the Maple program.

It should be noted, however, that since the Hopfcritical point at C− is rH = 1.0854 which is veryclose to the static bifurcation point rc = 1 at whichC+ and C− bifurcate from C0, the equilibrium C− ofthe controlled system almost does not have a chanceto exist.

Some numerical simulation results, obtainedfrom system (23), are given in Figs. 2 to 4,which are the projections of the trajectories fromthe four-dimensional space (x, y, z, v) onto thetwo-dimensional space (x, z). Figure 2 shows thetrajectories of the controlled system (23) for p = 4,c = 0.5, kc = 2.5 and kn = 0.009, which wereused in [Wang & Abed, 1995; Chen et al., 2001;Chen et al., 2000]. Figure 2(a) shows that whenr = 30 ∈ (1, 36.0043), the trajectory converges tothe equilibrium C+ even for an initial point notnear C+. This agrees with the analytical predic-tion obtained in this paper as well as the numer-ical result given in [Chen et al., 2000] (see Fig. 12therein). When r = 37.5 > rH = 36.0043, the bifur-cating limit cycle is unstable and diverges to infinity[see Fig. 2(b)]. This agrees with the results shownin Fig. 12 of [Chen et al., 2000], where a subcrit-ical Hopf bifurcation occurs from C+. However, itwas stated in [Chen et al., 2000] that the systemhas been stabilized by the feedback control. Thisstatement does not imply Hopf bifurcation, but onlymeans that the stability interval, given in terms ofthe parameter r for the equilibrium C+, has beenincreased.

Figure 3 shows the numerical simulation re-sults when p = 4, c = 0.5, kc = 2.5, whichare the same as that for Fig. 2, but kn = 10 ∈(0.0329, 41.5690). It is seen that the periodic solu-tions bifurcating from C+ for r > rH = 36.0043are all stable. For the values of r close to r = rH ,see Fig. 3(a), in which r = 37.5 [same as thatused in Fig. 2(b)], the bifurcating limit cycle is sta-ble as expected by the analytical prediction. Even


-10

-6

-2

2

6

2 4 6 8 10

z

x

(a)

-10

-6

-2

2

6

2 4 6 8 10

z

x

(b)

Fig. 2. Trajectories of the controlled Lorenz system (23) with washout filter for p = 4, c = 0.5, kc = 2.5 and kn =0.009: (a) convergent to the C+ when r = 30 and (x0, y0, z0, v0) = (10, 8,−5.5, 18); (b) divergent to ∞ when r = 37.5 and(x0, y0, z0, v0) = (6.5, 7.0,−1.1, 12.5).

-1.8

-1.4

-1.0

-0.6

-0.2

5.8 5.9 6 6.1 6.2 6.3

z

x

(a)

-1.8

-1.4

-1.0

-0.6

-0.2

6.4 6.5 6.6 6.7 6.8

z

x

(b)

Fig. 3. Stable limit cycles for the controlled Lorenz system (23) with washout filter for p = 4, c = 0.5, kc = 2.5 and kn = 10with (x0, y0, z0, v0) = (6.1, 6.5,−1.1, 12.0), when (a) r = 37.5 and (b) r = 45.

for larger values of r, shown in Fig. 3(b) wherer = 45, the numerical result still shows that theperiodic solution is stable.

Similarly, Fig. 4 shows the numerical results re-lated to C−, when p = 4, c = 0.5, kc = 2.5, kn = 10(same as that used for Fig. 3), but for smaller r,since rH = 1.0854 for C−. It is shown that whenr = 1.08 < rH , the trajectory converges to C−[Fig. 4(a)], but to a stable limit cycle when r = 1.11[Fig. 4(b)]. It has been noticed that unlike the peri-odic solutions bifurcating from C+, the limit cycles

bifurcating from C− are stable only if the value of ris close to the critical point rH = 1.0854. For exam-ple, when r = 1.15, the numerical simulation showsthat the trajectory diverges to infinity.

4.2.2. Control using formula (9)

The disadvantage of using the washout filter forcontrol is that it increases the dimension of theoriginal system by one, unnecessarily increases thecomplexity of the system and difficulty in analysis.


-1.4

-1.0

-0.6

-0.2

-0.7 -0.5 -0.3 -0.1 0.1

z

x

(a)

-1.5

-1.0

-0.5

0.0

0.5

1.0

-1.2 -0.8 -0.4 0.0

z

x

(b)

Fig. 4. Trajectories of the controlled Lorenz system (23) with washout filter for p = 4, c = 0.5, kc = 2.5 and kn = 10 when(a) r = 1.08 and (x0, y0, z0, v0) = (−0.5,−0.5,−1.5,−1.5); (b) r = 1.11 and (x0, y0, z0, v0) = (−0.3,−0.3,−1.1,−0.6).

One may use the control formula proposed in the previous section instead. By using formula (9), we canexplicitly write a component of the controller as

u = A1(x− x+e )(x− x−

e ) + A2(y − y+e )(y − y−e ) + A3(z − z+

e )(z − z−e )

+ B11(x− x+e )2(x− x−

e ) + B21(y − y+e )2(y − y−e ) + B31(z − z+

e )2(z − z−e )

+ B12(x− x+e )(x− x−

e )2 + B22(y − y+e )(y − y−e )2 + B32(z − z+

e )(z − z−e )2

+ C11(x− x+e )3(x− x−

e ) + C21(y − y+e )3(y − y−e ) + C31(z − z+

e )3(z − z−e )

+ C12(x− x+e )(x− x−

e )3 + C22(y − y+e )(y − y−e )3 + C32(z − z+

e )(z − z−e )3

+ D1(x− x+e )2(x− x−

e )2 + D2(y − y+e )2(y − y−e )2 + D3(z − z+

e )2(z − z−e )2 . (32)

However, if z± = −1, we may omit the terms in-volving x± and y± and add a linear term for z.Furthermore, we may only choose a control compo-nent for the third equation. Thus, applying formula(10) yields

u3 = −k31(z + 1) − k32(z + 1)2

− k33(z + 1)3 , (33)

where only terms up to the third order are used,which are enough for Hopf bifurcation control. Fur-thermore, for Hopf bifurcation, the second-orderterm might not be necessary due to the presenceof the third-order coefficient k33, so we set k32 = 0,resulting in a simpler controller:

u3 = −k31(z + 1) − k33(z + 1)3 , (34)

The closed-loop system is now given by

x = −p(x− y) ,y = −xz − y ,

z = xy − z − r − k31(z + 1) − k33(z + 1)3 ,(35)

where the negative signs are used for kij ’s for con-sistence and comparison with that of the controllerbased on the washout filter.

Introducing the transformation (20) intoEq. (35) results in

˙x = −p(x− y) ,˙y = −xz + x− y ∓√

r − 1 z ,

˙z = xy ±√r − 1(x + y) − z − k31z − k33z

3 .

(36)

Then Oe = (x, y, z) = (0, 0, 0) is an equilibriumof system (36), corresponding to the equilibria C+

and C− of the original system (16). It is easy to usethe Jacobian of system (36), evaluated at the equi-librium Oe, to find the characteristic polynomial,resulting in

P (λ) = λ3 + (p + 2 + k31)λ2 + (p + r

+ k31 + pk31)λ + 2p(r − 1) , (37)


which shows that only the linear term of the con-troller u3 affects the linear stability. It follows fromEq. (25) that the stability conditions for Oe (underthe assumption p, r > 0) can be obtained as

p + 2 + k31 > 0 ,

p + r + k31(p + 1) > 0 ,

2p(r − 1) > 0 ,

p(p + 4) − r(p− 2 − kc) + k231(p + 1)

+ k31(p2 + 4p + 2) > 0 . (38)

If k31 > 0, then it only requires r > 1. The last con-dition in Eq. (38) implies a critical point at whichthe controlled system has Hopf bifurcation emerg-ing from the equilibrium Oe, defined by

rH =p(p + 4) + k2

31(p + 1) + k31(p2 + 4p + 2)p− 2 − k31

,

(39)

for 0 < k31 < p− 2.Setting k31 = 0 (i.e. there is no linear term in

the control) yields rH = p (p + 4)/(p − 2) (p > 2)which is the condition given in Eq. (19) for thesystem without control. This condition yields thecritical point rH = 16 at which p = 4 or 8. Forconsistency, we again take p = 4 in the followinganalysis.

It can be seen from Eq. (39) that the param-eter rH for the controlled system can reach verylarge values as long as k31 is chosen close to p − 2.For example, when p = 4, choosing k31 = 1.5gives rH = 188.5, and rH = 71 if k31 = 1. Thesevalues of rH are much larger than rH = 16 forthe uncontrolled system. If we choose r > 1 and0 < p− 2 < k31, then the equilibria C+ and C− arealways stable, and no Hopf bifurcations can occurfrom the two equilibria.

Next, we perform a nonlinear analysis to de-termine the stability of Hopf bifurcation. If p = 4,then k31 ∈ (0, 2), and for determination we choosek31 = (2

√1006 − 58)/5 ≈ 1.087, thus rH = 82. Let

r = rH + µ = 82 + µ, where µ is a perturbationfrom the critical point. Then, we have the closed-loop system

˙x = −8(x− y) ,

˙y = −xz + x− y ∓√81 + µ z ,

˙z = xy ±√81 + µ (x + y)

− 2√

1006 − 535

z − k33z3 .

(40)

The eigenvalues of the Jacobian of sys-tem (40), when evaluated at the equilibrium Oe,are: λ1,2 = ±

√2√

1006 + 28i ≈ 9.5621i andλ3 = −(2

√1006 − 28)/5 ≈ −7.0870. To apply the

method of normal forms [Guckenheimer & Holmes,1993; Yu, 1998, 2000], we introduce the followingtransformation

x = u− 24 +√

100643

w ,

y = u +2√

1006 + 284

v + w ,

z = ±√

1006 + 1418

u∓ 5(2√

1006 + 28)36

v

± 9√

1006 − 171215

w ,

(41)

to Eq. (40), and then employ the Maple program[Yu, 1998] to obtain an identical normal form forthe system associated with the two equilibria C+

and C−, given in the following polar coordinates upto the third order:

ρ=ρ

[1249 − 34

√1006

52942µ +

(4646315818 − 102399253

√1006

358010321904− 5746272 + 187233

√1006

4235360k33

)ρ2

], (42)

θ =√

2√

1006 + 28

[1 +

122602 − 773√

100617153208

µ

−(

21706679417 + 211691192√

10066444185794272

+34871 + 1594

√1006

16941440k33

)ρ2

]. (43)


-6

-4

-2

0

2

4

6

z

(a) (b)

-6

-4

-2

0

2

4

z

(c) (d)

-6

-4

-2

0

2

4

-12 -10 -8 -6 -4 -2 0

z

x

(e)

2 4 6 8 10 12x

(f)

Fig. 5. Stable equilibria C± of the controlled Lorenz system (38) with the new control law for initial conditions (x0, y0, z0) =(±3.0,±12.0,−2.5) when (a and b) r = 20; (c and d) r = 55; and (e and f) r = 81.

Steady-state solutions and their stabilities can be found from Eq. (42): The solution ρ = 0 representsthe initial equilibrium solution Oe (C+, or C−), which is stable when µ < 0 (i.e. r < rH = 82) and unstablewhen µ > 0 (r > 82). The supercritical Hopf bifurcation solution can be obtained, if

4646315818 − 102399253√

1006358010321904

− 5746272 + 187233√

10064235360

k33 < 0 ,

i.e. k33 >3672843514

√1006 − 115816173526

478327912875≈ 0.001416 . (44)

We choose k33 = 0.01. Then the controller is

u = −1.087(z + 1) − 0.01(z + 1)3 , (45)

and the controlled system described in the originalstates is finally obtained asx = −4(x− y) ,y = −xz − y ,

z = xy − z − r − 1.087(z + 1) − 0.01(z + 1)3 .(46)

Its normal form then becomes

ρ = ρ(0.003222µ − 0.023683ρ2) + · · · (47)

θ = 9.562165 + 0.054678µ − 0.046512ρ2 + · · · (48)

and the supercritical Hopf bifurcation solution isgiven by

ρ = 0.136070√µ = 0.136070

√r − 82 . (49)

Some numerical simulation results, obtainedfrom the controlled system (46), are given in Figs. 5and 6. Figure 5 depicts that the trajectories con-verge to the equilibria C+ and C− for 1 < r < 82,


-8

-6

-4

-2

0

2

4

6

z

(a) (b)

-8

-6

-4

-2

0

2

4

z

(c) (d)

-8

-6

-4

-2

0

2

4

-13 -12 -11 -10 -9 -8

z

x

(e)

7 8 9 10 11 12 13x

(f)

Fig. 6. Stable limit cycles around C± of the controlled Lorenz system (38) with the new control law for initial conditions(x0, y0, z0) = (±3.0,±12.0,−2.5) when (a and b) r = 83; (c and d) r = 90; and (e and f) r = 101.

while Fig. 6 demonstrates the stable limit cycles bi-furcated from the system when r > 82. By usingEq. (49), one can estimate the amplitudes of thethree limit cycles shown in Fig. 6 as 0.136, 0.385and 0.593, respectively. These approximations givea good prediction, confirmed by the numerical in-tegration results. It can be seen from Figs. 5 and 6that the symmetry of the two equilibria C+ and C−remain unchanged before and after the Hopf bifur-cation generated by using the simple control (34).

5. Rossler System

In the previous section we have considered Hopfbifurcation control for the Lorenz equation. Thesystem has two symmetrical equilibria from whichunstable limit cycles bifurcate at a critical point. Ithas been shown using the control formula proposedin Sec. 2 that a simple control given by Eq. (34) can

be obtained such that both equilibria are treatedequally. In this section we turn to the Rossler sys-tem, which can also have Hopf bifurcations from itstwo equilibria. However, these two equilibria are nolonger symmetrical and thus one may not be ableto obtain a simple control law as that used for theLorenz equation. Nevertheless, we will show that itis still not difficult to apply formula (9) or (10) tofind an appropriate control law that can simulta-neously stabilize the two equilibria of the Rosslersystem.

The Rossler system is described by

x = −y − z ,

y = x + ay ,

z = b + z(x− c) ,

(50)

where a, b and c are adjustable parameters. In thisstudy, we first analytically explore Hopf bifurcation


and then numerically show the route to chaotic mo-tion via period-doubling. Afterwards, we use for-mula (9) to control Hopf bifurcation of the system.For convenience we choose b = 2 and c = 4 inEq. (50).

First, consider the equilibria of system (50),determined from x = y = z = 0, which yields

y = −z , x = az ,

and 2 + z(az − 4) = 0 . (51)

One can find either (i) if a = 0, then ze = 1/2,so the equilibrium is E = (0,−1/2, 1/2); or (ii) ifa �= 0, then

z±e =1a

(2 ±√4− 2a) , (52)

which is real for a ≤ 2. So, for 0 < a ≤ 2, theequilibria are

E±e = (az±e ,−z±e , z±e ) , (53)

in which z±e is given by Eq. (52). By noting thatcase (i) is a limit case of case (ii), i.e.

lima→0

z−e = lima→0

1a(2 −√

4 − 2a)

= lima→0

22 +

√4 − 2a

=12,

we can include case (i) into case (ii).To study the stability of the equilibria, evalu-

ating the Jacobian of system (50) at the equilibriayields

JE =

0 −1 −11 a 0z±e 0 x±

e − 4

, (54)

which gives the characteristic polynomial

G(λ) = det(λI − JE)

= λ3 + a1λ2 + a2λ + a3 , (55)

wherea1 = 4 − a(1 + z±e ) ,

a2 = 1 − 4a + (a2 + 1)z±e ,

a3 = 2(2 − az±e ) .

(56)

The stability conditions for the equilibria are givenby

ai > 0 (i = 1, 2, 3) and a1a2 − a3 > 0 . (57)

If a = 0 (so ze = 1/2), it is easy to check that allthe conditions listed in Eq. (57) are satisfied. So,the equilibrium (0,−1/2, 1/2) is stable. Now, sup-pose 0 < a ≤ 2. Then, the requirements ai > 0(i = 1, 2, 3) are equivalent to

z±e <4a− 1 , z±e >

4a− 1a2 + 1

and z±e <2a. (58)

Since 0 < a ≤ 2 implies (4/a) − 1 ≥ 2/a, it followsfrom Eq. (58) that

4a− 1a2 + 1

< z±e <2a

(0 < a ≤ 2) . (59)

First, consider z+e = (1/a)(2 +

√4 − 2a). For

0 < a ≤ 2, we have z+e ≥ 2/a (or a3 = −2

√4 − 2a ≤

0, and in fact, a1 =√

2 − a (√

2 − a−√2) ≤ 0 too),

indicating that condition (58) is not satisfied andthus the equilibrium point E+

e = (az+e , − z+

e z+e ) is

unstable for any value of a ∈ (0, 2].Next, consider z−e = (1/a)(2 −√

4− 2a). Sincewe consider the increase of a from 0, we may restricta such that 0 < a < 2. Then

a1 = 4 − a(1 + z−e )

= 2 − a +√

2(2 − a) > 0 ∀ a < 2 ,

and

a3 = 2(2 − az−e )

= 2√

4 − 2a > 0 ∀ a < 2 ,

which imply that static bifurcation does not existin the Rossler system. Here, a2 can be rewritten as

a2 = 1 − 4a + (a2 + 1)1a(2 −√

4 − 2a)

= 1 − 4a +2(a2 + 1)

2 +√

4 − 2a,

for which it is easy to show that lima→0+ a2 =1 + (1/2) = 3/2 > 0, but a2|a=2 = 1 − 8 + 5 =−2 < 0. Further, it can be shown that da2/da < 0for a ∈ [0,

√3) and da2/da > 0 for a ∈ (

√3, 2],

da2/da = 0 at a =√

3 and a2(√

3) = −3. This indi-cates that a2 must cross zero once and only once inthe interval a ∈ [0, 2]. Thus, one of the conditionsgiven in Eq. (58) which first violates the stabilitycriterion is a1a2 − a3 > 0. In other words, whena is increasing from 0, a1a2 − a3 will first becomezero (where a2 = a3/a1 > 0 due to a1, a3 > 0), atwhich Hopf bifurcation occurs. Therefore, in orderfor the equilibrium E−

e to be stable, we only need


-6

-4

-2

0

2

y

(a) (b)

-6

-4

-2

0

2

-4 -2 0 2 4

y

x

(c)

-4 -2 0 2 4x

(d)

Fig. 7. Numerical simulation of trajectories of the uncontrolled Rossler system (50) for b = 2, c = 4 with the initial condition(x0, y0, z0) = (− 3.8, 0, 0): (a) a = 0, stable focus; (b) a = 0.15, periodic solution; (c) a = 0.35, quasi-periodic motion; and(d) a = 0.4, chaos.

to consider a1a2 − a3 > 0. This inequality yields

(az−e + a− 4)[(a2 + 1)z−e + 1 − 4a] − 2(az−e − 2) < 0

⇔ 4a2 − 9a + 2 > a(4 − a)√

4− 2a > 0 (due to a �= 0)

⇔ √2− a(1 − 4a) >

√2a(4 − a)

⇔ 2a2(4 − a)2 < (2 − a)(1 − 4a)2

⇔ 2a4 − 8a2 + 17a− 2 < 0

⇔ 2(a + 2.699298)(a − 0.124967)[(a − 1.287165)2 + 2.964504] , (60)

where the roots are found using a numerical ap-proach. Therefore, at the critical point defined byaH ≈ 0.125, the system has a pair of purely imagi-nary eigenvalues and Hopf bifurcation occurs fromthe equilibrium E−

e .Before adding control to the system, we present

numerical results for some typical solutions: stableequilibrium, stable limit cycle, stable quasi-periodicmotion and chaos. The results will be used later fora comparison with the controlled system. The equi-librium solution E−

e is stable for a < 0.125, so Hopfbifurcation occurs at a = 0.125 from E−

e , and bi-furcating limit cycle is stable for a ∈ (0.125, 0.34).When a further increases, the Hopf bifurcation

solution becomes unstable and the first period-doubling bifurcation occurs at a = 0.34. After thatperiod-doubling leads to chaos around a = 0.4. Thetrajectories of system (50), obtained using numer-ical simulation, and their projections on the x–yplane are shown in Fig. 7.

Now, we consider controlling the Rossler sys-tem (50) using feedback control. By the generalformula (9), we may apply a control component toeach equation of (50). Since the system has only onenonlinear (quadratic) term in the third equation of(50), we might only consider adding a control to thethird equation for the purpose of controlling Hopfbifurcation. Thus we may use Eq. (32) to obtain the


controlled system as follows:x = −y − z

y = x + ay

z = b + z(x− c) + u3 ,

(61)

which does not change the equilibria of the originalsystem (50).

In order to give a clear illustration, we shall usea fixed value of a. It has been shown that for theuncontrolled system the equilibrium E+

e is always

unstable for a ∈ (0, 2] while E−e is stable only

for a ∈ (0, 0.125]. When a = 2, E−e = E+

e

(unstable) does not have representative. Here, wechoose a = 0.4 = 2/5, at which the uncontrolledsystem (50) exhibits chaotic motions (see Fig. 7).Thus, z±e = 5 ± 2

√5, so

E±e =

(2± 4

5

√5,−(5 ± 2

√5), 5 ± 2

√5)

, (62)

and the controller becomes

u3 = A1

(x− 2− 4

5

√5)(

x− 2 +45

√5)

+ A2(y + 5 + 2√

5)(y + 5− 2√

5)

+ A3(z − 5 − 2√

5)(z − 5 + 2√

5) + B11

(x− 2 − 4

5

√5)2(

x− 2 +45

√5)

+ B21(y + 5 + 2√

5)2(y + 5 − 2√

5) + B31(z − 5 − 2√

5)2(z − 5 + 2√

5)

+ B12

(x− 2− 4

5

√5)(

x− 2 +45

√5)2

+ B22(y + 5 + 2√

5)(y + 5 − 2√

5)2

+ B32(z − 5 − 2√

5)(z − 5 + 2√

5)2 + C11

(x− 2− 4

5

√5)3(

x− 2 +45

√5)

+ C21(y + 5 + 2√

5)3(y + 5 − 2√

5) + C31(z − 5 − 2√

5)3(z − 5 + 2√

5)

+ C12

(x− 2− 4

5

√5)(

x− 2 +45

√5)3

+ C22(y + 5 + 2√

5)(y + 5 − 2√

5)3

+ C32(z − 5 − 2√

5)(z − 5 + 2√

5)3 + D1

(x− 2− 4

5

√5)2(

x− 2 +45

√5)2

+ D2(y + 5 + 2√

5)2(y + 5 − 2√

5)2 + D3(z − 5− 2√

5)2(z − 5 + 2√

5)2 . (63)

Since the original system (50) does not havestatic bifurcation, we only consider controlling Hopfbifurcation. There are two objectives: (1) to delaythe appearance of Hopf bifurcation from the equi-libria E±

e ; and (2) if Hopf bifurcation occurs, sta-bilize the bifurcating limit cycles. In the following,for a = 2/5, we use the feedback control (63) to sta-bilize the two equilibria and control possible Hopfbifurcations. There are four cases: (i) both equilib-ria become stable without Hopf bifurcations; (ii) E+

e

is stable while E−e has a supercritical Hopf bifurca-

tion; (iii) E−e is stable while E+

e gives a supercriticalHopf bifurcation; and (iv) both equilibria have su-percritical Hopf bifurcations.

First, consider objective (1) which is deter-mined by the linear terms involved in the controlu3. A direct calculation shows that the equilibriumE−

e of the controlled system is stable when the con-ditions given in Eq. (57) are satisfied, where the ai’sare now given by

a−1 =45(2 +

√5) + 4

√5A3 − 80B31 + 320

√5C31 ,

a−2 =25

(13 − 29

5

√5)− 8

5

√5(A1 + A3) +

645

B11 + 32B31 − 51225

√5C11 − 128

√5C31 ,

a−3 =85

√5 +

1625

√5A1 + 4

√5(A2 + A3) − 128

25B11 + 80(B21 −B31)

+1024125

√5C11 + 320

√5(C21 + C31) , (64)


in which the superscript − indicates the equilibrium E−e . Similarly, one can find similar conditions for the

equilibrium E+e and the ai’s are

a+1 =

45(2 −

√5) − 4

√5A3 − 80B32 − 320

√5C32 ,

a+2 =

25

(13 +

295

√5)

+85

√5(A1 + A3) +

645B12 + 32B32 +

51225

√5C12 + 128

√5C32 ,

a+3 = −8

5

√5− 16

25

√5A1 − 4

√5(A2 + A3) − 128

25B12 + 80(B22 −B32)

− 1024125

√5C12 − 320

√5(C22 + C32) . (65)

It is not difficult to find from Eq. (57) that thestability condition that may be first violated is ei-ther a3 > 0 or a1a2 − a3 > 0. The former gives astatic bifurcation while the latter leads to a Hopfbifurcation. For the controlled Rossler system, weonly consider Hopf bifurcation. To show this, notethat if a1 crosses zero first (i.e. positive a1 becomeszero) while a2 and a3 are still positive, then thefourth condition becomes −a3 < 0, which had al-ready crossed zero. Similar discussion applies to thecase where a2 crosses zero first.

It is observed from Eqs. (64) and (65) that onlythe coefficients Ai’s, Bij ’s and Cij ’s are in the sta-bility conditions. Further, except for Ai’s, Eq. (64)has Bij’s and Cij ’s different from that in Eq. (65).Thus, we may choose

B11 = B21 = B31 = B12 = B22 = B32 = 0

and C11 = C21 = C31 = 0 , (66)

and use the Ai’s to stabilize E−e and the Ci2’s to sta-

bilize E+e . In order to further simplify the analysis,

noting that A3 takes opposite signs for E−e and E+

e ,we may let A3 = −1/4. Then, for the equilibriumE−

e , from the conditions

a−1 =8 −√

55

> 0 (satisfied) ,

a−2 =2(65 − 24

√5)

25− 8

√5

5A1 > 0 ,

a−3 =3√

55

+16√

525

A1 + 4√

5A2 > 0 ,

a−1 a−2 − a−3 =1280 − 589

√5

125

− 8(2√

5 − 1)5

A1

− 4√

5A2 > 0 , (67)

we can obtain

A1 <13√

5 − 2420

and − 320

− 425

A1 < A2 <256

√5 − 589500

−20 − 2√

525

A1 , (68)

under which the equilibrium E−e is stable. Note that

the second condition of (68) for A2 depends uponA1 given in the first condition of Eq. (68). For de-termination, we choose A1 = −1/4. Then

− 11100

< A2 <246

√5− 489500

. (69)

Using the above chosen values of Ai’s, simi-larly we may find the following stability conditions(based on Ci2’s) for the equilibrium E+

e :

a+1 > 0 ⇒ C32 <

5 + 8√

58000

,

a+2 > 0 ⇒ C12 > −19 + 13

√5

256− 25

4C32 ,

a+3 > 0 ⇒ C22 < − 11

8000

− C32 − 16625

C12 − 180

A2 ,

a+1 a+

2 − a+3 > 0 ⇒ C22 > −489 + 246

√5

40000

+89 + 36

√5

25C32 + 128

√5C2

32

+512

√5

25C32C12

− 8(10 +√

5)625

C12

− 180

A2 . (70)


Let

C32 = − 11000

and C12 = 0 . (71)

Then

−15785 + 7462√

51000000

− 180

A2

< C22 < − 38000

− 180

A2 . (72)

Finally, the control u becomes

u3 = −14

(x− 2 − 4

5

√5)(

x− 2 +45

√5)

+ A2(y + 5 + 2√

5)(y + 5− 2√

5)

−14(z − 5− 2

√5)(z − 5 + 2

√5)

+ C22(y + 5 + 2√

5)(y + 5− 2√

5)3

− 11000

(z − 5 − 2√

5)(z − 5 + 2√

5)3

+ D1

(x− 2− 4

5

√5)2(

x− 2 +45

√5)2

+ D2(y + 5 + 2√

5)2(y + 5 − 2√

5)2

+ D3(z − 5 − 2√

5)2(z − 5 + 2√

5)2 . (73)

Summarizing the above discussions yields the re-sults for case (i) as follows:

Case (i). Both equilibria

E−e =

(2 − 4

5

√5,−(5 − 2

√5), 5 − 2

√5)

and

E+e =

(2 +

45

√5,−(5 + 2

√5), 5 + 2

√5)

(corresponding to a = 2/5) of the controlled system(61) are stable and no bifurcations occur under thecontrol law given by Eq. (73) when

− 11100

< A2 <246

√5− 489500

and −15785 + 7462√

51000000

− 180

A2

< C22 < − 38000

− 180

A2 . (74)

Next, consider objective (2), i.e. at least oneof the two equilibria becomes unstable at a criti-cal point and bifurcates into periodic solutions. We

want to use the feedback control to stabilize thebifurcating limit cycles. The stability of the limitcycles is determined by the second- and third-orderterms in the control u3. However, it should be notedthat the controller u3 given by Eq. (73) involvesfourth-order terms, which are necessary for the orig-inal equilibria of system (50) to remain unchanged.In particular, we shall find appropriate values ofA2, C22 and Di’s to stabilize the bifurcating limitcycles. There are three cases, to be further discussedbelow.

Case (ii). E+e is stable, but E−

e has a Hopf bi-furcation. For this case, condition (72) still holds,but A2 is taken as

A2 =246

√5 − 489500

+ µ ≈ 0.122145 + µ, (75)

which renders a−1 a−2 − a−3 = 0 at the critical pointµ = 0, and µ is a bifurcation parameter. By usingthe second condition given in Eq. (74), we concludethat E+

e is still stable in this case when

−0.033997− 0.0125µ < C22 <−0.001902− 0.0125µ .

Then, applying the following transformation

x = 2 − 45

√5− 2

5x + ω−y +

√5 − 105

z ,

y = −5 + 2√

5 + x + z ,

z = 5 − 2√

5 +105 − 38

√5

25x +

25ω−y

− 110 − 18√

525

z , (76)

where ω− =√

130 − 38√

5/5, into Eq. (61) andthen employing the Maple program [Yu, 1998], weobtain the following normal form up to the thirdorder:

ρ = ρ

[50(270 + 199

√5)

25021µ + a13ρ

2

],

θ = ω−

[1 +

125(40263 + 21613√

5)12110164

µ + (· · ·)ρ2

],

(77)

where (· · ·) denotes an expression given in termsof D1, D2 and D3, which is not important for the


current concern. The coefficient a13 is

a13 = (65 − 19√

5)

[21466846537137

28061974774900000+

201327607272867√

5701549369372500000

+

(546825

24220328+

270975√

524220328

)C22

+

(19199767511406848874736940725

+3614040638506

√5

169774947388145

)D1 −

(15462188314675271639915821032

− 9185058311955√

5271639915821032

)D2

+

(34728896532374576790997895525800

− 53727378573781103√

5169774947388145000

)D3

−(

93734256873472169774947388145

+186609876599808

√5

848874736940725

)D2

1

−(

34448854131250033954989477629

+152161069312500

√5

33954989477629

)D2

2

+

(2996786374220283086817225239

− 551881189633284√

515434086126195

)D2

3

−(

15060886415760033954989477629

+64337170768800

√5

33954989477629

)D1D2

−(

319361870801184169774947388145

+27985678253088

√5

169774947388145

)D1D3

−(

15092533174200033954989477629

− 3753797077200√

533954989477629

)D2D3

]. (78)

Under the conditions (72) and (75), it is easy to show that

−0.033997 − 180

µ ≈ − 8925000

− 3403√

5250000

− 180

µ < C22 <237

20000− 123

√5

20000− 1

80µ ≈ −0.001902 − 1

80µ

under which Eq. (78) yields

−0.000211 − 0.000595µ < a13 < 0.001316 − 0.000595µ

if D1 = D2 = D3 = 0. This does not guarantee a13 < 0 required for supercritical Hopf bifurcations. Thus,all or part of Di’s should be used. It will be shown in the next case that D3 must be set zero, and D1 andD3 can be chosen as small real values. Therefore, we may set

D1 = D2 = −0.1, D3 = 0 , (79)

under which

a13 ≈ −0.291181 + 0.047594C22 < 0

as long as C22 is given by Eq. (72). Thus, the control is finally reduced to

u3 = −14

(x− 2− 4

5

√5)(

x− 2 +45

√5)

+ A2(y + 5 + 2√

5)(y + 5 − 2√

5)

−14(z − 5 − 2

√5)(z − 5 + 2

√5) + C22(y + 5 + 2

√5)(y + 5 − 2

√5)3

− 11000

(z − 5 − 2√

5)(z − 5 + 2√

5)3 − 110

(x− 2− 4

5

√5)2(

x− 2 +45

√5)2

− 110

(y + 5 + 2√

5)2(y + 5 − 2√

5)2 , (80)


for which the equilibrium E+e is stable when

−(15785 + 7462√

5)/1000000) − (1/80)A2 < C22 <−(3/8000) − (1/80)A2, while the equilibrium E−

e

becomes unstable at A2 = (246√

5 − 489)/500 andthe Hopf bifurcation is supercritical. The bifur-cating limit cycles exist for µ > 0 (or A2 >(246

√5 − 489)/500).

Case (iii). E−e is stable, but E+

e has a Hopf bi-furcation. For this case, condition (69) is still valid,but C22 is perturbed as

C22 = −15785 + 7462√

51000000

− 180

A2 − µ , (81)

which gives a+1 a+

2 − a+3 = 0 at the critical point

µ = 0. Then, similarly we can find the third-ordernormal form given as follows:

ρ = ρ

[20000(1139

√5− 1910)

567701µ + a13ρ

2

],

θ = ω+

[1 +

12500(1388601√

5 − 1957235)1923938689

µ

+ (· · ·)ρ2

], (82)

where

ω+ =

√3250 + 870

√5

25,

a13 ≈ −0.138283 − 0.132608A2 (83)

− 0.004952A22 .

For −11/100 < A2 < (246√

5− 489)/500 [seeEq. (69)], −0.154554 < a13 < −0.123756. There-fore, for this case the equilibrium E−

e is stable undercondition (69) while the equilibrium E+

e becomesunstable at C22 = −(15785 + 7462

√5)/1000000 −

(1/80)A2, and the Hopf bifurcation is supercritical.Bifurcating limit cycles exist for µ > 0.

Case (iv). Both E+e and E−

e become unstable andhave Hopf bifurcations. For this case, we may per-turb A2 and C22 simultaneously as follows:

A2 =246

√5− 489500

+ µ−

and C22 = −15785 + 7462√

51000000

− 180

A2 − µ+ ,

(84)

where µ− and µ+ are perturbation parameters. Theequilibrium E−

e is stable (unstable) for µ− < 0(> 0)

and E+e is stable (unstable) for µ+ < 0(> 0). At the

critical point µ− = 0, Hopf bifurcation occurs fromthe equilibrium E−

e while Hopf bifurcation mergesfrom the equilibrium E+

e at µ+ = 0.The third-order normal form corresponding to

the equilibrium E−e can be found as follows:

ρ− = ρ−

[50(270 + 199

√5)

25021µ− + a−13ρ

2−

],

θ− = ω−

[1 +

125(40263 + 21613√

5)12110164

µ−

+ (· · ·)ρ2−

], (85)

where the subscript − and superscript − indicatethe equilibrium E−

e , and

a−13 ≈ −6.592289 − 0.013395µ−− 1.071566µ+ (86)

which indicates that a−13 < 0 for any real valuesµ− > 0 and µ+ > 0. Thus, the Hopf bifurcationfrom E−

e is supercritical. Note that Eq. (85) is ac-tually the same as Eq. (77).

Similarly, the third-order normal form for theequilibrium E+

e is given by

ρ+ = ρ+

[20000(1139

√5− 1910)

567701µ+ + a+

13ρ2+

],

θ+ = ω+

[1 +

12500(1388601√

5 − 1957235)1923938689

µ+

+ (· · ·)ρ2+

], (87)

where

a+13 = −018241633 − 0.015794198µ− , (88)

implying that a+13 < 0 for any positive value of µ−.

Therefore, the Hopf bifurcation from E+e is also su-

percritical. Again, note that Eq. (87) is the same asEq. (82).

Summarizing the above discussions shows thatthe controlled Rossler system, i.e.

x = −y − z ,

y = x + ay ,

z = b + z(x− c) + u3 ,

where the parameters are chosen as a = 0.4, b = 2and c = 4 (under which the uncontrolled system is


Table 1. Hopf bifurcation control for Rossler system.

CaseStability Conditionand Critical Point

Hopf Bifurcation

E−e Stable − 11

100< A2 <

246√

5 − 489

500No Bifurcation

(i)

E+e Stable

−15785 + 7462√

5

1000000− 1

80A2

< C22 < − 3

8000− 1

80A2

No Bifurcation

E−e Unstable A2 =

246√

5 − 489

500+ µ Supercritical

(ii)

E+e Stable

−15785 + 7462√

5

1000000− 1

80A2

< C22 < − 3

8000− 1

80A2

No Bifurcation

E−e Stable − 11

100< A2 <

246√

5 − 489

500No Bifurcation

(iii)

E+e Unstable C22 = −15785 + 7462

√5

1000000− 1

80A2 − µ Supercritical

E−e Unstable A2 =

246√

5 − 489

500+ µ− Supercritical

(iv)

E+e Unstable C22 = −15785 + 7462

√5

1000000− 1

80A2 − µ+ Supercritical

chaotic), and the feedback control is given by

u3 = −14

(x− 2 − 4

5

√5)(

x− 2 +45

√5)

+ A2(y + 5 + 2√

5)(y + 5 − 2√

5)

−14(z − 5− 2

√5)(z − 5 + 2

√5) + C22(y + 5 + 2

√5)(y + 5− 2

√5)3

− 11000

(z − 5 − 2√

5)(z − 5 + 2√

5)3 − 110

(x− 2 − 4

5

√5)2(

x− 2 +45

√5)2

− 110

(y + 5 + 2√

5)2(y + 5 − 2√

5)2 ,

can always be stabilized by the two appropriatelychoosing coefficients A2 and C22. The detailed re-sults are summarized in Table 1, where E+

e =(2 + (4/5)

√5,−(5 + 2

√5), 5 + 2

√5) and E−

e =(2 − (4/5)

√5,−(5 − 2

√5), 5 − 2

√5).

Numerical results obtained using Eqs. (61) and(80) for the selected parameter values, correspond-

ing to the four cases, are shown in Fig. 8, wherea = 0.4, b = 2 and c = 4 have been used.Figures 8(a) and 8(b) depict that both E+

e and E−e

are stable when the conditions given in Eq. (74)are satisfied [Case (i)]. Figure 8(c) shows a stablelimit cycle bifurcating from E−

e while E+e is still

stable [Case (ii)]. Figure 8(d) demonstrates a stable


0.4

0.5

0.6

0.7

0.1 0.15 0.2 0.25 0.3

z

x

(a) Case (i)

8

9

10

11

3.2 3.6 4 4.4

z

x

(b) Case (i)

0.35

0.45

0.55

0.65

0 0.1 0.2 0.3 0.4

z

x

(c) Case (ii)

5

8

11

14

2 3 4 5 6

z

x

(d) Case (iii)

0.44

0.48

0.52

0.56

0.6

0.1 0.15 0.2 0.25 0.3

z

x

(e) Case (iv)

7

8

9

10

11

12

2.8 3.3 3.8 4.3 4.8

z

x

(f) Case (iv)

Fig. 8. Trajectories of the controlled Rossler system (61) when a = 0.4, b = 2, c = 4 with the control given byEq. (80): (a and b) Stable E−

e and E+e when (A2, C22) = (0.0,−0.01), with (x0, y0, z0) = (0.3,−0.5, 0.7) for (a) and

(x0, y0, z0) = (3.5,−9.5, 11.0) for (b); (c) Stable limit cycle around unstable E−e for (A2, C22) = (0.152145, −0.01) with

(x0, y0, z0) = (0.4,−0.5, 0.5) (stable E+e ); (d) Stable limit cycle around unstable E+

e for (A2, C22) = (0.0,−0.033471)with (x0, y0, z0) = (6.0,−9.0, 9.0) (stable E−

e ); (e and f) Unstable E−e and E+

e , surrounded by stable limit cycles when(A2, C22) = (0.124145, −0.034522), with (x0, y0, z0) = (0.3,−0.52, 0.52) for (e) and (x0, y0, z0) = (3.7,−9.5, 10.5) for (f).

limit cycle bifurcating from E+e when E−

e is stable[Case (iii)]. Figures 8(e) and 8(f) show that bothE+

e and E−e are unstable [Case (iv)] and they are

locally surrounded by stable limit cycles. These re-sults indeed confirm the analytical predictions givenin Table 1.

It has been noted that unlike the uncontrolledsystem, the simulation for the controlled Rossler

system associated with E+e is very sensitive to

the control coefficient C22. This can be seen fromEq. (87) from which we obtain the approximate so-lution for the family of limit cycles as follows:

r2+ = 1230.00µ+

implying that µ+ must be a very small positive realvalue, i.e. the perturbation to C22 must be very


small. It should be noted that the control law givenin this paper for the Rossler system may not be thebest choice. We could choose more coefficients inthe general control formula (63) and even add morecontrol components to the first and second equa-tions. This can reduce the sensitivity of the systemto the feedback control. It is beyond the scope ofthis paper, so is not further discussed.

6. Concluding Remarks

A general explicit formula has been derived andpresented for bifurcation control using nonlinearstate feedback. The formula, which can be appliedto many kind of singularities, automatically satis-fies the necessary condition required for not chang-ing the equilibria of the original system. The linearpart of the control formula can change the stabil-ity of an equilibrium and shift existing bifurcationsfrom the equilibrium. The nonlinear part of the con-trol formula can be used to stabilize (or destabilize)bifurcations.

In this paper, the developed approach has beenapplied to controlling Hopf bifurcation. The Lorenzequation and Rossler system are used to illustratethat multiple equilibria can be treated simultane-ously and to gain a big increase of stability interval(region). Furthermore, limit cycles bifurcating fromequilibria are dealt with in the same way so thatthe coexisting multiple limit cycles become stablesimultaneously. It should be noted that since theintended bifurcation control tasks are complicated,the controllers turn out to be somewhat complex.These findings have not been reported in the lit-erature based on other methods such as using thewashout filter. The analytical predictions of thispaper have been verified by numerical simulationresults.

The method proposed in this paper can be ex-tended to consider other singularities such as doubleHopf, Hopf-zero and double zero.

Acknowledgments

This work was supported by the Natural Sci-ence and Engineering Research Council of Canada(NSERC No. R2686A02), and the Hong KongCERG (CityU No. 1004/02E).

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HOPF BIFURCATION CONTROL USING NONLINEAR ......International Journal of Bifurcation and Chaos, Vol. 14, No. 5 (2004) 1683–1704 c World Scientiﬁc Publishing Company HOPF BIFURCATION