Controlling Hopf Bifurcations: Discrete-Time Systemsdownloads.hindawi.com/journals/ddns/2000/201496.pdf · 2019-08-01 · approach can be extended to higher-dimensional dynamicalsystems

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Controlling Hopf Bifurcations: Discrete-Time SystemsGUANRONG CHENa’*, JIN-QING FANG b, YIGUANG HONG and HUASHU QIN

aDepartment of Electrical and Computer Engineering, University of Houston, Houston, TX 77204, USA;bChina Institute of Atomic Energy, P.O. Box 275-27, Beijing 102413, P.R. China;

Clnstitute of Systems Science, Academia Sinica, Beijing 100080, P.R. China

(Received 20 July 1999)

Bifurcation control has attracted increasing attention in recent years. A simple and unifiedstate-feedback methodology is developed in this paper for Hopf bifurcation control fordiscrete-time systems. The control task can be either shifting an existing Hopf bifurcationor creating a new Hopfbifurcation. Some computer simulations are included to illustrate themethodology and to verify the theoretical results.

Keywords." Feedback control, Hopf bifurcation, Limit cycle, Period-doubling bifurcation,Stability

1. INTRODUCTION

Bifurcation control means to design a controllerthat can modify the bifurcative properties of a givennonlinear system, so as to obtain some desireddynamical behaviors. Typical examples includedelaying the onset of an inherent bifurcation, relo-cating an existing bifurcation point, modifying theshape or type of a bifurcation chain, introducinga new bifurcation at a preferable parameter value,stabilizing a bifurcated periodic trajectory, chang-ing the multiplicity, amplitude, and/or frequency ofsome limit cycles emerging from bifurcation, opti-mizing the system performance near a bifurcationpoint, or a certain combination of some of these[1,2,4-6,9,10,14,17]. Bifurcation control is impor-tant not only in its own right, but also in providing

Corresponding author.

29

an effective strategy for chaos control. In particu-lar, period-doubling bifurcation is a typical route tochaos in many nonlinear discrete-time dynamicalsystems. Bifurcation control is useful in many engi-neering applications, as discussed in [10].

System bifurcations can be controlled byusing differ-ent methods, such as linear delayed state-feedback[7,8] or nonlinear state-feedback [2], using a wash-out filter 16], employing harmonic balance approx-imation [4,5,11,14,15], and applying the quadraticinvariants in the normal form [13]. In [10], a unifiedlinear as well as simple nonlinear state-feedbacktechnique was developed for Hopf bifurcation con-trol for continuous-time systems. In this paper, thisnew methodology is further extended to discrete-time systems. Discrete-time systems differ fromthe continuous ones in many aspects. For instance,

30 G. CHEN et al.

the former has the typical period-doubling bifur-cation but the latter generally does not. In theinvestigation of this paper, both problems of shift-

ing and creating a Hopf bifurcation are discussed.Computer simulations are included to illustrate themethodology and to verify the theoretical results.

This paper is organized as follows. Section 2briefly summarizes the classical Hopf bifurcationtheory for discrete-time systems. Sections 3 and 4study the state-feedback control problem forHopf bifurcations, including computer simulationresults. Section 5 concludes the investigation withsome discussion.

2. HOPF BIFURCATIONS:PRELIMINARIES

In this section, the classical criterion for discreteHopf bifurcations is briefly reviewed.

Consider a general two-dimensional parametri-zed system:

Xk+l f(xk, y;#)(1)

Y+l g(x, y; #),

with a real variable parameter # E R and an equilib-rium point (x *, y *), satisfying x =f(x *, y *; #) andy*= g(x*, y*; #) simultaneously for all #. Let J(#)be its Jacobian at this equilibrium, with eigenvaluesA1,2(#) satisfying A2(#) 1(#). If, despite someminor details,

IAl(#*)l and0#

> 0 (2)

then the system undergoes a Hopfbifurcation at thebifurcation point (x*,y*,#*) [3,12,14]. More pre-cisely, in any small left-neighborhood of #* (i.e.,# < #*), (x*, y*) is a stable focus; and in any smallright-neighborhood of #* (i.e., # > #*), this focuschanges to be unstable, usually surrounded by alimit cycle.

Here, the second condition in (2) refers to asthe transversality condition for the crossing of the

eigenlocus at the unit circle, namely, the eigenlocusis not tangent to the circle. Moreover, both super-critical and subcritical bifurcations can be furtherdistinguished, via however a rather complicatedseries coordinate transformations (see Theorem 9.7of [12]).

3. CONTROLLING THE HOPFBIFURCATION

Conceptually, Hopf bifurcations can be relativelyeasily created for a given two-dimensional system,in a way similar to that for the continuous-time case

studied in [10]. Both calculation and simulationhave confirmed this observation. Therefore, the cur-

rent interest is twisted to finding out if Hopf bifur-cations can also be created in a one-dimensionalsystem in some way, which is not intuitivelyobvious. For this purpose, consider a parametrizedone-dimensional system in a general form

#)

with a real parameter # E R and an equilibriumpoint x*. Clearly, this one-dimensional systemcannot have a classical Hopf bifurcation due tothe dimensional deficit. However, one may considerthe following bifurcation control problem instead:Design a state-feedback controller, uk =uk(x; #),to be added to the right-hand side of the givensystem (3), such that the controlled system displaysa Hopf bifurcation in the extended phase planeassociated with system (3), in a sense to be furtherdescribed below.To facilitate quantitative analysis and calcula-

tion, a specific form of controller, namely, the pop-ular delayed state-feedback controller [7,8]

Uk Uk(Xk Xk-1; #), (4)

is chosen in the following discussion. It is easily seenthat, in principle, the methodology can be appliedto other forms of controllers.By introducing a new state variable, y=

x-x_ 1, the controlled system can be written

CONTROLLING HOPF BIFURCATIONS 31

in the following extended form:

xA+I f(xl; #) + uk(Yk; #),

Yk+l Xk+l Xk.

If the controller (4) is designed to satisfyu(0;#)=0, then it will not change the originalequilibrium point, x*, of the given system (3). Thecontrolled system has the Jacobian at (x,y)(x*, 0) as

J(#) fx Uyfx- uy x=x*, y=yO=O

(6)

where fx- Of/Oxk and Uy- Ou//Oyk, witheigenvalues

iv/ 2A1,2(#) -- (fx + uy) + (fx + u.y) -4uy. (7)

Conditions for the controller to satisfy AI(#)-A2(#) and (2) are

(fx-{-b/y) 2 4Uy, I,l,2(#*)l-0la,,2(#*)l > 0.

0#

and(8)

4. SIMULATION RESULTS

In this section, some simple but illustrative numeri-cal examples are presented.

To show an example, the familiar Logistic map,

Xk+l #Xk(1 Xk), (9)

is used, which has equilibrium points x* =0 andx* =(#-1)/#. This one-dimensional system doesnot have Hopf bifurcation, but rather, has the well-known period-doubling bifurcation leading tochaos as # varies from 1.0 to 4.0 (see Fig. 1). Withthe controller uk(Yk; #) e/yk being added to it, theextended controlled system

x+ x( x) + (; u),

Yk+l Xk+l Xk,(10)

satisfies all the stated conditions, and, as expected,has a Hopf bifurcation at #0_ 0. The correspond-ing simulation results are shown in Figs. 2-4, wherein Fig. 2: #--0.01 < 0.0-#0, the controlled sys-tem has a stable focus, (x*, y*) (0, 0); in Fig. 3:

#-0.0-#0, the stable focus bifurcates to a limitcycle of period six,

(x*, *) {(0.5, 0.5), (0.5, 0.0), (0.0,-0.5),(-0.5, -0.5), (-0.5, 0.0), (0.0, 0.5)}

when the initial point is (x0, Y0) (0. 5, 0. 5); in

Fig. 4: # 0.01 > 0.0= #0, the focus (x*, y*)= (0, 0)becomes unstable.

This bifurcation control strategy has been testedon other systems; for instance, on the following

1.0 3.0 3.4

FIGURE Period-doubling bifurcation of the Logistic map (1 < # _< 4).

4.0 p

32 G. CHEN et al.

-05 -0.4 -03 -0.62 -l $1 02 03 0.4 05

FIGURE 2 Orbit of the controlled Logistic map (#=-0.01 < 0.0 #).

FIGURE 4 Orbit of the controlled Logistic map (#=0.01 > 0.0 #0).

O

-0.3

-1t.4

-05 ".-05 -0.4 -0.3 -02 -0.1 0.1 02 0.3 0.4

FIGURE 3 Orbit of the controlled Logistic map (#=0.0- #o).

system:

(11)

which are linear with respect to y/ (and xk), and

u/ (2#x/ + 2# + #2)yk

which is nonlinear with respect to (x/, y).

5. CONCLUSIONS

In this paper, a simple and unified state-feedbackcontrol methodology has been developed for Hopfbifurcations for discrete-time systems, for bothproblems of shifting and creating a Hopf bifurca-tion point in the controlled system. Although thediscussion here has been restricted to two-dimen-sional systems, the basic idea and the proposedapproach can be extended to higher-dimensionaldynamical systems. It is anticipated that some realapplications of the new control method can befound in the near future.

This system also has a period-doubling bifurcationsrather than a Hopf bifurcation. Yet, a new Hopfbifurcation can be created at (x,y, #)-(0, 0, 1),by various feedback controllers. For example,either one of the following controllers works quitewell:

Uk lZYk, Uk It2yk

References

[1] E.H. Abed and J.H. Fu, "Local feedback stabilization andbifurcation control," Sys. Contr. Lett., Part I: Hopf bifur-cation, 7, 11-17, 1986; Part II: Stationary bifurcation, 8,467-473, 1987.

[2] E.H. Abed, H.O. Wang and R.C. Chen, "Stabilization ofperiod doubling bifurcations and implications for control ofchaos," Physica D, 70, 154-164, 1994.

CONTROLLING HOPF BIFURCATIONS 33

[3] D.K. Arrowsmith and C.M. Place, An Introductionto Dynamical Systems, Cambridge University Press,New York, 1990.

[4] D.W. Berns, J.L. Moiola and G. Chen, "Feedback control oflimit cycle amplitudes from a frequency domain approach,"Automatica, 34, 1567-1573, 1998.

[5] J.L. Moiola, D.W. Berns and G. Chen, "Controllingdegenerate Hopf bifurcations," Latin American AppliedResearch Journal, 29, 213-220, 1999.

[6] G. Chen, "Chaos, bifurcation, and their control," in TheWiley Encyclopedia of Electrical and Electronics Engineer-ing, J. Webster (Ed.), Wiley, New York, February 1999(in press).

[7] M.E. Brandt and G. Chen, "Bifurcation control of twononlinear models of cardiac activity," IEEE Trans. on Circ.Sys.- I, 44, 1031-1034, 1997.

[8] G. Chen and X. Yu, "On time-delayed feedback control ofchaotic dynamical systems," IEEE Trans. on Circ. Sys.-I, 46,767-772, 1999.

[9] G. Chen and X. Dong, From Chaos to Order: Perspectives,Methodologies, and Applications, World Scientific Pub. Co.,Singapore, 1998.

[10] G. Chen, J.-Q. Fang, Y. Hong and H.S. Qin, "ControllingHopf bifurcations: Continuous-time systems," 1998(preprint).

[11] R. Genesio, A. Tesi, H.O. Wang and E.H. Abed, "Controlof period doubling bifurcations using harmonic balance,"Proc. of Conf. on Decis. Contr., San Antonio, TX, 1993,pp. 492-497.

[12] P. Glendinning, Stability, Instability and Chaos, CambridgeUniversity Press, New York, 1994.

[13] W. Kang, "Bifurcation and normal form of nonlinearcontrol systems," Parts and II, SIAM J. of Contr. Optim.,36, 193-232, 1998.

[14] J.L. Moiola and G. Chen, Hopf Bifurcation Analysis."A Frequency Domain Approach, World Scientific Pub. Co.,Singapore, 1996.

[15] A. Tesi, E.H. Abed, R. Genesio and H.O. Wang,"Harmonic balance analysis of period-doubling bifurca-tions with implications for control of nonlinear dynamics,"Automatica, 32, 1255-1271, 1996.

[16] H.O. Wang and E.H. Abed, "Bifurcation control of achaotic system," Automatica, 31, 1213-1226, 1995.

[17] H. Yabuno, "Bifurcation control of parametrically excitedDuffing system by a combined linear-plus-nonlinear feed-back control," Nonlin. Dynam., 12, 263-274, 1997.

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Controlling Hopf Bifurcations: Discrete-Time Systemsdownloads.hindawi.com/journals/ddns/2000/201496.pdf · 2019-08-01 · approach can be extended to higher-dimensional dynamicalsystems