Controlling Hopf Bifurcations : Discrete-Time Systems

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


INTRODUCTION
Bifurcation control means to design a controller that can modify the bifurcative properties of a given nonlinear system, so as to obtain some desired dynamical behaviors.Typical examples include delaying the onset of an inherent bifurcation, relo- cating an existing bifurcation point, modifying the shape or type of a bifurcation chain, introducing a new bifurcation at a preferable parameter value, stabilizing a bifurcated periodic trajectory, changing the multiplicity, amplitude, and/or frequency of some limit cycles emerging from bifurcation, opti- mizing the system performance near a bifurcation point, or a certain combination of some of these [1,2,[4][5][6]9,10,14,17]. Bifurcation control is impor- tant not only in its own right, but also in providing Corresponding author.

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an effective strategy for chaos control.In particu- lar, period-doubling bifurcation is a typical route to chaos in many nonlinear discrete-time dynamical systems.Bifurcation control is useful in many engi- neering applications, as discussed in [10].
System bifurcations can be controlled by using 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 quadratic invariants in the normal form [13].In [10], a unified linear as well as simple nonlinear state-feedback technique was developed for Hopf bifurcation con- trol for continuous-time systems.In this paper, this new methodology is further extended to discrete- time systems.Discrete-time systems differ from the continuous ones in many aspects.For instance, the former has the typical period-doubling bifur- cation but the latter generally does not.In the investigation of this paper, both problems of shift- ing and creating a Hopf bifurcation are discussed.Computer simulations are included to illustrate the methodology and to verify the theoretical results.
This paper is organized as follows.Section 2 briefly summarizes the classical Hopf bifurcation theory for discrete-time systems.Sections 3 and 4  study the state-feedback control problem for Hopf bifurcations, including computer simulation results.Section 5 concludes the investigation with some discussion.
Here, the second condition in (2) refers to as the transversality condition for the crossing of the eigenlocus at the unit circle, namely, the eigenlocus is not tangent to the circle.Moreover, both super- critical and subcritical bifurcations can be further distinguished, via however a rather complicated series coordinate transformations (see Theorem 9.7 of [12]).

CONTROLLING THE HOPF BIFURCATION
Conceptually, Hopf bifurcations can be relatively easily created for a given two-dimensional system, in a way similar to that for the continuous-time case studied in [10].Both calculation and simulation have confirmed this observation.Therefore, the cur- rent interest is twisted to finding out if Hopf bifur- cations can also be created in a one-dimensional system in some way, which is not intuitively obvious.For this purpose, consider a parametrized one-dimensional system in a general form #) with a real parameter # E R and an equilibrium point x*.Clearly, this one-dimensional system cannot have a classical Hopf bifurcation due to the dimensional deficit.However, one may consider the following bifurcation control problem instead: Design a state-feedback controller, uk =uk(x; #), to be added to the right-hand side of the given system (3), such that the controlled system displays a Hopf bifurcation in the extended phase plane associated with system (3), in a sense to be further described 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 seen that, in principle, the methodology can be applied to other forms of controllers.
By introducing a new state variable, y= x-x_ 1, the controlled system can be written in the following extended form: xA+I f(xl; #) + uk(Yk; #), Yk+l Xk+l Xk.

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

CONCLUSIONS
In this paper, a simple and unified state-feedback control methodology has been developed for Hopf bifurcations for discrete-time systems, for both problems of shifting and creating a Hopf bifurca- tion point in the controlled system.Although the discussion here has been restricted to two-dimen- sional systems, the basic idea and the proposed approach can be extended to higher-dimensional dynamical systems.It is anticipated that some real applications of the new control method can be found in the near future.
This system also has a period-doubling bifurcations rather than a Hopf bifurcation.Yet, a new Hopf bifurcation can be created at (x,y , #)-(0, 0, 1), by various feedback controllers.For example, either one of the following controllers works quite well: Uk lZYk, Uk It2yk