Controlling Discrete-Time Chaotic Systems

Themin efforts im curret resach on feedback control of (Hnear and nonlinea) dynamic ystem have been focuing on et s h an unsable system or c g a stable system for cetain ing purpos. Recently, it has been observed tha titing trol stratei for dcatic -systems is needed n som ara suh as biomedial sciences and neural . This can be seen frown for example, Freemn's aricle (1V9), where It stes: "Conceivaby, the ochas we have obseved is impy an inetbe by-product a the brain's comLity, including its riad . Yet or evidence suggesb that the l c of the hm is more than an accdental by-product Indeed, it may be the ciief property hatmkes the brain dHEeret frm an artiftcial-iteigence machine." At the present ste, however, wry lttle is knownashow to control a chaoic dynamic system, pa a a discretetime chaotic system for some practical purpo . T" isyet some current -research on the toi by Huble (1M7), Jacks (1991) and Ott, Grebogi and Yorke (1990), and the related references cited therein. Other related papen indu& Ditto, Rause and Spano (1990) ad Hunt (1991). May deep insights and new ideas -can be found from these papers as how to understand the dynamic behavior of a complex systemand how to control it, where tih mai of Ott, Grebog and Yorke is to use a small and caefuly chos perturbation to control ("ttabls") :an unstable periodic orbit and that a Hfibler and Jackson s to use cmtrol without fdbck. It states imJason (1991) that once the control is initiated there is no need to further moitor the system's dynaiis, no to feedback this inormatiam order to -sustain the controL This is -obviouly very important in system which have chatic dynamics, since their sensitivity to small erors makes -them very difficult, -and probably impobe, to control usi conventional fedback methods over all of their phase space." Despite this negtive coment on te use of conventional feedback cotrol tecniques for chaotic dynamic systems, we present in this paper some interesting observations, analysis and simulation results on the control of chaotic Hinon svstems using conventional feedback control strategies. We demonstrate that the conventional feedback approach not only works for discrete-time chaotic dynamic systems, as shown in Chen and Doug (1992) and in this p'per, but als applies to continuous-tine chotc systems (see Chen and Dong (1991)). 2. Feedback control of the Hinon system Consider the nlinear H sstem { zi(n + 1) p,(-n) +zdtn)+ z,2(n + 1) =qzr(n), with suitablychosenreal parawtrp andq, which display chaotic ashoainF lkwIthp= lJad q =02 for lar m -to u: (a=-)