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SYNCHRONIZATION OF SELF-OSCILLATIONS BY PARAMETRIC EXCITATION

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International Journal of Bifurcation and Chaos, Vol. 8, No. 7 (1998) 1605–1612c World Scientific Publishing CompanySYNCHRONIZATION OF SELF-OSCILLATIONSBY PARAMETRIC EXCITATIONV. ASTAKHOV, A. SHABUNIN and V. ANISHCHENKORadiophysics Department, Saratov State University,Astrahanskaya 83, Saratov, RussiaReceived July 15, 1997; Revised November 14, 1997This paper is devoted to the problem of synchronization of symmetrically coupled self-oscillatorsexhibiting chaos by means of chaos control technique. We use a nonfeedback method of control,in particular, high frequency periodic modulation of the coefficient of coupling. The model ofChua’s circuits coupled via capacity is considered in this work. We study the possibility ofsynchronization in dependence on amplitude and frequency of modulation for various values ofthe parameters of the self-oscillators. The dependence of the threshold values of the amplitudeof the synchronizing influence on coefficient is presented.1. IntroductionRecently, much of the interest in nonlinear dynam-ics has focused on the problem of synchronizationof chaotic oscillations. Unfortunately, there is nocommon definition of this subject. Different au-thors use different approaches: from the classicalview on synchronization such as the locking of fre-quencies [Anishchenko et al., 1991, 1992] and phases[Rosenblum et al., 1995] to the equality of oscil-lations in subsystems (x1(t)=x2(t)) [Fujisaka &Yamada, 1983; Pecora & Caroll, 1990]. Besidesinvestigations of the self-synchronization problem,there are also tasks of forced-synchronization andsynchronization by control [Lai & Grebogi, 1993;Chua et al., 1993; Astakhov et al., 1996a].In this paper we investigate the effect ofsynchronization of symmetrically coupled identicalself-oscillators applying a high frequency paramet-ric excitation to the coupling element. The term“synchronization” is used here in a narrow sense asmotions in the symmetric subspace (x1(t)=x2(t)).The study of periodic parametric excitation tononlinear oscillators is a traditional problem of clas-sical and modern physics. The majority of worksis devoted to resonance influence on an oscillatorwhen the ratio between the own time scale of thesystem and the period of the external force is closeto a rational number of a small order. Applica-tions of periodic parametric perturbations for mod-ification of chaotic dynamics were considered in theworks [Lima & Pettini, 1990; Cicogna & Fronzoni,1990; Fronzoni et al., 1991]. It has been shownboth theoretically and experimentally that resonantparametric perturbation can suppress chaotic be-havior. Another interesting task is an application ofhigh frequency parametric excitation when its fre-quency is much higher than the own characteristicfrequency of the oscillator. This is the nonresonantcase.We propose to apply high frequency periodicmodulation of the coupling coefficient to stabilizesynchronous motions. The idea of this approachwas induced by a classical task of mechanics: a pen-dulum with vibrating suspension. As is known, atcertain values of amplitude and frequency of vibra-tion it is possible to stabilize the upper equilibriumof the pendulum [Kapitza, 1951a, 1951b].The task of the stability of symmetric motionscan be reduced to the stability of the zero fixed16051606 V. Astakhov et al.point of some system. To illustrate this fact let usconsider the system in the form:˙x1= F(x1)+γ(x2−x1)˙x2=F(x2)+γ(x1−x2),(1)where the equation˙x = F(x) describes an oscil-lator without coupling, γ is a matrix of coupling.Linearizing the system (1) near the symmetric sub-space and using new variables:u =(x1+x2)/2, v=(x1−x2)/2,we obtain the equations as follows:˙u = F(u)(2)˙v=∂F∂xu−2γv(3)Here Eq. (2) describes the character of motions inthe symmetric subspace. The stability of the zeroequilibrium of Eq. (3) determines the stability ofthese motions.As high frequency parametric excitation canchange the stability of the equilibrium (in the caseof the pendulum) we suppose that it can alsochange the stability of the symmetric motions. Weapplied the high frequency perturbation to thecoupling parameter in the system of two couplednonautonomous nonlinear oscillators with chaos[Astakhov et al., 1996b] and found that such per-turbation can stabilize the in-phase chaotic motionsin this system. In this paper we present the ef-fect of synchronization of chaotic oscillations in aself-oscillatory system by means of high frequencyparametric modulation of the coupling coefficient.2. The System under ConsiderationAs a model of coupled self-oscillators we considertwo Chua’s circuits coupled via a capacity. TheUosin (t)CCLLGGCCVV22cRRCmmmB−B010ppRRiV=C+ C sin( t)o~a)b)Fig. 1. The scheme of the circuit (a) and the characteristic of the nonlinear element (b).Synchronization of Self-Oscillations 1607capacity is parametrically modulated by externalperiodic force. The scheme of the coupled circuitsis shown in Fig. 1. The normalized system of equa-tions that describes this scheme has the followingform:˙x1= α(y1− x1− f(x1))˙y1=1+γ1+2γ(x1−y1+z1)+γ1+2γ(x2−y2+z2)+γ11+2γ(y2−y1)˙z1=−βy1˙x2= α(y2−x2−f(x2))˙y2=1+γ1+2γ(x2−y2+z2)+γ1+2γ(x1−y1+z1)+γ11+2γ(y1−y2)˙z2=−βy2,(4)where f(x)=bx +0.5(a − b)(|x +1|−|x−1|),α=C2/C1,β=C2/(LG2),γ=γ0+ξsin Ωτ, γ0=Cc0/C2,ξ=Cc∼/C2,γ1=ξΩcosΩτ, a = m1/G,b = m0/G, x1,2=(Vc1)1,2/Bp,y1,2=(Vc2)1,2/Bp,z1,2=(iL)1,2/GBp, ˙x = dx/dτ, τ = tG/C2, Ω=C2/Gω (Fig. 1).In-phase motions are typical for a system withcoupling via a resistor in a wide range of its pa-rameters, especially in the case of strong coupling[Anishchenko et al., 1995], we use the coupling viaa capacity in the model system. In this case thein-phase motions lose their stability right away af-ter the first period-doubling bifurcation [Astakhovet al., 1997].Chua’s circuit is a generator with 1.5 degreesof freedom. It demonstrates the transition to chaosthrough the cascade of period-doubling bifurca-tions. The evolution of chaos leads to a so-calleddouble-scroll attractor [Chua et al., 1986]. The sys-tem of coupled oscillators demonstrates more com-plicated dynamics. It has period-doubling, toribirth and breakdown, symmetry breaking and in-creasing bifurcations [Astakhov et al., 1997]. Thesystem is also characterized by multistability, whendifferent stable regimes coexist in the phase space.In Fig. 2 the dashed regions represent the domain ofstable


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