IC/99/188 United Nations Educational Scientific and Cultural Organization and International Atomic Energy Agency THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS HARMONIC OSCILLATIONS, CHAOS AND SYNCHRONIZATION IN SYSTEMS CONSISTING OF VAN DER POL OSCILLATOR COUPLED TO A LINEAR OSCILLATOR

This paper deals with the dynamics of a model describing systems eonsisting of the classical Van der Pol oscillator coupled gyroscopically to a linear oscillator. Both the forced and autonomous cases are considered. Harmonic response is investigated along with its stability boundaries. Condition for quenching phenomena in the autonomous case is derived. Neimark bifurcation is observed and it is found that our model shows period doubling and period-m sudden transitions to chaos. Synchronization of two and more systems in their chaotic regime is presented. MIRAMARE TRIESTE December 1999 Regular Associate of the Abdus Salam ICTP. IIntroduction Due to their occurrence in various scientific fields, ranging from biology, chemistry, physics to engineering, coupled nonlinear oscillators have been a subject of particular interest in recent years [1,2,3], Among these coupled systems, a particular class is that containing self-sustained components such as the classical Van der Pol oscillator. The classical Van der Pol oscillator serves as a paradigm for smoothly oscillating limit cycle or relaxation oscillations [4]. In presence of an external sinusoidal excitation, it leads to various phenomena: harmonic, subharmonic and superharmonic frequency entrainment [5], devil's staircase in the behavior of the winding number [6], chaotic behavior in small range of control parameters [6-8], The generalization of the classical Van der Pol oscillator to include cubic nonlinear term (the so-called Duffing-Van der Pol or Van der PolDuffing oscillator) has also been investigated and various bifurcation structures observed (see ref. [9] and references therein). The particular dynamics of the Van der Pol oscillator has raised the question of the behavior of coupled Van der Pol oscillators or that of systems consisting of Van der Pol oscillator coupled to another type of oscillator. As concerns coupling between Van der Pol oscillators, some interesting works have been carried out. Rand et al. [10] have investigated various bifurcations of motions of two coupled classical Van der Pol oscillators. They derived criteria or parameters space regions for phase locked periodic motions, phase entrainment and phase drift. The case of two Van der Pol-Duffing oscillators with linear coupling has also been considered by Polianshenko et al. [11]. They showed that nonisochronism (the dependence of oscillation frequencies on amplitudes) substantially changes the dynamics of the system by generating not only one frequency gap, but also displays two frequencies and chaotic dynamics (the transition to chaos being through period doubling or via type I-intermittency). Additionally, the system exhibits hysteresis between two and one-frequency regimes and between one frequency and suppressed oscillations. The same group also extended their study to include nonlinear coupling terms in ref.[12]. They observed multistability, three frequency oscillations and found the parameters boundaries for chaos using Shilnikov theorem. Transition to hyperchaos has also been reported in coupled Van der Pol-Duffing oscillators [13]. More recently, another nonlinear coupling model has been analyzed in ref.[14]. It was found that the structure of attraction basins is related to the symmetry of the attractors and can be understood using discrete transformations similar to logistic maps. For the coupling between Van der Pol oscillator and other types of oscillators (e.g. oscillators that cannot sustain their own oscillations), we tackled the problem in ref.[2] by investigating the dynamics of a system consisting of a Van der PolDuffing oscillator coupled dissipatively and elastically to a Duffing oscillatorUsing the multiple time scales method, we analyzed the oscillatory states both in the resonant and non-resonant cases. Chaos was also found using the Shilnikov theorem. In this paper, we are dealing with a model consisting of a classical Van der Pol oscillator coupled gyroscopically to a linear oscillator. Three major problems are analyzed. First, the regular dynamics of the system is considered using analytical methods. In the forced case, we find and study the stability of harmonic oscillations using the method of harmonic balance and the Hill determinant procedure. In the autonomous system, the averaging method leads to two types of oscillatory states and indicates the criteria for quenching phenomena in the system. The second problem consists of analyzing the appearance of the chaotic states. Two types of transition indicators are used: the projection of the attractors in the Poincare section onto the system coordinates and the largest Lyapunov exponent, both versus a control parameter (one of the coupling coefficient). It is found that there are domains where chaotic and regular states appear and disappear randomly following very small changes in the coupling coefficient. The synchronization of two or more devices described by our model is also discussed as the third issue of the paper. The paper is organized as follows. Section 2 presents the model, the resulting harmonic solutions with the stability boundary equation and the oscillatory states in the autonomous system. In section 3, we use the numerical simulation to analyze different types of transition of the system behavior as the coupling coefficient varies. Then, the continuous control strategy is used to synchronize two or more devices in a chaotic state. We conclude in section 4. IIModel and oscillatory states