Ukrainian Mathematical Journal, Vol. 59, No. 2, 2007 VAN DER POL OSCILLATOR UNDER PARAMETRIC AND FORCED EXCITATIONS Nguyen Van Dao, 1 Nguyen Van Dinh, 2 and Tran Kim Chi 2 UDC 517.9 We study a van der Pol oscillator under parametric and forced excitations. The case where a sys- tem contains a small parameter and is quasilinear and the general case (without the assumption of the smallness of nonlinear terms and perturbations) are studied. In the first case, equations of the first approximation are obtained by the Krylov–Bogolyubov–Mitropol’skii technique, their aver- aging is performed, frequency-amplitude and resonance curves are studied, and the stability of the given system is considered. In the second case, the possibility of chaotic behavior in a deter- ministic system of oscillator type is shown. Introduction It is well known that there always exists an interaction of some kind between nonlinear oscillating systems. Minorsky stated that “Perhaps the whole theory of nonlinear oscillations could be formed on the basis of interac- tion” [1]. In the monograph [1], we have investigated different interesting cases of interaction by using the effi- cient asymptotic method of nonlinear mechanics created by Krylov, Bogolyubov, and Mitropol’skii. The present paper introduces our study of the behavior of a van der Pol oscillator under parametric and forced excitations. The dynamical system under consideration is described by an ordinary nonlinear differential equation of the type (1.1). Section 1 is devoted to the case of small parameters. The amplitudes of nonlinear de- terministic oscillations and their stability are studied. Analytical calculations in combination with a computer are used to obtain amplitude curves, which show a very complicated form in Figs. 1.2–1.4. In Sec. 2, we study the chaotic phenomenon occurring in the system described by Eq. (1.1) without the assumption of the smallness of parameters. As is known, the fundamental characteristic of a chaotic system is its sensitivity to initial conditions. The diagnostic tool used in this work is the Lyapunov exponents. The positiveness of the largest Lyapunov exponent will help us to determine the values of parameters for which the chaotic motions are occurred. Chaotic attractors and associated power spectra will be presented. 1. Case of Small Parameters In this section, we consider the case where the parameters are small. The opposite case is investigated in the next section. The smallness of parameters is characterized by introducing a small positive parameter ε. In this case, the Krylov–Bogolyubov–Mitropol’skii asymptotic method [2, 3] is used for seeking approximate solu- tions and studying their stability. 1 Deceased. 2 Vietnam Academy of Sciences, Hanoi, Vietnam. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 2, pp. 206 – 216, February, 2007. Original article submitted August 14, 2006. 0041–5995/07/5902–0215 © 2007 Springer Science+Business Media, Inc. 215