In the first case, equations of the first approximation are obtained by means of the Krylov-Bogoliubov-Mitropolskii technique, their averaging is performed, frequencyamplitude and resonance curves are studied, on the stability of the given system is considered. In the second case, the possibility of chaotic behavior in a deterministic system of oscillator type is shown. | Vietnam Journal of Mechanics, VAST, Vol. 29, No. 3 (2007), pp. 207 - 219 Special Issue Dedicated to the Memory of Prof. Nguyen Van Dao VAN DER POL'S OSCILLATOR UNDER THE PARAMETRIC AND FORCED EXCITATIONS NGUYEN VAN DAO Vietnam National University, Hanoi NGUYEN VAN DINH, TRAN KIM Cm Vietnam Academy of Sciences, Hanoi (This paper has been published in: MaTeMaT11YH11i1 >Kypnan 2007 , ToM 59, N° 2) Abstract. Van der Pol's oscillator under parametric forced excitations is studied. The case where the system contains a small parametrer being quasilinear and the general case (without assumption on the smallness of nonlinear terms and perturbations) are studied. In the first case, equations of the first approximation are obtained by means of the Krylov-Bogoliubov-Mitropolskii technique, their averaging is performed, frequencyamplitude and resonance curves are studied, on the stability of the given system is considered. In the second case, the possibility of chaotic behavior in a deterministic system of oscillator type is shown. 1. INTRODUCTION It is well-known that there always exists an interaction of some kind between nonlinear oscillating systems. N. Minorsky stated that "Perhaps the whole theory of nonlinear oscillations could be formed on the basis of interaction" [3]. Different interesting cases of interaction have been investigated by us and published in the monograph [3], using the effective asymptotic method of nonlinear mechanics created by Krylov N. M., Bogoliubov N. N. and Mitropolskii Yu. A. · The present paper introduces our research on the behaviour of a Van der Pol's oscillator under the parametric and forced excitations. The dynamic system urider consideration is described by an ordinary nonlinear differential equation of type ( ). The section 1 is devoted to the case of small parameters. The amplitudes of nonlinear deterministic oscillations and their stability are studied. Analytical calculations in combination with a computer
