Van der Pol o~cill atos with various linear and cubic restoring elements are examined. The evolution of stationary self-excited oscillations is described and the role of self-excitation as that of restoring elements is shown. Numerical method on computer is used, | Vietnam Journal of Mechanics, VAST, Vol. 28, No. 2 (2006), pp. 67 - 73 VAN DER POL SYSTEMS WITH VARIOUS RESTORING ELEMENTS NGUYEN VAN DINH AND TRAN KI M CHI Institute of M echanics, VAST Abstract. Van der Pol o~cillators with various linear and cubic restoring elements are examined. The evolution of stationary self-excited oscillations is described and the role of self-excitation as that of restoring elements is shown. Numerical method on computer is used. 1. INTRODUCTION The classical Van der Pol oscillator with linear restoring element has been presented in every book on nonlinear oscillations [1, 2]. It is known t hat, when the self-excitation is absent, closed orbits are observed in the phase plane (free oscillation) encircling a stable center 0 (stable equilibrium state) . With the presence of self-excitation, 0 becomes an unstable focus. A unique closed orbit S (stationary self-excited oscillation) exists and all other orbits are spirals, asymptotically approaching S. The situation becomes complicated with the presence of nonlinear restoring elements, especially if linear and nonlinear elements are of different (attracting, pushing) type. The aforesaid stable center 0 can be transformed into saddle. Other critical points (centers, foci , saddles) together with critical orbits (homo - heteroclinic) can appear; closed orbits (stationary self-excited oscillations) can not exist or a great number of them can appear simultaneously. Below, in order to obtain useful informations, some Van der Pol systems with various linear and cubic restoring elements will be considered. The evolution and qualitative characteristics of closed orbits (stationary self-excited oscillations) are of special interest . The numerical method on computer is used. Some results, especially the reestablishment of homo-heteroclinic orbits, can be proved by t he Melnikov criterion. 2. VAN DER POL SYSTEMS WITH LINEAR AND CUBIC ATTRACTING ELEMENTS The simplest case is that of Van der Pol .
