Self-sustained oscillations of Rayleigh and Van der Pol oscillators with moderately large feedback factors

Abstract

Periodic motions of essentially non-linear self-sustained oscillatory systems, described by Rayleigh and Van der Pol equations, are constructed and investigated. The period and initial value of the velocity of the system, which determine the self-sustained oscillations of the oscillators for small and moderately large values of the feedback factors, are calculated by the Lyapunov-Poincaré method using a developed accelerated-convergence algorithm and a continuation with respect to a parameter. The trajectories and limit cycles are also constructed with a guaranteed relative and absolute error. The qualitative features of the self-sustained oscillations due to an increase in the self-excitation factors are established and the oscillators are compared. The results of a numerical investigation of periodic solutions of the Van der Pol equation are compared with familiar solutions.