Van der Pol, Rayleigh and Duffing oscillators in discrete time dynamics

Abstract

Using a combination of the method of invariance of pulse characteristics of dynamic systems and the method of parametric synthesis is proposed for time sampling in nonlinear models of oscillatory systems. The difference equation of motion and the block diagram of a linear dissipative oscillator form the basis for the solution of the problem of designing a nonlinear discrete system (discrete mapping). Thus, Van der Pol, Rayleigh, Duffing oscillators are introduced as objects of nonlinear dynamics with discrete time mappings. Characteristics of self-oscillations of discrete and analog Van der Pol oscillators are compared, their similarity and difference at high levels of excitation is established. The results of a numerical experiment with a discrete Van der Pol Duffing oscillator made it possible to establish that it can operate in two modes of self-oscillations differing in amplitudes and frequencies. It is shown that the transition between the modes can be followed by the generation of chaotic self-oscillations as the system parameters vary. Possible areas of application of the discrete dynamic systems described in the article are identified.