Non-stationary responses of self-excited driven system

Abstract

This is the first attempt to determine the effects of a broad range of linear non-stationary regimes related to the frequencies of external excitation v(t) ranging from evolving to robust, on dynamic responses of self-excited systems, governed by the Van der Pol differential equation. The results obtained, time histories and phase plane plots, are revealing. Because of continuous variations of v(t), a representative point in the (v, K) plane, where K is the amplitude of excitation, continually crosses various regions and boundaries of periodic, aperiodic, stable, and unstable system motions, thus exhibiting a variety of new dynamic forms, which lead in some cases to a possible chaotic motion. The non-stationary responses are sensitive to the sweep rates α : the faster the sweep, the earlier are the appearances of new waveforms (patterns) and the shorter are the time intervals between the changed patterns. The responses are also sensitive to the variations of other system parameters: ϵ, which indicates the degree of non-linearity; K, amplitude of external excitation; σ, detuning, and σ 0 , initial ( t=0) detuning; and p 0 , initial amplitude of the response. The Rayleigh-Van der Pol oscillator is the basis of a model for a number of physical, biological, chemical, and engineering phenomena. This paper is an initial contribution to further theoretical and applied studies in non-stationary processes.