Self-oscillations in a system with hysteresis: the small parameter approach

Abstract

The paper investigates a modified van der Pol equation with hysteresis nonlinearity formalized within the operator approach, namely by means of the Preisach operator, a continual analog of the converter consisting of a family of non-ideal relays connected in parallel. The system considered in the work is a mathematical model of an electrical system similar to the classical van der Pol system where characteristics of the nonlinear part are of hysteresis nature. The main method for studying this system is the classical small parameter approach. Within this method, an analytical solution to the equation describing the system under consideration was obtained for both cases of the presence and absence of external harmonic excitation. Numerical results for the oscillator dynamics are presented, and a comparative analysis of the dynamics of the system under consideration with the dynamics of the classical van der Pol oscillator is carried out. The dynamic modes of the modified oscillator are investigated depending on the parameters of the system. The spectral characteristics are compared to the corresponding characteristics of the classical van der Pol oscillator. Bifurcation diagrams that illustrate the transition from regular to chaotic dynamics through a cascade of bifurcations and period doubling are constructed. The dependence of the amplitude of forced oscillations on the amplitude of the external harmonic force is plotted for various parameters of the system.