Dynamics of Hysteretic-Related Van-Der-Pol Oscillators: the Small Parameter Method

The main results of this paper are related to studying the dynamics of the Van-der-Pol oscillator, which is under the influence of a driving force and a hysteretic effect, formalized by the Bouc–Wen phenomenological model. The regularizing role of the hysteretic link in the reduction of chaotic regimes is established. The main results are obtained as part of the asymptotic expansion method (small parameter method). A system of two coupled Van-der-Pol oscillators is considered under the conditions of a driving force acting on one of them: the relationship between the amplitudes and frequencies of the natural oscillations of the oscillators is established in an analytical form. The paper also describes a system of cross-related Van-der-Pol oscillators, in which the driving force acting on one of them is determined by the output of a hysteretic transducer, to the input of which the mismatch between the oscillators’ velocities is applied. It is found that the coupling parameter between the oscillators and the parameter that determines the influence of the hysteretic link have a significant effect on the synchronization in the system.