Impact of combined coupling on the dynamics of a self-sustained oscillatory system with two degrees of freedom

Abstract

Traditionally, research on coupled oscillators has focused on synchronization problems with a fixed type of coupling (most often inductive or resistive). However, a comparative analysis of bifurcation mechanisms and steady states in the “oscillator–circuit” system when transitioning from conservative coupling to dissipative or mixed coupling within the framework of a single experimental task often remains outside the scope of detailed consideration. Such a simplest classical self‑oscillatory system with two degrees of freedom is the van der Pol oscillator with an additional oscillatory circuit. If the role of capacitive and inductive coupling between the generator and the circuit has been studied in considerable detail, then much less is known about the influence of resistive (dissipative or diffusion) coupling, and even less about combined coupling. This work aims to examine possible structurally stable types of dynamics observed in the “generator–circuit” system when it contains a complex combined coupling represented by a linear combination of capacitive and diffusion couplings. During the study, a numerical modeling method was applied to the original system using the Matlab program; in addition, shortened equations for the amplitudes and phase differences were derived analytically. Due to the complexity of obtaining an explicit analytical solution for the equilibrium states, these equations were also solved using numerical methods. The study established that, depending on the values of the control parameters of the generator and the circuit and the strength of the coupling between them, the system can exhibit a stable equilibrium state and stable periodic oscillations. In the case of capacitive coupling, a hysteresis effect occurs, when, for the same parameter values, stable in‑phase and anti‑phase limit cycles coexist in the phase space of the system under study; these cycles are realized under different initial conditions. This phenomenon is not observed in the case of resistive coupling. The correctness of the obtained results for the equilibrium state was confirmed analytically using the Routh–Hurwitz method. The obtained numerical results are presented in the form of mode maps on the planes of the system’s parameters, one‑parameter diagrams, projections of phase portraits, and various dependencies observed in the system when the control parameters and coupling coefficients are varied.

Speaker

Igor M. Nazin
Vladimir V. Astakhov, Galina I. Strelkova
Russia

Discussion

Ask question