Impact of anharmonicity on multistability in a self-sustained oscillatory system with two degrees of freedom

Abstract

Background and Objectives: The van der Pole oscillator with an additional oscillatory circuit represents one of the simplest self-sustained oscillator system with two degrees of freedom. It is characterized by the phenomenon of frequency pulling, caused by the appearance of bistability and hysteresis. The bifurcation mechanism of pulling and bistability was previously identified, and the bifurcation analysis was carried out for the case of weak excitation when the system exhibits quasi-harmonic self-sustained oscillations. However, the question remains open about the influence of anharmonicity, which develops in the system with increasing excitation parameter, on the phenomenon of multistability and on the bifurcation mechanism of its formation. Is the effect of frequency pulling and the corresponding bistable states preserved over a wide range of values of the control parameters? Are new multistable states being formed? What does the bifurcation structure of the control parameter plane look like? In this paper, the above issues are studied using as an example a self-sustained oscillatory system consisting of the Rayleigh oscillator with an additional linear oscillator. Materials and Methods: Numerical simulation and bifurcation analysis of equilibrium states and limit cycles were performed using the XPPAUTO software package. Results: The results of a two-parameter analysis in a wide range of excitation and frequency detuning parameters have been presented, typical modes of self-sustained oscillations and their bifurcations have been described. Conclusion: It has been shown that the classical phenomenon of frequency pulling is observed only at small values of the excitation parameter of the system. The bistability region, where two limit cycles coexist, corresponding to in-phase and anti-phase oscillation modes in coupled oscillators, is bounded by both the detuning parameter and the excitation parameter.

Keywords

• rayleigh oscillator
• frequency pulling
• multistability
• hysteresis
• bifurcation analysis