Stochastic bifurcations induced by Lévy noise in a fractional trirhythmic van der Pol system

Abstract

Trirhythmical nature has aroused considerable interest in describing dynamic behaviors of a fractional self-sustained system. In this paper, the fractional trirhythmic system is considered and a bifurcation analysis in a stochastic fractional trirhythmic self sustained system subjected to Lévy noise perturbation is presented. The fractional electronic circuit has been used to model the system and the oscillations are described by a nonlinear fractional differential equation to show a new bifurcation parameter. Then, based on the minimum mean square error principle, the fractional derivative term is found to be equivalent to the linear combination of the damping force and restoring force, and the original system is further simplified to an equivalent integer order system. Using the predictor–corrector simulation method, it is shown a good agreement between the fractional and the equivalent integer order system. The detailed parameter space study reveals that the multi-rhythmic properties of the oscillation of this system can be efficiently controlled by fractional order parameter and Lévy noise parameters. Finally, the analytical solutions are validated by numerical results of the Monte Carlo simulation of the original fractional modified van der Pol oscillator. The trirhythmic region can be detected in the parameter space (α,β,γ,δ) by choosing an approximate fractional order. The stability index and the noise intensity can in any case govern the dynamics of the self sustained system, but regulating the skewness parameter cannot bring about transitions between unimodal, bimodal and trimodal in this multi-rhythmic system. More bifurcations appear by adjusting the fractional order in this system. These results may be conducive to further exploring bifurcations in the real-world applications. This study sheds new insight lights on the understanding nontrivial effects of fractional derivative on a trirhythmic self sustained system.