Coherence and stochastic resonance in the fractional-birhythmic self-sustained system subjected to fractional time-delay feedback and Lévy noise

Abstract

This paper treats the nonlinear dynamics of an enzymatic-substrate modeled by the fractional multi-limit cycles Van der Pol oscillator with fractional time-delay feedback device subjected to Lévy noise perturbation and periodic excitation. The fractional electronic circuit has been used to model the system and the oscillations are described by a nonlinear fractional differential equation and show a new bifurcation parameter. The robustness of the stochastic resonance is examined by the use of standard measures within a continuous and a two-state description of the system. Firstly, the electronic circuit with the fractional-order operator and fractional delay feedback is used as a prototype of a fractional self-sustained system. Secondly, 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. The stochastic bifurcation of a bistable Van der Pol system with fractional-order and time delay without and under Lévy noise excitation is studied where we show the considerable effect of these parameters on birhythmic region, escape time and energy barriers. Additionally, we study the multi-effects of fractional order and fractional time-delay feedback in the self-sustained system driven by Lévy noise. The effects of fractional-order parameter, time delay feedback parameter on the autocorrelation function, power spectral density and signal-to-noise-ratio used in this investigation are shown to be maximized for an appropriate choice of the Lévy noise intensity and for a convenient choice of fractional-order and time delay feedback parameters. For a choice of a control parameter in the birhythmic region, by varying the other parameters of the system like fractional and time delay parameters, it appears that, for a fixed value of skewness Lévy noise parameter, the initial selection of the attractor seems to have a large effect on the resonance and coherence.