Dynamics of a new modified self-sustained biological trirythmic system with fractional time-delay feedback under correlated noise

The dynamics of a new modified Van der Pol (VDP) self-sustained oscillator, driven by fractional time-delay feedback under correlated noise, is addressed in this paper. The studied system presents a tristability mode with the coexistence of three stable limit cycles in the deterministic case. Under the generalized harmonic balance technique, the fractional derivative simultaneously includes an equivalent quasi-linear dissipative force and quasi-linear restoring force, which reduces the whole problem to an equivalent VDP equation without a fractional derivative. The stochastic averaging method investigates analytical solutions for the equivalent stochastic equation. The critical parametric conditions for stochastic P-bifurcation of amplitude are obtained via the singularity theory for the system switch among the three steady states. The analytical solutions are confronted with direct numerical simulations, in a process where the dynamical features of the system are characterized using the stationary probability density function (PDF) of amplitude and joint PDF of displacement and velocity. A satisfactory agreement is obtained between both approaches, therefore confirming the accuracy of the theoretical predictions. Changing the fractional order, the fractional coefficient, the time delay parameter, and the correlation time also appears to induce the occurrence of the stochastic P-bifurcation.