Fractional stochastic vibration system under recycling noise

The fractional stochastic vibration system is quite different from the traditional one, and its application potential enormous if noise can be deployed correctly connection between order property unlocked. This article uses a modification of well-known van der Pol oscillator with multiplicative additive recycling noises as an example to study stationary response bifurcation. First, based on principle minimum mean square error, derivative equivalent linear combination damping restoring forces, original simplified into integer system. Second, Itô differential equations One-dimensional Markov process are obtained according averaging method, using Oseledec ergodic theorem maximal Lyapunov exponent judge local stability, judging global stability done by singularity theory. Lastly, D-bifurcation behavior model analyzed dynamical invariant measure, probability density function solved FPK equation. results show that greatly affect system’s properties. paper offers profound, original, challenging window for investigating systems.