Fractional stochastic vibration system under recycling noise

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