## Hopf bifurcations of a stochastic fractional-order Van der Pol system.(English)Zbl 1474.34397

Summary: The Hopf bifurcation of a fractional-order Van der Pol (VDP for short) system with a random parameter is investigated. Firstly, the Chebyshev polynomial approximation is applied to study the stochastic fractional-order system. Based on the method, the stochastic system is reduced to the equivalent deterministic one, and then the responses of the stochastic system can be obtained by numerical methods. Then, according to the existence conditions of Hopf bifurcation, the critical parameter value of the bifurcation is obtained by theoretical analysis. Then, numerical simulations are carried out to verify the theoretical results.

### MSC:

 34F05 Ordinary differential equations and systems with randomness 34A08 Fractional ordinary differential equations 34C23 Bifurcation theory for ordinary differential equations 37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior

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