Stochastic jump and bifurcation of Duffing oscillator with fractional derivative damping under combined harmonic and white noise excitations

Stochastic dynamics and fractional optimal control of quasi integrable Hamiltonian systems with fractional derivative damping

“…This is essentially a typical P-bifurcation and it is called the bifurcation of stochastic jump. The procedure has been extended to the Duffing oscillator with fractional derivative damping subjected to combined harmonic and white excitations [26] or bounded noise excitation [37]. Fig.6 shows the stationary probability density of amplitude for different fractional derivative order [26].…”

“…In recent years, this stochastic averaging method was extended to the quasi integrable Hamiltonian systems with fractional derivative damping under excitations of Gaussian white noise [39], combined harmonic function and Gaussian white noise [20], wide-band real noise [27] and bounded narrow-band noise [37] and the generalized stochastic averaging method was applied to study the stochastic response [27], [37], [39], stochastic stability [17], [19], stochastic bifurcation [26], [36], [37], first passage time and reliability [20], [21], [25], [28], and fractional stochastic optimal control [38] of strongly nonlinear stochastic systems with fractional derivative damping.…”

First passage of stochastically dynamical system with fractional derivative and power-form restoring force under Gaussian excitation

“…For example, in the famous Duffing system, the restoring force is a threeorder polynomial function [5]. However, a purely nonlinear function is more reasonable to describe the restoring force, that is why we want to consider the power-form restoring force in our system.…”

On simulation of a bistable system with fractional damping in the presence of stochastic coherence resonance

“…Their analysis results show that the fractional-order-damped Duffing system could be treated as a bifurcation parameter. By continuing these studies, Chen et al [20] and Hu et al [21] analysed such a system with a bounded noise excitation term composed of harmonic excitation with an additional random phase. The authors investigated the appearance of bimodal amplitude through a corresponding probability density.…”