Stationary response of strongly non-linear oscillator with fractional derivative damping under bounded noise excitation

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

“…The stationary response of SDOF strongly nonlinear oscillator with fractional derivative damping under bounded noise excitations in the case of external resonance has been studied by Hu et al [37]. In Fig.2, the stationary probability density of amplitude of fractionally damped Duffing oscillator under bounded noise excitation [37] is shown. It is seen that the analytical results agree well with those from the Monte Carlo simulation of original system.…”

“…Then, establishing and solving the corresponding FPK equations, one can obtain the stochastic response of system (58). The stationary response of SDOF strongly nonlinear oscillator with fractional derivative damping under bounded noise excitations in the case of external resonance has been studied by Hu et al [37]. In Fig.2, the stationary probability density of amplitude of fractionally damped Duffing oscillator under bounded noise excitation [37] is shown.…”

“…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.…”

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.…”