Stochastic Excited Hamiltonian Systems

Zhu, Wei; Cai, Guorong; Ye, Lin

doi:10.1007/978-3-642-84789-9_47

Cited by 8 publications

(5 citation statements)

References 12 publications

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“…A great deal of effort has been devoted to finding exact probability densities for the FP equations of various nonlinear systems. The early results can be found in a review paper by Fuller [1] and great progress has been achieved [2][3][4][5][6][7][8][9][10][11]. The recent results can been found in [12-16, 18, 32-36].…”

Section: Introductionmentioning

confidence: 99%

Exact stationary solutions of the Fokker-Planck equation for nonlinear oscillators under stochastic parametric and external excitations

Wang

Zhang

2000

Nonlinearity

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A systematic procedure is developed to obtain the stationary probability density function for the response of general single-degree-of-freedom nonlinear oscillators under parametric and external Gaussian white noise excitations. In a previous paper (Wang and Zhang 1998 J. Eng. Mech. ASCE 18 18-23) we expressed a nonlinear function of oscillators using a polynomial formula. The nonlinear system described here has the following form: ẍ) ẋi and ξ 1 , ξ 2 are Gaussian white noise functions. Thus, this paper is a generalization of the results studied in our previous paper. The stationary Fokker-Planck equation is employed to obtain the governing equation of the probability density function. Based on this procedure, the exact stationary probability densities of many nonlinear stochastic oscillators are obtained and it is shown that some of the exact stationary solutions described in the literature are only particular cases of the presented generalized results.

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Section: Introductionmentioning

confidence: 99%

Exact stationary solutions of the Fokker-Planck equation for nonlinear oscillators under stochastic parametric and external excitations

Wang

Zhang

2000

Nonlinearity

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“…This is in principle achievable by employing the multidimensional analog of the scalar nonlinear Fokker-Planck equations used here. The main difficulty in this direction is the fact that analytical stationary solutions for multidimensional nonlinear Fokker-Planck equations are often unavailable, even for relatively simple systems such as stochastic oscillators [42,[59][60][61]. Another interesting direction is to derive the appropriate nonlinear Fokker-Planck equations that correspond to the controlled SDDE, without the small time-delay limitation.…”

Section: Discussionmentioning

confidence: 99%

Mitigation of rare events in multistable systems driven by correlated noise

Mamis,

Farazmand

2021

Preprint

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We consider rare transitions induced by colored noise excitation in multistable systems. We show that undesirable transitions can be mitigated by a simple time-delay feedback control if the control parameters are judiciously chosen. We devise a parsimonious method for selecting the optimal control parameters, without requiring any Monte Carlo simulations of the system. This method relies on a new nonlinear Fokker-Planck equation whose stationary response distribution is approximated by a rapidly convergent iterative algorithm. In addition, our framework allows us to accurately predict, and subsequently suppress, the modal drift and tail inflation in the controlled stationary distribution. We demonstrate the efficacy of our method on two examples, including an optical laser model perturbed by multiplicative colored noise.

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“…Some remarks are pertinent at this point. (i) The cases of non-resonant and resonant are classi"ed in the present paper based on the internal resonance of the Hamiltonian system associated with Ito( equation (10). Since deterministic excitations g G are harmonic functions t with frequencies G which are the same as the frequencies of the associated Hamiltonian system, system (9) is often resonant externally.…”

Section: Resonant Casementioning

confidence: 99%