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Stochastic averaging of quasi-non-integrable Hamiltonian systems under fractional Gaussian noise excitation. (English) Zbl 1349.70036

Summary: A stochastic averaging method for quasi-non-integrable Hamiltonian systems under excitation of fractional Gaussian noise (fGn) with Hurst index \(1/2<H<1\) is proposed. First, the definitions and the basic properties of fGn and fractional Brownian motion are briefly introduced. Then, the averaged stochastic differential equation for the total energy of the quasi-non-integrable Hamiltonian system under fGn excitation is derived. The simulation and asymptotic analysis for two examples are conducted to illustrate the proposed stochastic averaging method. It is shown that the simulation results obtained from averaged equation and from original systems agree well and that the displacement response process of linear system to fGn has the same long-range dependence index and the same \(H\) self-similarity as those of the fGn excitation.

MSC:

70H08 Nearly integrable Hamiltonian systems, KAM theory
70K65 Averaging of perturbations for nonlinear problems in mechanics
37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
70H07 Nonintegrable systems for problems in Hamiltonian and Lagrangian mechanics
34A08 Fractional ordinary differential equations
37H05 General theory of random and stochastic dynamical systems
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