谐和与白（宽带）噪声联合激励下具有分数导数型阻尼的拟可积哈密顿系统的随机动力学研究
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摘要
本文研究谐和与白噪声或谐和与宽带噪声联合激励下具有分数导数型阻尼的拟可积哈密顿系统的随机响应、首次穿越破坏及随机稳定性。首先,基于广义谐和函数,推导了谐和与白(宽带)噪声联合激励下具有分数导数型阻尼的拟可积哈密顿系统的随机平均方程,指出了平均It(?)方程与系统共振性之间的关系,给出了平均It(?)方程的漂移与扩散系数。在此基础上,建立了响应过程的概率密度所满足的Fokker-Planck-Kolmogorov(FPK)方程、支配条件可靠性函数的后向Kolmogorov方程及支配平均首次穿越时间的Pontryagin方程,分别求解这些方程得到系统的响应统计量、条件可靠性函数、首次穿越时间条件概率密度以及平均首次穿越时间。通过一系列的算例分别讨论了分数阶数对系统随机响应以及可靠性的影响。另外,本文还研究了谐和与白噪声联合激励下两个非线性耦合Duffing振子的首次穿越破坏。最后,用最大Lyapunov指数研究谐和与白(宽带)噪声联合激励下具有分数导数型阻尼的拟可积哈密顿系统在非共振与共振情形的概率为1渐近稳定性,以及分数阶数对单自由度系统概率为1渐近稳定性的影响。
The stochastic response, first passage failure and stochastic stability of quasi integrable Hamiltonian systems with fractional derivative damping subjected to combined harmonic and Gauss white noise excitations or combined harmonic and wide band noise excitations are investigated. First, the stochastic averaged equations for quasi integrable Hamiltonian systems with fractional derivative damping under combined harmonic and Gauss white (wide band) noise excitations are derived by using the generalized harmonic functions. It is pointed out that the form and dimension of averaged Ito equations depend upon the resonance of systems. The drift and diffusion coefficients of the averaged Ito equation are given. Then, the Fokker-Planck-Kolmogorov ( FPK) equations governing the probability density of response processes, the backward Kolmogorov governing the conditional reliability function and the Pontryagin equation governing the mean first passage time are established. The response statistics, the conditional reliability function, the conditional probability density and mean of first passage time of the system are obtained by solving these equations, respectively. The effects of fractional order on the stochastic response and reliability are discussed via a series of examples, respectively. In addition, the first passage failure of two nonlinearly coupled Duffing oscillators subjected to combined harmonic and Gaussian white noises excitations is investigated. Finally, the asymptotic Lyapunov stability with probability one of combined harmonic and white (wide band) noise excited quasi integrable Hamiltonian systems with fractional derivative damping in the cases of non-resonance and resonance is studied by using the largest Lyapunov exponent and the effect of fractional order on the asymptotic stability of a single-degree-of-freedom system is investigated.
The stochastic response, first passage failure and stochastic stability of quasi integrable Hamiltonian systems with fractional derivative damping subjected to combined harmonic and Gauss white noise excitations or combined harmonic and wide band noise excitations are investigated. First, the stochastic averaged equations for quasi integrable Hamiltonian systems with fractional derivative damping under combined harmonic and Gauss white (wide band) noise excitations are derived by using the generalized harmonic functions. It is pointed out that the form and dimension of averaged Ito equations depend upon the resonance of systems. The drift and diffusion coefficients of the averaged Ito equation are given. Then, the Fokker-Planck-Kolmogorov ( FPK) equations governing the probability density of response processes, the backward Kolmogorov governing the conditional reliability function and the Pontryagin equation governing the mean first passage time are established. The response statistics, the conditional reliability function, the conditional probability density and mean of first passage time of the system are obtained by solving these equations, respectively. The effects of fractional order on the stochastic response and reliability are discussed via a series of examples, respectively. In addition, the first passage failure of two nonlinearly coupled Duffing oscillators subjected to combined harmonic and Gaussian white noises excitations is investigated. Finally, the asymptotic Lyapunov stability with probability one of combined harmonic and white (wide band) noise excited quasi integrable Hamiltonian systems with fractional derivative damping in the cases of non-resonance and resonance is studied by using the largest Lyapunov exponent and the effect of fractional order on the asymptotic stability of a single-degree-of-freedom system is investigated.
引文
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