Abstract
An approximate method for predicting the stationary response of stochastically excited nonlinear systems with continuous-time Markov jump is proposed. By using the stochastic averaging method, the original system is reduced to one governed by a 1D averaged Itô equation for the total energy with the Markov jump process as parameter. A Fokker-Planck- Kolmogorov (FPK) equation is then deduced, from which the approximate stationary probability density of the response of the original system is obtained for different jump rules. To illustrate the effectiveness of the proposed method, a stochastically excited Markov jump Duffing system is worked out in detail.
摘要
目的
提出一种预测随机激励下连续时间马尔科夫跳 变非线性系统的平稳响应的近似方法。
创新点
1. 得到了含有马尔科夫跳变参数的关于能量的 平均Itô 方程;2. 建立了含有马尔科夫跳变参数 的平均Itô 方程相应的FPK 方程。
方法
1. 将一个随机激励的马尔科夫跳变非线性系统 由状态方程转化为等价的Itô 方程,并根据Itô 微分法则给出哈密顿量(系统总能量)的Itô 方 程;2. 通过随机平均法,得到关于系统能量的 平均Itô 方程;3. 推导并求解相应的FPK 方程。
结论
1. 跳变规律对马尔科夫跳变非线性系统随机响 应具有重要影响;2. 理论结果与数字模拟结果 吻合验证了理论方法的准确性。
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Project supported by the National Natural Science Foundation of China (Nos. 11272279, 11272201, 11321202, 11372271, 11432012, and 51175474)
ORCID: Rong-hua HUAN, http://orcid.org/0000-0003-4648-3805
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Pan, Ss., Zhu, Wq., Hu, Rc. et al. Stationary response of stochastically excited nonlinear systems with continuous-time Markov jump. J. Zhejiang Univ. Sci. A 18, 83–91 (2017). https://doi.org/10.1631/jzus.A1600176
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DOI: https://doi.org/10.1631/jzus.A1600176