发表机构
Academy of Mathematics and Systems Science, Chinese Academy of Sciences; University of Chinese Academy of Sciences（中国科学院数学与系统科学研究院; 中国科学院大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究随机自参数块摆系统离散化的动力学特征，证明所提离散化方法的随机吸引子收敛性及单模态解数值李雅普诺夫指数的符号一致性，保留原系统的全局渐近动力学与稳定性。
AI 中文摘要
对于随机自参数块摆系统，其长期动力学表现出两个基本特征：由李雅普诺夫指数表征的单模态解的几乎必然稳定性，以及该单模态解失稳时的全局渐近动力学。这自然引出一个问题：这些动力学特征在离散化过程中是否能被保留，因为这种保留对离散系统能否忠实地捕捉连续系统的定性行为至关重要。为解决该问题，我们首先针对受乘性随机激励的连续系统建立随机吸引子的存在性，为全局渐近动力学提供严格刻画；随后提出一种数值离散化方法，该方法诱导出离散随机动力系统，并证明当步长趋于零时，其随机吸引子收敛到连续系统的随机吸引子。此外，我们还表明，对于足够小的步长，单模态解的数值李雅普诺夫指数与连续系统对应指数符号相同，从而保留了相应的几乎必然稳定性或失稳分类。这些结果表明，所提出的离散化方法既捕捉了潜在随机自参数系统的全局渐近动力学，又保留了其稳定性特征。
英文摘要
For a stochastic autoparametric block-and-pendulum system, the long-time dynamics exhibit two fundamental features: the almost-sure stability of the single mode solution, characterized by its Lyapunov exponent, and the global asymptotic dynamics when this single mode solution loses stability. This naturally raises the question of whether these dynamical features are preserved under discretization, since such preservation is essential for the resulting discrete system to faithfully capture the qualitative behavior of the continuous system. To address this question, we first establish the existence of a random attractor for the continuous system subject to multiplicative stochastic excitation, providing a rigorous characterization of the global asymptotic dynamics. We then propose a numerical discretization that induces a discrete random dynamical system and prove the convergence of its random attractor to the continuous one as the step size tends to zero. In addition, we show that the numerical Lyapunov exponent of the single mode solution has the same sign as its continuous counterpart for sufficiently small step sizes, thus preserving the corresponding almost-sure stability or instability classification. These results demonstrate that the proposed discretization captures both the global asymptotic dynamics and the stability characteristics of the underlying stochastic autoparametric system.
