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首页> 外文期刊>Zeitschrift fur Angewandte Mathematik und Mechanik >Invariant Measures of Perturbed Dynamical Systems
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Invariant Measures of Perturbed Dynamical Systems

机译：扰动动力系统的不变测度

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摘要

For stability investigations of dynamical systems with parametric perturbations, one can apply the multiplicative ergodic theorem of OSCELEDEC in order to calculate the top LYAPUNOV exponent of the system. The numerical results, obtained by this procedure, are considerably improved by utilizing the invariant measure of the system. They are derived by KHASMINSKII 's projection on hyperspheres leading to nonlinear projection equations and associated linear FOKKER-PLANCK equations, respectively. The paper explains special iterative schemes to solve such higher-dimensional partial differential equations and checks the obtained results by means of Monte-Carlo simulations. Both methods are applied to oscillators excited by generalized perturbation models. They are modelled in such a way that they include the limiting case of stochastic white noise as well as the deterministic case of harmonic or periodic or almost periodic parametric excitations. Finally, a typical application in structural dynamics is discussed.
机译：对于带有参数摄动的动力系统的稳定性研究，可以应用OSCELEDEC的可乘遍历定理来计算系统的最高LYAPUNOV指数。通过使用系统的不变度量，可以大大改善通过此过程获得的数值结果。它们是由KHASMINSKII在超球面上的投影得出的，分别导致非线性投影方程和相关的线性FOKKER-PLANCK方程。本文介绍了解决此类高维偏微分方程的特殊迭代方案，并通过蒙特卡洛仿真检查了所得结果。两种方法都适用于由广义扰动模型激发的振荡器。它们的建模方式使得它们包括随机白噪声的极限情况以及谐波或周期性或几乎周期性的参数激励的确定性情况。最后，讨论了结构动力学的典型应用。

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