随机敏感度函数法在非自治非线性系统中的应用

摘要

Distribution characters of periodic attractors in non-autonomous differential dynamic systems disturbed by weak Gaussian white noise were studied.Based on stroboscopic mapping,differential dynamic systems were discretized into maps.Through solving stochastic sensitivity functions of periodic attractors in maps,confidence ellipses were constructed to describe the distributions of random attractors.In this way,solving boundary value problems of matrix differential equations was avoided,and only matrix algebraic equations needed to be solved.Distributions of stochastic periodic attractors in Duffing equation were studied.The results showed that confidence ellipses achieve a good agreement with Monte-Carlo simulation.Finally,noise-induced chaos in Duffing equation were investigated qualitatively,it was shown that stochastic sensitivity functions can reveal the mechanism of this kind of phenomena.%研究非自治微分动力系统周期吸引子受弱高斯白噪声扰动后的分布特性。基于频闪映射将微分动力系统离散为映射，通过求解映射系统周期吸引子的随机敏感度函数，构造置信椭圆来刻画随机吸引子的分布情况，从而避免了求解矩阵微分方程的边值问题，只需求解矩阵代数方程即可。研究了 Duffing 方程随机周期吸引子的分布情况，结果表明置信椭圆与 Monte －Carlo 模拟取得了很好的一致。最后对 Duffing 方程的噪声诱导混沌现象进行了定性研究，证明了通过随机敏感度函数可以揭示这类现象的机理。