Deterministic Properties of Stochastically Perturbed Dynamic Systems
Deterministic Properties of Stochastically Perturbed Dynamic Systems
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DOI：
10.1137/1133095
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发表时间：
1989
影响因子：
0.6
通讯作者：
M. Blank
中科院分区：
文献类型：
--
作者：
M. Blank
2.本文给出了离散时间未扰动动力系统的Onsager-Machlup泛函的Perron-Frobenius算子表示（定理1）。利用这种表示，我们能够证明MPP是“平均”的6-路的原始动力系统（定理2）。在双曲系统的情况下，它表明，后者的性质导致存在的路径的原始系统跟踪MPP”的平均值。“独立均匀分布扰动的研究对于应用是很重要的，它使人们能够澄清文献中描述的一些现象，例如，在数值模拟中，当噪声强度增加时，李雅普诺夫指数减小.本文对原动力系统f和随机扰动构造了一个新的确定性动力系统f，在两倍于原始尺寸的空间中。发现随机扰动系统f的MPP是f_i的路径，因此很自然地认为f_i是扰动系统的确定性分量。应用Aubry关于Onsager-Machlup泛函的理论[14]，我们可以研究MPPs的渐近性质。特别是，我们发现在某些附加假设下，具有强随机性质的一维动力系统（例如，fx 2x（mod 1））的MPP是周期或准周期路径（定理6和7）。我们注意到
2. In this paper we obtain a representation of the Onsager-Machlup functional in terms of the Perron-Frobenius operator 10], 13] of an unperturbed dynamic system with discrete time (Theorem 1). Using this representation, we are able to show that the MPP is" in the mean" a 6-path for the original dynamic system (Theorem 2). In the case of a hyperbolic system, it is shown that the latter property leadsto the existence of a path of the original system tracking the MPP" in the mean." The investigation of the case of independent uniformly distributed perturbations which is important for applications has permitted one to clarify a number of the phenomena described in the literature, eg, the diminution of the Lyapunov exponent as the noise intensity increases in numerical modeling.In this paper we construct for the original dynamic system f and a stochastic perturbation a newdeterministic dynamic system f, in the space of twice the original dimension. It is found that the MPPs of a stochastically perturbed system f are the paths of f,, and therefore it is natural to consider f, as the deterministic component of the perturbed system. Application of Aubry’stheory 14] for theOnsager-Machlup functional in the one-dimensional case permits us to investigate the asymptotic proper-ties of MPPs. In particular, we findthat under certain additional assumptions, the MPPs ofone-dimensional dynamic systems with strong stochastic properties (eg, fx 2x (mod 1)) are periodic or quasiperiodic paths (Theorems 6 and 7). We remark