Noise-induced unstable dimension variability and transition to chaos in random dynamical systems

作     者：Ying-Cheng Lai Zonghua Liu Lora Billings Ira B. Schwartz 

作者机构：Department of Mathematics Center for Systems Science and Engineering Research Arizona State University Tempe Arizona 85287 Departments of Electrical Engineering and Physics Arizona State University Tempe Arizona 85287 Department of Mathematical Sciences Montclair State University Upper Montclair New Jersey 07043 Plasma Physics Division Naval Research Laboratory Code 6792 Washington DC 20375 

出 版 物：《Physical Review E》 (物理学评论E辑：统计、非线性和软体物理学)

年 卷 期：2003年第67卷第2期

页      面：026210-026210页

核心收录：

学科分类：07[理学] 070203[理学-原子与分子物理] 0702[理学-物理学] 

主　　题：.variability noisy DIMENSION Dynamical Systems unstable Chaotic Lyapunov exponent noise-induced Orbits periodic window saddle periodic attractor chaos in random 

摘      要：Results are reported concerning the transition to chaos in random dynamical systems. In particular, situations are considered where a periodic attractor coexists with a nonattracting chaotic saddle, which can be expected in any periodic window of a nonlinear dynamical system. Under noise, the asymptotic attractor of the system can become chaotic, as characterized by the appearance of a positive Lyapunov exponent. Generic features of the transition include the following: (1) the noisy chaotic attractor is necessarily nonhyperbolic as there are periodic orbits embedded in it with distinct numbers of unstable directions (unstable dimension variability), and this nonhyperbolicity develops as soon as the attractor becomes chaotic; (2) for systems described by differential equations, the unstable dimension variability destroys the neutral direction of the flow in the sense that there is no longer a zero Lyapunov exponent after the noisy attractor becomes chaotic; and (3) the largest Lyapunov exponent becomes positive from zero in a continuous manner, and its scaling with the variation of the noise amplitude is algebraic. Formulas for the scaling exponent are derived in all dimensions. Numerical support using both low- and high-dimensional systems is provided.