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首页> 外文期刊>Physical review, E. Statistical physics, plasmas, fluids, and related interdisciplinary topics >Noise-induced unstable dimension variability and transition to chaos in random dynamical systems - art. no. 026210
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Noise-induced unstable dimension variability and transition to chaos in random dynamical systems - art. no. 026210

机译：随机动力学系统中由噪声引起的不稳定尺寸可变性和过渡到混沌-艺术。没有。 026210

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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. [References: 81]
机译：报告了有关随机动力系统向混沌过渡的结果。特别是考虑了周期性吸引子与不吸引人的混沌鞍共存的情况，这在非线性动力学系统的任何周期窗口中都是可以预期的。在噪声下，系统的渐近吸引子会变得混乱，以正Lyapunov指数的出现为特征。过渡的一般特征包括：（1）嘈杂的混沌吸引子必定是非双曲线的，因为其中嵌入了周期轨道，且轨道的不稳定方向数不一（尺寸不稳定），并且吸引子一旦变得混沌，这种非双曲线就会发展。 ; （2）对于用微分方程描述的系统，不稳定的尺寸可变性破坏了流体的中性方向，即在嘈杂的吸引子变得混乱之后，不再有零的Lyapunov指数; （3）最大的李雅普诺夫指数连续从零变为正，并且随着噪声幅度的变化其缩放比例是代数的。比例指数的公式是在所有维度上得出的。提供了使用低维和高维系统的数值支持。 [参考：81]

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