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

Ying Cheng Lai, Zonghua Liu, Lora Billings, Ira B. Schwartz

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    39 Scopus citations

    Abstract

    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.

    Original languageEnglish
    Article number026210
    Pages (from-to)262101-2621017
    Number of pages1
    JournalPhysical Review E - Statistical, Nonlinear, and Soft Matter Physics
    Volume67
    Issue number2
    DOIs
    StatePublished - 1 Feb 2003

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