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

Ying-Cheng Lai, Z H Liu, L Billings, I B Schwartz

Research output: Contribution to journalArticle

35 Citations (Scopus)

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
Number of pages17
JournalPhysical Review. E, Statistical, Nonlinear and Soft Matter Physics
Volume67
Issue number2
DOIs
Publication statusPublished - 19 Feb 2003

Keywords

  • on-off intermittency
  • generalized dimensions
  • coherence resonance
  • strange attractors
  • stochastic chaos
  • fractal measures
  • synchronization
  • brain
  • sets
  • oscillators

Cite this

Noise-induced unstable dimension variability and transition to chaos in random dynamical systems. / Lai, Ying-Cheng; Liu, Z H ; Billings, L ; Schwartz, I B .

In: Physical Review. E, Statistical, Nonlinear and Soft Matter Physics, Vol. 67, No. 2, 026210, 19.02.2003.

Research output: Contribution to journalArticle

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AB - 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.

KW - on-off intermittency

KW - generalized dimensions

KW - coherence resonance

KW - strange attractors

KW - stochastic chaos

KW - fractal measures

KW - synchronization

KW - brain

KW - sets

KW - oscillators

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