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Transition to chaos in continuous-time random dynamical systems.

Zonghua Liu1, Ying-Cheng Lai, Lora Billings

  • 1Department of Mathematics, Center for Systems Science and Engineering Research, Arizona State University, Tempe, Arizona 85287, USA.

Physical Review Letters
|March 23, 2002
PubMed
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Noise can induce chaos in dynamical systems with coexisting attractors. This study reveals a scaling law for the largest Lyapunov exponent and identifies topological disturbances caused by noise-induced changes in unstable directions.

Area of Science:

  • Dynamical Systems
  • Nonlinear Dynamics
  • Chaos Theory

Background:

  • Coexistence of nonchaotic attractors and chaotic saddles in continuous-time systems.
  • Periodic windows as examples of such coexistence.
  • Potential for noise to induce chaotic behavior.

Purpose of the Study:

  • Investigate the dynamical mechanism of noise-induced chaos.
  • Derive a general scaling law for the largest Lyapunov exponent.
  • Analyze topological changes in the flow due to noisy chaos.

Main Methods:

  • Analysis of continuous-time dynamical systems.
  • Perturbation analysis under the influence of noise.
  • Calculation of Lyapunov exponents.
  • Topological analysis of flow dynamics.

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Main Results:

  • A general scaling law for the largest Lyapunov exponent is obtained.
  • Noisy chaos fundamentally disturbs the topology of the flow.
  • Topological disturbance is linked to changes in unstable eigendirections.

Conclusions:

  • Noise can transition a system from nonchaotic to chaotic states.
  • The onset of noisy chaos involves significant topological alterations.
  • Understanding these topological changes is crucial for characterizing noisy chaotic systems.