Transition to Chaos in Continuous-Time Random Dynamical Systems

Zonghua Liu; Ying-Cheng Lai; Lora Billings; Ira B. Schwartz

doi:10.1103/PhysRevLett.88.124101

Transition to Chaos in Continuous-Time Random Dynamical Systems

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

Phys. Rev. Lett. 88, 124101 – Published 11 March 2002

Abstract

We consider situations where, in a continuous-time dynamical system, a nonchaotic attractor coexists with a nonattracting chaotic saddle, as in a periodic window. Under the influence of noise, chaos can arise. We investigate the fundamental dynamical mechanism responsible for the transition and obtain a general scaling law for the largest Lyapunov exponent. A striking finding is that the topology of the flow is fundamentally disturbed after the onset of noisy chaos, and we point out that such a disturbance is due to changes in the number of unstable eigendirections along a continuous trajectory under the influence of noise.

Received 6 November 2001

DOI:https://doi.org/10.1103/PhysRevLett.88.124101

Authors & Affiliations

Zonghua Liu¹, Ying-Cheng Lai^1,2, Lora Billings³, and Ira B. Schwartz⁴

¹Department of Mathematics, Center for Systems Science and Engineering Research, Arizona State University, Tempe, Arizona 85287
²Department of Electrical Engineering and Physics, Arizona State University, Tempe, Arizona 85287
³Department of Mathematical Sciences, Montclair State University, Upper Montclair, New Jersey 07043
⁴Special Project in Nonlinear Science, Code 67003, Plasma Physics Division, Naval Research Laboratory, Washington, D.C. 20375