Open Access
June 2010 Study of noise-induced transitions in the Lorenz system using the minimum action method
Xiang Zhou, Weinan E
Commun. Math. Sci. 8(2): 341-355 (June 2010).


We investigate noise-induced transitions in non-gradient systems when complex invariant sets emerge. Our example is the Lorenz system in three representative Rayleigh number regimes. It is found that before the homoclinic explosion bifurcation, the only transition state is the saddle point, and the transition is similar to that in gradient systems. However, when the chaotic invariant set emerges, an unstable limit cycle continues from the homoclinic trajectory. This orbit, which is embedded in a local tube-like manifold around the initial stable stationary point as a relative attractor, plays the role of the most probable exit set in the transition process. This example demonstrates how limit cycles, the next simplest invariant set beyond fixed points, can be involved in the transition process in smooth dynamical systems.


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Xiang Zhou. Weinan E. "Study of noise-induced transitions in the Lorenz system using the minimum action method." Commun. Math. Sci. 8 (2) 341 - 355, June 2010.


Published: June 2010
First available in Project Euclid: 25 May 2010

zbMATH: 1202.34104
MathSciNet: MR2664454

Primary: 34D10 , 82C26 , 82C35

Keywords: limit cycle , Lorenz system , minimum action path , Noise-induced transitions , transition set

Rights: Copyright © 2010 International Press of Boston

Vol.8 • No. 2 • June 2010
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