Contents Online
Communications in Mathematical Sciences
Volume 8 (2010)
Number 2
Special Issue on the Occasion of Andrew Majda’s Sixtieth Birthday: Part II
Study of noise-induced transitions in the Lorenz system using the minimum action method
Pages: 341 – 355
DOI: https://dx.doi.org/10.4310/CMS.2010.v8.n2.a3
Authors
Abstract
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.
Keywords
Noise-induced transitions, Lorenz system, limit cycle, transition set, minimum action path
2010 Mathematics Subject Classification
34D10, 82C26, 82C35
Published 1 January 2010