Study of noise-induced transitions in the lorenz system using the minimum action method

Xiang Zhou, E. Weinan

Research output: Contribution to journalArticlepeer-review

30 Scopus citations


Dedicated to the sixtieth birthday of Professor Andrew Majda 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.

Original languageEnglish (US)
Pages (from-to)341-355
Number of pages15
JournalCommunications in Mathematical Sciences
Issue number2
StatePublished - Jun 2010

All Science Journal Classification (ASJC) codes

  • General Mathematics
  • Applied Mathematics


  • Limit cycle
  • Lorenz system
  • Minimum action path
  • Noise-induced transitions
  • Transition set


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