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

Abstract

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.

Crossing the quasi-threshold manifold of a noise-driven excitable system

“…Moreover, the minimizing path of the action functional gives the pathway in whose vicinity the transitions occur with maximal probability. In past decades, a great deal of work has been devoted to problems of noise-induced transitions or escapes from equilibria [12], limit cycles [13,14] and chaotic attractors [15,16]. However, most works studied cases where a well-defined threshold manifold can be found.…”