Hyperbolic periodic orbits in nongradient systems and small-noise-induced metastable transitions
For researchers studying rare events in stochastic dynamical systems, this work extends the theory of metastable transitions to nongradient systems with periodic orbits, though the results are partly incremental and based on numerical observations.
The paper studies small-noise-induced metastable transitions in nongradient systems, showing that maximum likelihood paths can cross the separatrix at hyperbolic periodic orbits rather than saddle points. A string method is proposed to identify such orbits, and numerical experiments reveal that in general cases the separatrix crossing may not determine a unique local maximum of the transition rate.
Small noise can induce rare transitions between metastable states, which can be characterized by Maximum Likelihood Paths (MLPs). Nongradient systems contrast gradient systems in that MLP does not have to cross the separatrix at a saddle point, but instead possibly at a point on a hyperbolic periodic orbit. A numerical approach for identifying such unstable periodic orbits is proposed based on String method. In a special class of nongradient systems (`orthogonal-type'), there are provably local MLPs that cross such saddle point or hyperbolic periodic orbit, and the separatrix crossing location determines the associated local maximum of transition rate. In general cases, however, the separatrix crossing may not determine a unique local maximum of the rate, as we numerically observed a counter-example in a sheared 2D-space Allen-Cahn SPDE. It is a reasonable conjecture that there are always local MLPs associated with each attractor on the separatrix, such as saddle point or hyperbolic periodic orbit; our numerical experiments did not disprove so.