Topological features of large fluctuations to the interior of a limit cycle

Abstract

We investigate the pattern of optimal paths along which a dynamical system driven by weak noise moves, with overwhelming probability, when it fluctuates far away from a stable state. Our emphasis is on systems that perform self-sustained periodic vibrations, and have an unstable focus inside a stable limit cycle. We show that in the vicinity of the unstable focus, the flow field of optimal paths generically displays a pattern of singularities. In particular, it contains a switching line that separates areas to which the system arrives along optimal paths of topologically different types. The switching line spirals into the focus and has a self-similar structure. Depending on the behavior of the system near the focus, it may be smooth, or have finite-length branches. Our results are based on an analysis of the topology of the Lagrangian manifold for an auxiliary, purely dynamical, problem that determines the optimal paths. We illustrate our theory by studying, both theoretically and numerically, a van der Pol oscillator driven by weak white noise.