TOPOLOGICAL FEATURES OF LARGE FLUCTUATIONS TO THE INTERIOR OF A LIMIT-CYCLE

We investigate the pattern of optimal paths along which a dynamical sy stem driven by weak noise moves, with overwhelming probability,when it fluctuates far away from a stable state. Our emphasis is on systems t hat perform self-sustained periodic vibrations, and have an unstable f ocus 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 t he system near the focus, it may be smooth, or have finite-length bran ches. Our results are based on an analysis of the topology of the Lagr angian manifold for an auxiliary, purely dynamical, problem that deter mines the optimal paths. We illustrate our theory by studying, both th eoretically and numerically, a van der Pol oscillator driven by weak w hite noise.