On stochastic behavior of perturbed Hamiltonian systems

Abstract

We consider deterministic perturbations $\ddot q^\varepsilon(t)+F'(q^\varepsilon(t))=\varepsilon b(\dot q^\varepsilon(t),q^\varepsilon(t))$ of an oscillator $\ddot q+F'(q)=0$, $q\in{\mathbb R}^1$. Assume that $\lim_{|q|\to\infty}F(q)=\infty$ and that $F'(q)$ has a finite number of nondegenerate zeros. For a generic $F$, if $\partial b/\partial\dot q<0$ (as in the case of friction), then typical orbits are attracted to points where $F$ has a local minimum. For $0<\varepsilon\ll1$, the equilibrium to which the trajectory is attracted is ‘random’. To study this randomness, which is caused by the sensitive behavior of trajectories near the saddle points, we consider the graph $\Gamma$ homeomorphic to the space of connected components of the level sets of the Hamiltonian $H(p,q)=p^2/2+F(q)$. We show that, as $\varepsilon\to0$, the slow component of $(p^\epsilon(t/\epsilon),q^\epsilon(t/\epsilon))$ tends to a certain stochastic process on $\Gamma$ which is deterministic inside the edges and branches at the interior vertices into adjacent edges with probabilities which can be calculated through the Hamiltonian $H$ and the perturbation $b$.