Noise-induced stabilization of a Gaussian-curve class of Hamiltonian systems

Abstract. In this article, we analyze a class of Hamiltonian systems whose phase portrait consists of Gaussian curves. The class of Hamiltonian systems is unstable because the trajectories approach infinity along the x-axis. We consider perturbing the systems in two distinct ways: (1) a small deterministic push toward the origin that preserves the qualitative behavior of the systems and (2) constant white noise. We prove that neither perturbation by itself is sufficient to stabilize the class of Hamiltonian systems, but that the combination of both perturbation types results in stabilization by creating quasi-periodic behavior.