Phase space structure of nonautonomous one-degree-of-freedom Hamiltonian systems with nontrivial time dependence

Abstract

It has been recently argued that near-integrable nonautonomous one-degree-of-freedom Hamiltonian systems are constrained by KAM theory even when the time-dependent (nonintegrable) part of the Hamiltonian is given in the form of a superposition of time-periodic functions with incommensurate frequencies. Furthermore, such systems are constrained by one fewer integral of motion than is required to render the system completely integrable. As a consequence, the phase space in systems of this type is expected to be partitioned into nonintersecting regular and chaotic regions. In this note we provide numerical evidence of the existence of such a characteristic mixed phase space structure. This is done by considering the problem of acoustic ray dynamics in deep ocean environments, which is naturally described as a nonautonomous one-degree-of-freedom Hamiltonian system with a multiply periodic Hamiltonian in the independent (time-like) variable. Also, we discuss the implications of a mixed phase space for the dynamics of that geophysical system and another one which describes Lagrangian motion in the ocean. The latter is also naturally described as a nonautonomous one-degree-of-freedom Hamiltonian system with a multiply time-periodic Hamiltonian.