Statistical properties of the circle map

Abstract

The circle map provides a generic model for the response of damped, nonlinear oscillators to periodic perturbations. As the parameters are varied this dynamical system exhibits transitions from regular, ordered behavior to chaos. In this paper the regular and irregular behavior of the circle map are examined for a broad range of nonlinearities and frequencies which span not only the regular and transition regimes, which have been studied extensively, but also the strongly nonlinear, chaotic regime. We discuss the bistable behavior (split bifurcations) associated with maps with multiple extrema that gives rise to two disjoint attractors which can be periodic or chaotic and to the low order periodic orbits emerging from chaos via tangent bifurcations. To describe the chaotic behavior we use a statistical description based on a path integral formulation of classical dynamics. This path integral method provides a convenient means of calculating statistical properties of the nonlinear dynamics, such as the invariant measure of the average Lyapunov exponent, which in many cases reduce to analytic expressions or to numerical calculations which can be completed in a fraction of the time required to explicitly iterate the map.