Cycle Expansions for Intermittent Maps

In a generic dynamical system chaos and regular motion coexist side by side, in different parts of the phase space. The border between these, where trajectories are neither unstable nor stable but of marginal stability, manifests itself through intermittency, dynamics where long periods of nearly regular motions are interrupted by irregular chaotic bursts. We discuss the Perron-Frobenius operator formalism for such systems, and show by means of a 1-dimensional intermittent map that intermittency induces branch cuts in dynamical zeta functions. Marginality leads to long-time dynamical correlations, in contrast to the exponentially fast decorrelations of purely chaotic dynamics. We apply the periodic orbit theory to quantitative characterization of the associated power-law decays. In fluid dynamics one often observes long periods of regular dynamics (laminar phases) interrupted by irregular chaotic bursts, with the distribution of laminar phase intervals well described by a power law. This phenomenon is called intermittency 1), 2) , and it is a very general aspect of dynamics, a shadow cast by nonhyperbolic and marginally stable phase space regions. Complete hyperbolicity is indeed the exception rather than the rule: almost any dynamical system of interest exhibits a mixed phase space where islands of stability or regular regions coexist with hyperbolic regions with dynamics mixing exponentially fast. The trajectories on the border between chaos and regular dynamics are marginally stable, and trajectories from chaotic regions which come close to marginally stable regions can stay ‘glued’ there for arbitrarily long times. These intervals of regular motion are interrupted by irregular bursts as the trajectory is re-injected into the chaotic part of the phase space. How the trajectories are precisely ‘glued’ to the marginally stable region is often hard to describe. What coarsely looks like a border of an island in a Hamiltonian system with a mixed phase space, will under magnification dissolve into infinities of island chains of decreasing sizes and cantori 3) . The existence of marginal or nearly marginal orbits is due to incomplete intersections of stable and unstable manifolds in a Smale horseshoe type dynamics. Following