Sticky orbits in a kicked-oscillator model

Abstract

We study a four-fold symmetric kicked-oscillator map with sawtooth kick function. For the values of the kick amplitude λ =2 cos (2π p/q) with rational p / q, the dynamics is known to be pseudochaotic, with no stochastic web of non-zero Lebesgue measure. We show that this system can be represented as a piecewise affine map of the unit square—the so-called local map—driving a lattice map. We develop a framework for the study of long-time behaviour of the orbits, in the case in which the local map features exact scaling. We apply this method to several quadratic irrational values of λ, for which the local map possesses a full Legesgue measure of periodic orbits; these are promoted to either periodic orbits or accelerator modes of the kicked-oscillator map. By constrast, the aperiodic orbits of the local map can generate various asymptotic behaviours. For some parameter values the orbits remain bounded, while others have excursions which grow logarithmically or as a power of the time. In the power-law case, we derive rigorous criteria for asymptotic scaling, governed by the largest eigenvalue of a recursion matrix. We illustrate the various behaviours by performing exact calculations with algebraic numbers; the hierarchical nature of the symbolic dynamics allows us to sample extremely long orbits with high efficiency, i.e. uniformly on a logarithmic time scale.