Global transport in a nonautonomous standard map

TL;DR

This paper introduces and analyzes a non-autonomous version of the standard map with periodic parameter variation, exploring its symmetry properties, invariant sets, and conditions for global transport and invariant circle destruction.

Contribution

It presents a novel non-autonomous standard map, investigates its symmetry, and details the critical boundaries for global transport and invariant circle breakdown.

Findings

Symmetry properties relate rotation numbers of invariant sets.

Critical boundaries for transport have a horn-shaped structure.

Non-autonomous dynamics significantly affect stability and bifurcations.

Abstract

A non-autonomous version of the standard map with a periodic variation of the parameter is introduced and studied. Symmetry properties in the variables and parameters of the map are found and used to find relations between rotation numbers of invariant sets of the autonomous realization of the period-two case of the map. The role of the nonautonomous dynamics on period-one orbits, stability and bifurcation is studied. The critical boundaries for the global transport and the destruction of invariant circles with fixed rotation number are studied in detail using direct computation and a continuation method. In the case of global transport, the critical boundary has a particular symmetrical horn shape. The results are contrasted with similar calculations found in the literature.