Global transport in a nonautonomous periodic standard map

•An non-autonomous version of the standard map is introduced and the two periodic case is studied via an autonomous map of two parameters.•Symmetry properties, period-one orbits and reductions of autonomous map are found and 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 symmetric star shape.

A non-autonomous version of the standard map with a periodic variation of the perturbation parameter is introduced and studied via an autonomous map obtained from the iteration of the nonautonomous map over a period. Symmetry properties in the variables and parameters of the map are found and used to find relations between rotation numbers of invariant sets. The role of the nonautonomous dynamics on period-one orbits, stability and bifurcation is studied. The critical boundaries for the global transport and for 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.