Nontwist Area Preserving Maps with Reversing Symmetry Group

The aim of this paper is to give a theoretical explanation of the rich phenomenology exhibited by nontwist mappings of the cylinder, in numerical experiments reported in [del-Castillo et al., 1996; Howard & Humpherys, 1995], and to give new insights on the dynamics of nontwist standard-like maps. Our approach is based on the reversing symmetric properties of the nontwist standard-like systems. We relate the bifurcation of periodic orbits of such maps to their position with respect to a group-invariant circle, and to a critical circle, at whose points the twist property is violated. Moreover, we show that appearance of the meanders, i.e. homotopically nontrivial invariant circles of the cylinder, that are not graphs of real functions of angular variable, is a global bifurcation of a curve on the cylinder, with nodal-type singularities, and invariant with respect to the action of a reversing symmetry group.