KAM Theory and a Partial Justification of Greene's Criterion for Nontwist Maps

Abstract

We consider perturbations of integrable, area preserving nontwist maps of the annulus (those are maps in which the twist condition changes sign). These maps appear in a variety of applications, notably transport in atmospheric Rossby waves. We show in suitable two-parameter families the persistence of critical circles (invariant circles whose rotation number is the maximum of all the rotation numbers of points in the map) with Diophantine rotation number. The parameter values with critical circles of frequency $\omega_0$ lie on a one-dimensional analytic curve. Furthermore, we show a partial justification of Greene's criterion: If analytic critical curves with Diophantine rotation number $\omega_0$ exist, the residue of periodic orbits (that is, one fourth of the trace of the derivative of the return map minus 2) with rotation number converging to $\omega_0$ converges to zero exponentially fast. We also show that if analytic curves exist, there should be periodic orbits approximating them and indicate how to compute them. These results justify, in particular, conjectures put forward on the basis of numerical evidence in [D. del Castillo-Negrete, J.M. Greene, and P.J. Morrison, Phys. D., 91 (1996), pp. 1--23]. The proof of both results relies on the successive application of an iterative lemma which is valid also for 2d-dimensional exact symplectic diffeomorphisms. The proof of this iterative lemma is based on the deformation method of singularity theory.