On the Diagram for the Onset of Global Chaos and Reconnection Phenomena in the Quadratic Nontwist Map

Abstract

The quadratic nontwist map is studied by defining the “indicator points” for the phase space. The indicator points enable us to obtain useful information concerning the onset of global chaos and reconnection phenomena. The diagram obtained by using the indicator points clearly reveals the effects of the reconnection phenomena on the transition to global chaos. A discussion is given of the last KAM curve for the quadratic nontwist map. In this paper, we consider area-preserving maps that violate the twist condition. They are often called area-preserving “nontwist” maps. Violation of the twist condition is encountered in models from a variety of fields such as fluid dynamics, plasma physics, celestial mechanics, etc. Moreover, violation of the twist condition generically occurs in the neighborhood of tripling bifurcations of a Hamiltonian system. 5),11) Originally, the twist condition is assumed in order to prove important theorems concerning the persistence of KAM curves, e.g. the Moser twist theorem 1) and the Aubry-Mather theorem. 2) However, the precise effects of violation of the twist condition on the persistence of KAM curves have not yet been elucidated. Numerical studies of nontwist systems revealed that violation of the twist condition generally leads to complicated phase space phenomena, called reconnection phenomena. 3) - 11) The reconnection phenomena drastically change the phase space structure and thus strongly influence the breakup process of KAM curves. In order to study in detail the properties arising from violation of the twist condition, we focus here on the quadratic nontwist map (QNM), which is one of the simplest prototypes of an area-preserving nontwist map. Previously, we showed that the onset of global chaos and the reconnection phenomena in the QNM could be systematically studied by defining the “indicator points” for the phase space. After summarizing the properties of the indicator points, we present the diagram for the onset of global chaos and reconnection phenomena. The diagram clearly reveals how reconnection phenomena influence the transition to global chaos. On the basis of the numerical results obtained by using the indicator points, a discussion is given of the last KAM curve for the QNM.