Indicators of Reconnection Processes and Transition to Global Chaos in Nontwist Maps

Abstract

Reconnection processes of twin-chains are systematically studied in the quadratic twist map. By using the reversibility and symmetry of the mapping, the location of the indicator points is theoretically determined in the phase space. The indicator points enable us to obtain useful information about the reconnect ion processes and the transition to global chaos. We succeed in deriving the general conditions for the reconnection thresholds. In addition, a new type of reconnect ion process which generates shearless curves is studied.(1·1) (1·2)

On reconnection phenomena in the standard nontwist map

“…As the perturbation is increased (by increasing b at a fixed value of a), the hyperbolic orbits on the s 3 The threshold for reconnection of even-period orbits, which is the same as the point of collision of their hyperbolic points, can be obtained by using the numerical observation [18] that at the point of bifurcation two of the indicator points belong to the periodic orbit. .…”

“…1 Of particular significance are indicator points, [18] fixed points of some of the symmetries of the SNM, given by…”

“…[18], Shinohara and Aizawa obtained the four points of (3) and used them to propose exact expressions for the reconnection threshold of even-period hyperbolic orbits. These indicator points were independently re-discovered by Petrisor [2] in the analysis of the reversing symmetry group [22] of nontwist standard-like area-preserving maps.…”

“…We aim to deepen the understanding of the behavior of the SNM in different regions in (a, b)-parameter space by extending the discussions of Refs. [20,21,14,19,18].…”

“…2, we describe the reconnection of hyperbolic orbits of even period, applying the method of Ref. [18] to Eq. (1).…”