Hierarchical fragmentation of regular islands in a discontinuous nontwist map

TL;DR

This paper investigates how discontinuities in a nontwist map lead to hierarchical fragmentation of regular islands, showing the absence of a global transport barrier and revealing detailed trapping and escape dynamics.

Contribution

It demonstrates that discontinuity causes hierarchical fragmentation of regular islands and the loss of a global invariant curve, contrasting with the continuous case.

Findings

Regular islands fragment into smaller components connected by chaotic channels

No global transport barrier exists in the discontinuous map

Finite-time RTE reveals spatial organization and trapping phenomena

Abstract

The destruction of regular regions in two-dimensional, area-preserving maps is traditionally described in terms of the breakup of invariant curves and the persistence of transport barriers. Here, we investigate how this scenario changes when continuity is lost. We study the extended standard nontwist map with a perturbation whose period differs from a full revolution on the cylinder. In this setting, the map becomes discontinuous on this cylinder while remaining smooth on the real line. Using escape times, the smaller alignment index (SALI), Lyapunov exponents, and finite-time recurrence time entropy (RTE), we find that regular islands are not enclosed by a single invariant curve but instead undergo hierarchical fragmentation into smaller regular components connected by chaotic channels. We show that trajectories initialized near elliptic points exhibit long trapping followed by escape, ruling out the existence of a global transport barrier or a last invariant curve. We demonstrate that finite-time RTE exhibits broad, asymmetric distributions with a clear spatial organization, with low values near island centers and high values along chaotic channels, persisting at fine scales. We also find persistent partial barriers, where trajectories remain trapped for extremely long times before escaping. By restoring continuity in a modified formulation, we recover smooth invariant curves and eliminate fragmentation, demonstrating that the hierarchical structure originates from discontinuity rather than twist violation alone.