Hierarchical fragmentation of regular islands in a discontinuous nontwist map

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.

Figures (8)

• Figure 1: Phase space of the extended standard nontwist map [Eq. \ref{['eq:esnm']}] for $a = b = 0.53$, $c = 0.005$ and (a) $m = 0.0$, (b) $m = 0.4$, (c) $m = 0.8$, (d) $m = 1.0$, (e) $m = (\sqrt{5} + 1) / 2$, (f) $m = 1.8$, (g) $m = 2.0$, (h) $m = 2.3$, and (i) $m = 3.0$.
• Figure 2: Magnifications around the upper (top row) and lower (bottom row) central islands for $a = b = 0.53$, $m = 0.8$, and (a) and (b) $c = 0$, (c) and (d) $c = 1.0\times10^{-5}$, (e) and (f) $c = 5.0\times10^{-5}$, (g) and (h) $c = 1.0\times10^{-4}$, (i) and (j) $c = 5.0\times10^{-4}$, and (k) and (l) $c = 1.0\times10^{-3}$.
• Figure 3: (a)--(h) Escape time and (i)--(p) the smaller alignment index (SALI) for a uniformly distributed grid of initial conditions around the (a)--(d) and (i)--(l) upper island and the (e)--(h) and (m)--(p) lower island for $a = b = 0.53$, $m = 0.8$, and (a, e, i, m) $c = 5.0\times10^{-5}$, (b, f, j, n) $c = 1.0\times10^{-4}$, (c, g, k, o) $c = 5.0\times10^{-4}$, and (d, h, l, p) $c = 1.0\times10^{-3}$. For the escape time, each initial condition is iterated (up to $10^8$ iterations) until it escapes the region defined by $y \in [-1, 1]$. The non-escaping initial conditions are colored as gray.
• Figure 4: Conservative generalized bifurcation diagrams (CGBDs) Manchein2013 illustrating the effect of the perturbation parameter $c$ on the central islands for $x_0 = 0.5$ and $m = 0.8$. Panels (a) and (b) correspond to one-dimensional scans of initial conditions $y_0$ along vertical lines intersecting the upper and lower resonant islands, respectively. For each pair $(c, y_0)$, the smaller alignment index (SALI) was computed up to $10^6$ iterations. The color scale represents the final SALI value, revealing the transition from regular motion (non-vanishing SALI) to chaotic behavior ($\mathrm{SALI} \to 0$) as the perturbation strength $c$ increases.
• Figure 5: The absolute value of the largest Lyapunov exponent $\lambda_1$ for different values of $c$, computed from an initial condition located at the elliptic point of the central islands for $c = 0$: (a) upper island and (b) lower island. When $c > 0$, the elliptic point may shift or vanish; therefore, the initial condition is kept fixed at the phase-space location of the elliptic point at $c = 0$. The black dashed line corresponds to the slope of $n^{-1}$.