Carlos E. P. Abreu

In this work, we investigate the global transport properties and the destruction of invariant structures in the extended nontwist standard map. We propose a robust methodology based on the statistics of Poincaré recurrences and Slater's theorem to identify the most resilient transport barrier in the phase space: the shearless curve. By detecting quasi-periodic trajectories characterized by exactly three distinct recurrence times, we accurately determine the critical perturbation parameter c_r associated with the global transition to chaos. To validate our approach, we introduce a geometric similarity measure based on the Hausdorff distance, demonstrating that the boundary curve identified via Slater's theorem strictly coincides with the theoretical shearless curve obtained via the standard indicator-point method. A major advantage of this recurrence-based framework is its complete independence from spatial symmetries, enabling the precise identification of transport barriers in both symmetric (odd m) and non-symmetric (even m) perturbation regimes. Furthermore, we systematically map the (a,b) parameter space to construct detailed robustness diagrams, delineating regimes where invariant curves persist from fully chaotic, barrier-less domains. Our results establish recurrence-time statistics as a highly effective and universal diagnostic tool for studying transport barriers in nontwist Hamiltonian systems.

We consider the standard nontwist map with strong dissipation that leads the system to a 1D circular map with a quadratic sinusoidal oscillation and two control parameters. The 2D Lyapunov and isoperiodic diagrams reveal a complex interplay between domains of periodicity embedded in regions dominated by quasiperiodic and chaotic behaviors. Arnold tongues and shrimp-like, among other sets of periodicities, compose this rich dynamical scenario in the parameter space. Cobwebs and bifurcation diagrams help reveal the behavior of attractors, including multistability, period-doubling, pitchfork bifurcations, as well as boundary, merging, and interior crises that influence the structures of periodicity. Furthermore, we bring to light the global organization of shrimp-like structures by carrying out a new concept of orbits, the extreme orbits, and announce that the fractal dimension, believed to be universal in the parameter space for decades, has its symmetry breaking in the vicinity of shrimp-like cascades.

In the parameter spaces of nonlinear dynamical systems, we investigate the boundaries between periodicity and chaos and unveil the existence of fractal sets characterized by a singular fractal dimension that deviates greatly from the fractal sets in their vicinity. This extreme fractal dimension stands out from the typical value previously considered universal for these parameter boundaries. We show that such singular fractal sets dwell along parameter curves, called extreme curves, that intersect periodicity cascades at their centers of stability across all scales of parameter spaces. The results reported here are generally demonstrated for the class of one-dimensional maps with at least two control parameters. Generalizations to other classes of systems are possible.