arXiv subjects

Leonardo Costa de Souza

Publications and source records attributed to Leonardo Costa de Souza.

5 recordsLinked to original sources

arXiv2026

Adiabatic confinement and escape in an open billiard

We introduce a stationary, field-free billiard that reproduces the adiabatic confinement mechanism of a magnetic mirror through boundary geometry alone. Rapid transverse reflections generate an approximately conserved action that plays the role of the magnetic moment, producing an effective longitudinal barrier and a nominal loss cone for escape. For the half-width profile considered here, confinement is controlled by a competition between mirror strength and adiabaticity: shortening the profile increases the mirror ratio but also reduces the accuracy of the adiabatic approximation. As a result, the escaping fraction is nonmonotonic and confinement is strongest at an intermediate profile length. We characterize the associated phase-space structure, escape-time statistics, and escape basins, revealing long-lived transient motion and a strong sensitivity of the final escape direction to the initial condition that persists as the uncertainty in the initial condition is reduced.

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arXiv2026

Recurrence time entropy and weak chaos in Hamiltonian flows

Stickiness in mixed Hamiltonian systems causes chaotic trajectories to remain temporarily trapped near regular structures, making it difficult to distinguish regular, weakly chaotic, and strongly chaotic motion over finite times. We show that the recurrence time entropy (RTE), previously used in discrete maps, also characterizes weak chaos in Hamiltonian flows. In the Hénon-Heiles system, the RTE reproduces the structures identified by the largest Lyapunov exponent, taking intermediate values in the sticky layers, and yields a proportion of chaotic trajectories consistent with the smaller alignment index. The finite-time RTE identifies low-entropy episodes near regular islands, whose durations decay algebraically, whereas high-entropy episodes decay exponentially. The same characterization holds for a periodically driven system and for a Hamiltonian system with three degrees of freedom, whose section is four-dimensional. The RTE is thus an effective diagnostic of weak chaos and stickiness in Hamiltonian flows.

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arXiv2026

Hierarchical fragmentation of regular islands in a discontinuous nontwist map

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 induced map on the cylinder becomes discontinuous, even though the map remains smooth on the real line. Using complementary chaos diagnostics, 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. The fragmentation occurs when island chains are centered on the discontinuity line, while island chains away from it preserve the conventional islands-around-islands structure. By restoring continuity of the induced map on the cylinder 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. Similar behavior is also observed in other two-dimensional area-preserving maps, indicating that the phenomenon is not restricted to nontwist systems.

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arXiv2025

pynamicalsys: A Python toolkit for the analysis of dynamical systems

Since Lorenz's seminal work on a simplified weather model, the numerical analysis of nonlinear dynamical systems has become one of the main subjects of research in physics. Despite of that, there remains a need for accessible, efficient, and easy-to-use computational tools to study such systems. In this paper, we introduce pynamicalsys, a simple yet powerful open-source Python module for the analysis of nonlinear dynamical systems. In particular, pynamicalsys implements tools for trajectory simulation, bifurcation diagrams, Lyapunov exponents and several others chaotic indicators, period orbit detection and their manifolds, as well as escape and basins analysis. It also includes many built-in models and the use of custom models is straighforward. We demonstrate the capabilities of pynamicalsys through a series of examples that reproduces well-known results in the literature while developing the mathematical analysis at the same time. We also provide the Jupyter notebook containing all the code used in this paper, including performance benchmarks. pynamicalsys is freely available via the Python Package Index (PyPI) and is indented to support both research and teaching in nonlinear dynamics.

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arXiv2024

Ratchet current and scaling properties in a nontwist mapping

We investigate the transport of particles in the chaotic component of phase space for a two-dimensional, area-preserving nontwist map. The survival probability for particles within the chaotic sea is described by an exponential decay for regions in phase space predominantly chaotic and it is scaling invariant in this case. Alternatively, when considering mixed chaotic and regular regions, there is a deviation from the exponential decay, characterized by a power law tail for long times, a signature of the stickiness effect. Furthermore, due to the asymmetry of the chaotic component of phase space with respect to the line $I = 0$, there is an unbalanced stickiness which generates a ratchet current in phase space. Finally, we perform a phenomenological description of the diffusion of chaotic particles by identifying three scaling hypotheses, and obtaining the critical exponents via extensive numerical simulations.

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