arXiv subjects

Iberê Luiz Caldas

Publications and source records attributed to Iberê Luiz Caldas.

At least 19 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.

nlin.CD↗

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.

nlin.CD↗

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.

nlin.CD↗

arXiv2026

Discrete dynamical systems with scaling and inversion symmetries

In this work, we investigate scale invariance in the temporal evolution and chaotic regime of discrete dynamical systems. By exploiting the close interrelation between scaling and inversion transformations, we formulate scale symmetry in terms of inversion symmetry. As applications of our approach, we determine fractal dimensions and compute Lyapunov exponents for paradigmatic dynamical systems using scaling and inversion symmetries. By comparing our method with standard approaches, we obtain identical numerical values for the Lyapunov exponents using only a small number of iterations. Furthermore, our geometric-based framework naturally provides access to the fractal dimension. The agreement with standard results demonstrates that the proposed method is efficient and can be effectively employed in the study of dynamical systems.

physics.gen-ph↗

arXiv2025

Shearless barriers in the conservative Ikeda map

We investigate the dynamics of the Ikeda map in the conservative limit, where it is represented as a two-dimensional area-preserving map governed by two control parameters, $θ$ and $ϕ$. We demonstrate that the map can be interpreted as a composition of a rotation and a translation of the state vector. In the integrable case ($ϕ= 0$), the map reduces to a uniform rotation by angle $θ$ about a fixed point, independent of initial conditions. For $ϕ\ne 0$, the system becomes nonintegrable, and the rotation angle acquires a coordinate dependence. The resulting rotation number profile exhibits extrema as a function of position, indicating the formation of shearless barriers. We analyze the emergence, persistence, and breakup of these barriers as the control parameters vary.

nlin.CD↗

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.

nlin.CD↗

arXiv2025

Hamiltonian chaos for one particle with two waves: Self-consistent dynamics

A simple model of wave-particle interaction is studied in its self-consistent form, that is, where the particles are allowed to feedback on the waves dynamics. We focus on the configurations of locked solutions (equilibria) and how the energy-momentum exchange mechanism induces chaos in the model. As we explore the system, we analyse the mathematical structure that gives rise to locked states and how the model's non-linearity enables multiple equilibrium amplitudes for waves. We also explain the predominance of regularity as we vary the control parameters and the mechanism behind the emergence of chaos under limited parameter choices.

nlin.CD↗

arXiv2024

Identifying ballistic modes via Poincaré sections

Exploring chaotic systems via Poincaré sections has proven essential in dynamical systems, yet measuring their characteristics poses challenges to identify the various dynamical regimes considered. In this paper, we propose a new approach that uses image processing to distinguish chaotic and regular regions of area-preserving dynamics, and then classify the transport regime. We characterize different transport regimes in the standard map with the proposed method based on image reconstruction techniques, identifying the superdiffusion much faster than the usual mean square displacement method. The procedure is also applied to a two-wave, time-dependent Hamiltonian to investigate superdiffusion in function of two parameters.

nlin.CD↗

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.

nlin.CD↗

arXiv2023

Chaotic saddles and interior crises in a dissipative nontwist system

We consider a dissipative version of the standard nontwist map. Nontwist systems present a robust transport barrier, called the shearless curve, that becomes the shearless attractor when dissipation is introduced. This attractor can be regular or chaotic depending on the control parameters. Chaotic attractors can undergo sudden and qualitative changes as a parameter is varied. These changes are called crises, and at an interior crisis the attractor suddenly expands. Chaotic saddles are nonattracting chaotic sets that play a fundamental role in the dynamics of nonlinear systems, they are responsible for chaotic transients, fractal basin boundaries, chaotic scattering and they mediate interior crises. In this work we discuss the creation of chaotic saddles in a dissipative nontwist system and the interior crises they generate. We show how the presence of two saddles increase the transient times and analyze the phenomenon of crisis induced intermittency.

nlin.CD↗

arXiv2020

Growth and performance of the periodic orbits of a nonlinear driven oscillator

Periodic orbits are fundamental to understand the dynamics of nonlinear systems. In this work, we focus on two aspects of interest regarding periodic orbits, in the context of a dissipative mapping, derived from a prototype model of a non-linear driven oscillator with fast relaxation and a limit cycle. For this map, we show numerically the exponential growth of periodic orbits quantity in certain regions of the parameter space and provide an analytical bound for such growth rate, by making use of the transition matrix associated with a given periodic orbit. Furthermore, we give numerical evidence to support that optimal orbits, those that maximize time averages, are often unstable periodic orbits with low period, by numerically comparing their performance under a family of sinusoidal functions.

nlin.CD↗

arXiv2020

Mathematical model of brain tumour growth with drug resistance

Brain tumours are masses of abnormal cells that can grow in an uncontrolled way in the brain. There are different types of malignant brain tumours. Gliomas are malignant brain tumours that grow from glial cells and are identified as astrocytoma, oligodendroglioma, and ependymoma. We study a mathematical model that describes glia-neuron interaction, glioma, and chemotherapeutic agent. In this work, we consider drug sensitive and resistant glioma cells. We show that continuous and pulsed chemotherapy can kill glioma cells with a minimal loss of neurons.

q-bio.TO↗

arXiv2020

Controlling Escape in the Standard Map

We investigate how the diffusion exponent is affected by controlling small domains in the phase space.The main Kolomogorov-Arnold-Moser - KAM island of the Standard Map is considered to validate the investigation. The bifurcation scenario where the periodic island emits smaller resonance regions is considered and we show how closing paths escape from the island shore by controlling points and hence making the diffusion exponent smaller. We notice the bigger controlled area the smaller the diffusion exponent. We show that controlling around the hyperbolic points associated to the bifurcation is better than a random control to reduce the diffusion exponent. The recurrence plot shows us channels of escape and a control applied there reduces the diffusion exponent.

nlin.CD↗

arXiv2019

Real-time vulnerability of synchronized states

The time-dependent vulnerability of synchronized states is shown for a complex network composed of electronic circuits. We demonstrate that disturbances to the local dynamics of network units can produce different outcomes to synchronization depending on the event timing. We address such time dependence by systematically perturbing the synchronized system at instants of time equally distributed along its trajectory. We find the time instants at which the perturbation desynchronizes the network to be complicatedly mixed with the ones that restore synchronization. Additionally, we characterize perturbation sets obtained for consecutive instants of time by defining a safety index between them. Finally, we demonstrate that the vulnerability is due to state space sensitivities occurring along synchronized trajectories.

nlin.AO↗

arXiv2018

Riddling: Chimera's dilemma

We investigate the basin of attraction properties and its boundaries for chimera states in a circulant network of Hénon maps. Chimera states, for which coherent and incoherent domains coexist, emerge as a consequence of the coexistence of basin of attractions for each state. It is known that the coexisting basins of attraction lead to a hysteretic behaviour in the diagrams for the density of incoherent and coherent states as a function of a varying parameter. Consequently, the distribution of chimera states can remain invariant by a parameter change, as well as it can suffer subtle changes when one of the basin ceases to exist. A similar phenomenon is observed when perturbations are applied in the initial conditions. By means of the uncertainty exponent, we characterise the basin boundaries between the coherent and chimera states, and between the incoherent and chimera states, and uncover fractal and riddled boundaries, respectively. This way, we show that the density of chimera states can be not only moderately sensitive but also highly sensitive to initial conditions. This chimera's dilemma is a consequence of the fractal and riddled nature of the basins boundaries.

nlin.CD↗

arXiv2016

An efficient method for tracing high-resolution invariant manifolds of three-dimensional flows

In Hamiltonian systems subjected to periodic perturbations the stable and unstable manifolds of the unstable periodic orbits provide the dynamical "skeleton" that drives the mixing process and bounds the chaotic regions of the phase space. Determining the behavior of these objects is valuable in physical applications involving asymmetric solenoidal fields or time-dependent Hamiltonian systems. Here we introduce a simple method to calculate an unstable periodic orbit given an initial guess on its position. Then we present an efficient adaptive method to build its high-resolution invariant manifolds to arbitrary length and compare it to a random sampling method with the same computational cost. The adaptive method gives a high-quality representation of the manifolds and reveals fine details that become lost in the random sampling method. Finally, we introduce an approximation to the adaptive method to build the manifolds avoiding redundant calculations and reducing logarithmically the number of computations needed to represent these surfaces.

physics.plasm-ph↗

arXiv2016

Synchronised firing patterns in a random network of adaptive exponential integrate-and-fire

We have studied neuronal synchronisation in a random network of adaptive exponential integrate-and-fire neurons. We study how spiking or bursting synchronous behaviour appears as a function of the coupling strength and the probability of connections, by constructing parameter spaces that identify these synchronous behaviours from measurements of the inter-spike interval and the calculation of the order parameter. Moreover, we verify the robustness of synchronisaton by applying an external perturbation to each neuron. The simulations show that bursting synchronisation is more robust than spike synchronisation.

q-bio.NC↗

arXiv2012

Stickiness in a bouncer model: A slowing mechanism for Fermi acceleration

Some phase space transport properties for a conservative bouncer model are studied. The dynamics of the model is described by using a two-dimensional measure preserving mapping for the variables velocity and time. The system is characterized by a control parameter $ε$ and experiences a transition from integrable ($ε=0$) to non integrable ($ε\ne 0$). For small values of $ε$, the phase space shows a mixed structure where periodic islands, chaotic seas and invariant tori coexist. As the parameter $ε$ increases and reaches a critical value $ε_c$ all invariant tori are destroyed and the chaotic sea spreads over the phase space leading the particle to diffuse in velocity and experience Fermi acceleration (unlimited energy growth). During the dynamics the particle can be temporarily trapped near periodic and stable regions. We use the finite time Lyapunov exponent to visualize this effect. The survival probability was used to obtain some of the transport properties in the phase space. For large $ε$, the survival probability decays exponentially when it turns into a slower decay as the control parameter $ε$ is reduced. The slower decay is related to trapping dynamics, slowing the Fermi Acceleration, i.e., unbounded growth of the velocity

nlin.CD↗