arXiv subjects

Edson D. Leonel

Publications and source records attributed to Edson D. Leonel.

At least 19 recordsLinked to original sources

arXiv2026

A Symplectic Map Approach to Magnetic Field-Line Dynamics in Tokamaks

Magnetic field-line transport in tokamaks is governed by the interplay between chaotic dynamics and invariant phase-space structures that act as partial transport barriers. We investigate the conservative Tokamap, an exact symplectic mapping for magnetic field-line dynamics, using a unified geometrical, dynamical, and statistical framework. Phase-space portraits and Lyapunov exponents characterize the mixed Hamiltonian dynamics, while ensemble-averaged transport exhibits universal dynamic scaling with growth, crossover, and saturation regimes connected through a homogeneous scaling theory. Poincar\'e recurrence statistics reveal that decreasing the magnetic-shear parameter systematically slows transport, with the characteristic transport time following the algebraic scaling $\tau_c\propto x_q^{-0.213}$. This behavior reflects increasingly dominant stickiness associated with KAM islands, resonance chains, and cantori. Our results establish a direct connection between the geometrical organization of Hamiltonian phase space and macroscopic transport properties, providing a comprehensive framework for understanding long-time magnetic field-line transport in tokamaks and other Hamiltonian systems with mixed phase space.

nlin.CD↗

arXiv2026

Integrability-breaking phase transitions in stadium-like billiards

We investigate integrability-breaking transitions in two classes of stadium-like billiards with parabolic boundaries. While focusing boundaries generate a mixed phase space in which regular islands coexist with a chaotic sea, dispersing boundaries produce a fully chaotic phase space for any finite boundary deformation. By analyzing the scaling behavior of the roughness $\omega$, we identify two qualitatively distinct transitions: a continuous transition for the focusing geometry and a first-order transition for the dispersing one. We determine the corresponding critical exponents and establish the associated scaling laws. For the continuous transition, we further provide a complete characterization within the framework of critical phenomena by identifying the broken symmetry, the order parameter and its diverging susceptibility, the elementary excitations responsible for chaotic diffusion, and the topological defects governing transport. These results establish a statistical-mechanics framework for describing integrability-breaking transitions in Hamiltonian billiards and suggest that the concepts of critical phenomena naturally extend to deterministic nonlinear dynamical systems.

nlin.CD↗

arXiv2026

Recurrence and anti-recurrence patterns reveal an antiperiodic fingerprint that survives into chaos in the Duffing--Holmes oscillator

The periodically forced Duffing--Holmes oscillator possesses a discrete symmetry under sign reversal of the coordinate combined with a half-period shift of the drive. When this symmetry is dynamically realized, the system supports \emph{antiperiodic} solutions, whose state at any instant is the point reflection of the state half a driving period earlier. We show that a standard recurrence plot (RP) is blind to this symmetry, whereas a complementary \emph{anti-recurrence plot} (anti-RP), built from the cross recurrence between a trajectory and its point-reflected image, detects it directly. Across four regimes -- periodic and chaotic single-well motion, and antiperiodic and chaotic two-well motion -- the anti-RP is empty when the attractor occupies one well and densely diagonal when the motion respects the symmetry. Crucially, the antiperiodic fingerprint persists into the chaotic two-well regime, where the anti-recurrence rate stays high relative to the ordinary one ($\mathrm{RR}_a/\mathrm{RR}\approx0.8$) despite the chaos. Recurrence quantification of both matrices separates order from chaos, while the anti-RP independently distinguishes one- from two-well, symmetry-respecting dynamics, giving a compact classification of all regimes. Requiring only a time series and the symmetry operation, the anti-RP is a model-free probe of dynamical symmetry for any system with a sign-reversal invariance, including experimental signals where phase-averaged observables fail.

nlin.CD↗

arXiv2026

Antiperiodic orbits and spontaneous symmetry breaking in the Duffing--Holmes oscillator

We investigate the origin and distribution of antiperiodicity -- oscillations satisfying $x(t+T)=-x(t)$ -- in the periodically driven Duffing--Holmes oscillator, combining analytical arguments with extensive numerical exploration. We first establish the minimal conditions, in terms of nonlinearity and symmetry, required for the existence of nontrivial antiperiodic trajectories, and we map how the antiperiodic, periodic, and chaotic regimes are organized in both phase space and parameter space. Antiperiodic orbits are shown to be precisely the periodic orbits that remain invariant under the half-period shift symmetry $S:(x,\dot{x},t)\mapsto(-x,-\dot{x},\,t+T_d/2)$, with $T_d$ the driving period, of the equations of motion. This invariance imposes a parity selection rule, verified without exception across our parameter sweeps: antiperiodic orbits lock to the drive only at odd multiples of the forcing period. Periodic orbits that lack the antisymmetry occur instead as conjugate pairs related by $S$, each orbit being the point reflection of its twin; the spontaneous symmetry breaking that takes place near the underlying bifurcations selects one member of each pair, while the pair as a whole restores the symmetry lost by each orbit individually. Antiperiodicity thus emerges not as an accidental property of particular waveforms but as the orbit-level manifestation of a discrete symmetry of the driven system.

nlin.CD↗

arXiv2026

Critical parameters of an oval billiard with an elliptical component

We explore the critical parameters responsible for the transition from integrability to chaos in a family of billiards combining elliptical and oval deformations. Unlike standard oval billiards, where a known critical parameter governs the destruction of the last invariant curve, the introduction of an integrable elliptic component yields a second deformation axis. We derive an analytical expression for the critical parameter in this combined system and validate it numerically using Slater's theorem, showing that increasing the elliptical component lowers the critical threshold for global chaos. Moreover, we uncover a previously unexplored regime: when the two deformation components are in phase, the elliptic contribution progressively suppresses chaos, leading to the restoration of invariant curves and periodic orbits. A first-order analytical approximation confirms this behavior, supported by numerical validation. Our results reveal how the interplay between distinct boundary deformations enriches phase-space organization and offers enhanced controllability of chaotic dynamics in billiard systems.

nlin.CD↗

arXiv2026

Phase-space organization of the elastic pendulum: chaotic fraction, energy exchanges, and the order-chaos-order transition

We study the phase-space organization of the planar elastic pendulum as a function of its two dimensionless control parameters: the reduced energy $R$ and the squared frequency ratio $\mu$. By randomly sampling the isoenergetic volume to classify trajectories as oscillatory, rotational, or chaotic across the $(\mu, R)$ parameter plane, we obtain a global portrait of the coexistence and competition between dynamical regimes. The chaotic fraction is not uniformly distributed across the parameter plane but concentrates in a well-defined central cloud whose ridge follows a linear relation in the $(\mu, R)$ plane and whose maximum does not exceed $70\%$ of the available phase space. The order-chaos-order transition is not a global property of the parameter plane but occurs specifically in the central region surrounding this cloud: along paths that traverse it, oscillatory orbits progressively give way to chaotic trajectories, which in turn yield to rotational orbits as the energy grows, revealing a clear sequential mechanism underlying the transition. The onset of rotational motion is gradual rather than sharp, reflecting a strong dependence on initial conditions. By decomposing the total energy into spring-like, pendulum-like, and coupling contributions, we establish a direct correspondence between the coupling power and the abundance of chaotic trajectories, showing that enhanced inter-mode energy exchange is a reliable indicator of dynamical complexity. These results provide a comprehensive and quantitative map of the dynamical regimes of the elastic pendulum, clarifying the structure of the chaotic cloud and connecting it to the underlying mode-coupling mechanisms.

nlin.CD↗

arXiv2026

Stochastic Web Map: Survival probability and escape frequency

We study transport and escape in the Stochastic Web Map (SWM), an area-preserving system with phase-space structure controlled by a symmetry parameter $q$ and nonlinearity $K$. By analyzing the survival probability $P_{\text{S}}(n)$ and escape frequency $P_{\text{E}}(\ln n)$, we show that in the chaotic regime escape dynamics is governed by a single time scale $n_{\text{typ}}\propto K^{-2}h^{2}$; here $h$ is the size of the escape horizon. Deviations at large $K$ and small $h$ indicate a breakdown of the quasilinear approximation. Then, upon rescaling the time by $n_{\text{typ}}$, escape statistics becomes universal, independent of $q$. These results demonstrate that escape is controlled by global transport rather than symmetry.

nlin.CD↗

arXiv2026

Jacobian determinant as a deformation field in static billiards

We develop a deformation-based framework for analyzing static billiard systems through the Jacobian determinant computed in noncanonical angular coordinates. Although these systems are conservative, the determinant is not identically equal to unity, generating structured domains of local phase-space expansion and contraction. We show numerically that these domains balance globally, providing a geometric manifestation of area preservation in noncanonical variables. The curves defined by det J = 1 act as deformation boundaries that intersect unstable periodic points and correlate with invariant manifolds. We prove analytically that period-two orbits restore exact unit determinant under composition, while higher-period orbits exhibit angular modulation consistent with reversibility. The Jacobian determinant thus reveals an additional geometric layer in phase-space organization and offers a complementary perspective on conservative billiard dynamics.

nlin.CD↗

arXiv2026

Universal Second-Order Phase Transition from Integrability to Chaos

We report a dynamical phase transition from integrability to non-integrability in a simple oval-like billiard with boundary $R(\theta)=1+\epsilon\cos(p\theta)$. For $\epsilon=0$, the phase space is {\it foliated} by invariant curves corresponding to periodic or quasiperiodic motion, whereas for small $\epsilon$ a thin chaotic layer separates rotational and librational trajectories. As $\epsilon$ increases, this layer grows according to a well-defined scaling law whose chaotic dispersion follows $\omega_{\rm rms,sat}\sim\epsilon^{\tilde{\alpha}}$, where the exponent $\tilde{\alpha}$ coincides with those of the Fermi-Ulam model, periodically corrugated waveguides, and a family of discrete mappings, revealing a universal mechanism for the onset of chaos in weakly perturbed integrable systems. The deviation of the reflection angle in the billiard, $\omega_{\rm rms,sat}$, acts as an order parameter: it vanishes continuously as $\epsilon\to 0$, signalling an ordered (integrable) phase, while its susceptibility $\chi=d\omega_{\rm rms,sat}/d\epsilon$ diverges, indicating a second-order phase transition. A symmetry breaking and an analytically solvable diffusion process complete the near-critical phenomenology. These results establish a unified framework for the emergence of chaos from integrability.

nlin.CD↗

arXiv2026

Describing a Universal Critical Behavior in a transition from order to chaos

We present a comprehensive discussion of a transition from integrability to non-integrability in an oval billiard with a static boundary. This transition is controlled by a deformation parameter $\epsilon$, which modifies the boundary shape from circular, corresponding to $\epsilon=0$ and an integrable dynamics, to oval for $\epsilon\neq 0$, where non-integrability emerges. The deformation of the circular billiard gives rise to a chaotic layer that develops along a well-defined stripe in phase space. By introducing a set of transformations that isolate this chaotic stripe, we characterise the diffusive spreading of ensembles of trajectories and identify an observable, $\omega_{rms,{\rm sat}}$, which plays the role of an order parameter for the transition. For small deformations, the saturation value of the diffusion obeys the scaling law $\omega_{rms,{\rm sat}}\propto\epsilon^{\tilde{\alpha}}$, with a critical exponent $\tilde{\alpha}=0.507(2)$, vanishing continuously as $\epsilon\rightarrow 0$. The associated susceptibility, $\chi=d\omega_{rms,{\rm sat}}/d\epsilon$, diverges in the same limit, signalling the presence of critical behavior analogous to that observed in second-order (continuous) phase transitions in statistical mechanics.

nlin.CD↗

arXiv2026

Scaling invariance: a bridge between geometry, dynamics and criticality

Scale invariance is a central organizing principle in physics, underlying phenomena that range from critical behaviour in statistical mechanics to transport and chaos in nonlinear dynamical systems. Here we present a unified and physically motivated exploration of scaling concepts, emphasizing how invariance under rescaling transformations emerges across systems of increasing dynamical complexity. Rather than adopting a purely abstract approach, we combine simple geometrical constructions, analytical arguments, and prototypical dynamical models to build physical intuition. We begin with elementary, easily reproducible examples governed by a single control parameter, showing how power-law behaviour naturally arises when characteristic scales are absent. We then extend the discussion to nonlinear dynamical systems exhibiting local bifurcations, where two scaling variables control the relaxation toward stationary states. In this context, scaling invariance manifests through critical exponents, crossover phenomena, and critical slowing down, allowing systems of different dimensionality to be grouped into universality classes. Finally, we address continuous phase transitions in chaotic dynamical systems, including transitions from integrability to non-integrability and from bounded to unbounded diffusion. By drawing on concepts traditionally associated with statistical mechanics, such as order parameters, susceptibilities, symmetry breaking, elementary excitations, and topological defects, we show how these transitions can be interpreted within a coherent scaling framework. Taken together, the examples discussed here demonstrate that scaling invariance provides a unifying language for understanding structure, transport, and criticality in nonlinear systems, bridging deterministic dynamics and nonequilibrium statistical physics in a transparent and physically intuitive manner.

cond-mat.stat-mech↗

arXiv2026

A phenomenological description of critical slowing down at period-doubling bifurcations

We present a phenomenological description of the critical slowing down associated with period-doubling bifurcations in discrete dynamical systems. Starting from a local Taylor expansion around the fixed point and the bifurcation parameter, we derive a reduced description that captures the convergence towards stationary state both at and near criticality. At the bifurcation point, three universal critical exponents are obtained, characterising the short-time behaviour, the asymptotic decay, and the crossover between these regimes. Away from criticality, a fourth exponent governing the relaxation time is identified. We show this phenomenology, well established for one-dimensional maps, extends naturally to two-dimensional mappings. By projecting the dynamics onto the centre manifold, we demonstrate that the local normal form of a two-dimensional period-doubling bifurcation reduces to the same universal structure found in one dimension. The theoretical predictions are validated numerically using the H\'enon and Ikeda maps, showing excellent agreement for all scaling laws and critical exponents.

nlin.CD↗

arXiv2025

Convergence Dynamics and Scaling Laws in the Dissipative Relativistic Kicked Rotator

We investigate the convergence dynamics of this system near period-doubling bifurcations by combining analytical derivations and large-scale numerical simulations. At the bifurcation threshold ($K = K_c$), the dynamics reduce to a normal form that produces a power-law decay $d(n) \propto n^{-1/2}$, from which the critical exponents $\alpha = 1$, $\beta = -1/2$, and $z = -2$ are derived. These analytical predictions are confirmed numerically and shown to satisfy the homogeneous scaling relation $z = \alpha / \beta$. Linearization of the map near the fixed point yields an exponential relaxation law $d_n = d_0 e^{-n/\tau}$ for $K < K_c$, with $\tau \propto (K_c - K)^{-1}$, leading to the relaxation exponent $\delta = -1$. The remarkable agreement between theory and simulation demonstrates that the dissipative relativistic kicked rotator shares the same universality class as one-dimensional unimodal maps, despite its higher dimensionality and relativistic corrections.

nlin.CD↗

arXiv2025

High Energy States of Recurrent Chaotic Trajectories in Time-Dependent Potential Well

In this numerical study, recurrence quantification analysis of chaotic trajectories is explored to detect atypical dynamical behaviour in non-linear Hamiltonian systems. An ensemble of initial conditions is evolved up to a maximum iteration time, and the recurrence rate of each orbit is computed, allowing a subset of trajectories exhibiting significantly higher recurrences than the ensemble average to be identified. These special trajectories are determined through a suitable statistical distribution, within which peak detection reveals the respective initial condition that is evolved into a highly recurrent chaotic orbit, a phenomenon known as stickiness. By applying this methodology to a model of a classical particle in a time-dependent potential well, it is demonstrated that, for specific parameter values and initial conditions, such recurrent chaotic trajectories can give rise to transient high-energy states.

nlin.CD↗

arXiv2025

Scaling invariance for the diffusion coefficient in a billiard system

We investigated the unbounded diffusion observed in a time-dependent oval-shaped billiard and its suppression owing to inelastic collisions with the boundary. The main focus is on the behavior of the diffusion coefficient, which plays a key role in describing the scaling invariance characteristic of this transition. For short times, the low-action regime is characterized by a constant diffusion coefficient, which begins to decay after a crossover iteration, thereby suppressing the unlimited growth of velocity. We demonstrate that this behavior is scaling-invariant concerning the control parameters and can be described by a homogeneous generalized function and its associated scaling laws. The critical exponents are determined both phenomenologically and analytically, including the decay exponent beta = -1, previously identified in the diffusion coefficient of the dissipative standard map.

nlin.CD↗

arXiv2025

Scaling Invariance: A Gateway to Phase Transitions

We explore the concept of scaling invariance in a type of dynamical systems that undergo a transition from order (regularity) to disorder (chaos). The systems are described by a two-dimensional, nonlinear mapping that preserves the area in the phase space. The key variables are the action and the angle, as usual from Hamiltonian systems. The transition is influenced by a control parameter giving the form of the order parameter. We observe a scaling invariance in the average squared action within the chaotic region, providing evidence that this change from regularity (integrability) to chaos (non-integrability) is akin to a second-order or continuous phase transition. As the order parameter approaches zero, its response against the variation of the control parameter (susceptibility) becomes increasingly pronounced (indeed diverging), resembling a phase transition. These findings could not be obtained without a seminal paper on Phys. Rev. Lett. {\bf 2004}, {\em 93}, 014101.

nlin.CD↗

arXiv2024

Rare events for low energy domain in bouncing ball model

The probability distribution for multiple collisions observed in the chaotic low energy domain in the bouncing ball model is shown to be scaling invariant concerning the control parameters. The model considers the dynamics of a bouncing ball particle colliding elastically with two rigid walls. One is fixed, and the other one moves periodically in time. The dynamics is described by a two-dimensional mapping for the variables velocity of the particle and phase of the moving wall. For a specific combination of velocity and phase, the particle may experience a type of rare collision named successive collisions. We show that a power law describes the probability distribution of the multiple impacts and is scaling invariant to the control parameter.

nlin.CD↗

arXiv2024

Scaling invariance for the diffusion coefficient in a dissipative standard mapping

The unbounded diffusion observed for the standard mapping in a regime of high nonlinearity is suppressed by dissipation due to the violation of Liouville's theorem. The diffusion coefficient becomes important for the description of scaling invariance particularly for the suppression of the unbounded action diffusion. When the dynamics start in the regime of low action, the diffusion coefficient remains constant for a long time, guaranteeing the diffusion for an ensemble of particles. Eventually, it evolves into a regime of decay, marking the suppression of particle action growth. We prove it is scaling invariant for the control parameters and the crossover time identifying the changeover from the constant domain, leading to diffusion, for a regime of decay marking the saturation of the diffusion, scales with the same critical exponent $z=-1$ for a transition from bounded to unbounded diffusion in a dissipative time dependent billiard system.

nlin.CD↗