Scaling properties of the action in the Riemann-Liouville fractional standard map.

The Riemann-Liouville fractional standard map (RL-fSM) is a two-dimensional nonlinear map with memory given in action-angle variables (I,θ). The RL-fSM is parameterized by K and α∈(1,2], which control the strength of nonlinearity and the fractional order of the Riemann-Liouville derivative, respectively. In this work we present a scaling study of the average squared action 〈I^{2}〉 of the RL-fSM along strongly chaotic orbits, i.e., for K≫1. We observe two scenarios depending on the initial action I_{0}, I_{0}≪K or I_{0}≫K. However, we can show that 〈I^{2}〉/I_{0}^{2} is a universal function of the scaled discrete time nK^{2}/I_{0}^{2} (n being the nth iteration of the RL-fSM). In addition, we note that 〈I^{2}〉 is independent of α for K≫1. Analytical estimations support our numerical results.

Characterizing a transition from limited to unlimited diffusion in energy for a time-dependent stochastic billiard.

We explore Fermi acceleration in a stochastic oval billiard which shows unlimited to limited diffusion in energy when passing from the free to the dissipative case. We provide evidence for a transition from limited to unlimited energy growth taking place while detuning the corresponding restitution coefficient responsible for the degree of dissipation. A corresponding order parameter is suggested, and its susceptibility is shown to diverge at the critical point. We show that this order parameter is also be applicable to the periodically driven oval billiard and discuss the elementary excitation of the controlled diffusion process.

Application of the Slater criteria to localize invariant tori in Hamiltonian mappings.

We investigate the localization of invariant spanning curves for a family of two-dimensional area-preserving mappings described by the dynamical variables I and θ by using Slater's criterion. The Slater theorem says there are three different return times for an irrational translation over a circle in a given interval. The returning time, which measures the number of iterations a map needs to return to a given periodic or quasi periodic region, has three responses along an invariant spanning curve. They are related to a continued fraction expansion used in the translation and obey the Fibonacci sequence. The rotation numbers for such curves are related to a noble number, leading to a devil's staircase structure. The behavior of the rotation number as a function of invariant spanning curves located by Slater's criterion resulted in an expression of a power law in which the absolute value of the exponent is equal to the control parameter γ that controls the speed of the divergence of θ in the limit the action I is sufficiently small.

Information geometry theory of bifurcations? A covariant formulation.

The conventional local bifurcation theory (CBT) fails to present a complete characterization of the stability and general aspects of complex phenomena. After all, the CBT only explores the behavior of nonlinear dynamical systems in the neighborhood of their fixed points. Thus, this limitation imposes the necessity of non-trivial global techniques and lengthy numerical solutions. In this article, we present an attempt to overcome these problems by including the Fisher information theory in the study of bifurcations. Here, we investigate a Riemannian metrical structure of local and global bifurcations described in the context of dynamical systems. The introduced metric is based on the concept of information distance. We examine five contrasting models in detail: saddle-node, transcritical, supercritical pitchfork, subcritical pitchfork, and homoclinic bifurcations. We found that the metric imposes a curvature scalar R on the parameter space. Also, we discovered that R diverges to infinity while approaching bifurcation points. We demonstrate that the local stability conditions are recovered from the interpretations of the curvature R, while global stability is inferred from the character of the Fisher metric. The results are a clear improvement over those of the conventional theory.

Multiple Reflections for Classical Particles Moving under the Influence of a Time-Dependent Potential Well.

We study the dynamics of classical particles confined in a time-dependent potential well. The dynamics of each particle is described by a two-dimensional nonlinear discrete mapping for the variables energy en and phase ϕn of the periodic moving well. We obtain the phase space and show that it contains periodic islands, chaotic sea, and invariant spanning curves. We find the elliptic and hyperbolic fixed points and discuss a numerical method to obtain them. We study the dispersion of the initial conditions after a single iteration. This study allows finding regions where multiple reflections occur. Multiple reflections happen when a particle does not have enough energy to exit the potential well and is trapped inside it, suffering several reflections until it has enough energy to exit. We also show deformations in regions with multiple reflection, but the area remains constant when we change the control parameter NC. Finally, we show some structures that appear in the e0e1 plane by using density plots.

Investigation of pollen release by poricidal anthers using mathematical billiards.

Buzz pollination is described using a mathematical model considering a billiard approach. Applications to a rough morphology of a typical poricidal anther of a tomato flower (Solanum lycopersicum) experiencing vibrations applied by a bumblebee (Bombus terrestris) are made. The anther is described by a rectangular billiard with a pore on its tip while the borders are perturbed by specific oscillations according to the vibrational properties of the bumblebee. Pollen grains are considered as noninteracting particles that can escape through the pore. Our results not only recover some observed data but also provide a possible answer to an open problem involving buzz pollination.

Dynamical aspects of a bouncing ball in a nonhomogeneous field.

We study some dynamical properties of a charged particle that moves in a nonhomogeneous electric field and collides against an oscillating platform. Depending on the values of parameters, the system presents (i) predominantly regular dynamics or (ii) structures of chaotic behavior in phase space conditioned to the initial conditions. The localization of the fixed points and their stability are carefully discussed. Average properties of the chaotic sea are investigated under a scaling approach. We show that the system belongs to the same universality class as the Fermi-Ulam model.

Leaking of orbits from the phase space of the dissipative discontinuous standard mapping.

We investigate the escape of particles from the phase space produced by a two-dimensional, nonlinear and discontinuous, area-contracting map. The mapping, given in action-angle variables, is parametrized by K and γ which control the strength of nonlinearity and dissipation, respectively. We focus on two dynamical regimes, K<1 and K≥1, known as slow and quasilinear diffusion regimes, respectively, for the area-preserving version of the map (i.e., when γ=0). When a hole of hight h is introduced in the action axis we find both the histogram of escape times P_{E}(n) and the survival probability P_{S}(n) of particles to be scale invariant, with the typical escape time n_{typ}=exp〈lnn〉; that is, both P_{E}(n/n_{typ}) and P_{S}(n/n_{typ}) define universal functions. Moreover, for γ≪1, we show that n_{typ} is proportional to h^{2}/D, where D is the diffusion coefficient of the corresponding area-preserving map that in turn is proportional to K^{5/2} and K^{2} in the slow and the quasilinear diffusion regimes, respectively.

Instantaneous frequencies in the Kuramoto model.

Using the main results of the Kuramoto theory of globally coupled phase oscillators combined with methods from probability and generalized function theory in a geometric analysis, we extend Kuramoto's results and obtain a mathematical description of the instantaneous frequency (phase-velocity) distribution. Our result is validated against numerical simulations, and we illustrate it in cases in which the natural frequencies have normal and Beta distributions. In both cases, we vary the coupling strength and compare systematically the distribution of time-averaged frequencies (a known result of Kuramoto theory) to that of instantaneous frequencies, focusing on their qualitative differences near the synchronized frequency and in their tails. For a class of natural frequency distributions with power-law tails, which includes the Cauchy-Lorentz distribution, we analyze the tails of the instantaneous frequency distribution by means of an asymptotic formula obtained from a power-series expansion.

Diffusion phenomena in a mixed phase space.

We show that, in strongly chaotic dynamical systems, the average particle velocity can be calculated analytically by consideration of Brownian dynamics in a phase space, the method of images, and the use of the classical diffusion equation. The method is demonstrated on the simplified Fermi-Ulam accelerator model, which has a mixed phase space with chaotic seas, invariant tori, and Kolmogorov-Arnold-Moser islands. The calculated average velocities agree well with numerical simulations and with an earlier empirical theory.

Dynamical thermalization in time-dependent billiards.

Numerical experiments of the statistical evolution of an ensemble of noninteracting particles in a time-dependent billiard with inelastic collisions reveals the existence of three statistical regimes for the evolution of the speed ensemble, namely, diffusion plateau, normal growth/exponential decay, and stagnation. These regimes are linked numerically to the transition from Gauss-like to Boltzmann-like speed distributions. Furthermore, the different evolution regimes are obtained analytically through velocity-space diffusion analysis. From these calculations, the asymptotic root mean square of speed, initial plateau, and the growth/decay rates for an intermediate number of collisions are determined in terms of the system parameters. The analytical calculations match the numerical experiments and point to a dynamical mechanism for "thermalization," where inelastic collisions and a high-dimensional phase space lead to a bounded diffusion in the velocity space toward a stationary distribution function with a kind of "reservoir temperature" determined by the boundary oscillation amplitude and the restitution coefficient.

Diffusion entropy analysis in billiard systems.

In this work we investigate how the behavior of the Shannon entropy can be used to measure the diffusion exponent of a set of initial conditions in two systems: (i) standard map and (ii) the oval billiard. We are interested in the diffusion near the main island in the phase space, where stickiness is observed. We calculate the diffusion exponent for many values of the nonlinear parameter of the standard map where the size and shape of the main island change as the parameter varies. We show that the changes of behavior in the diffusion exponent are related with the changes in the area of the main island and show that when the area of the main island is abruptly reduced, due to the destruction of invariant tori and, consequently, creation of hyperbolic and elliptic fixed points, the diffusion exponent grows.

An investigation of the parameter space for a family of dissipative mappings.

The parameter plane investigation for a family of two-dimensional, nonlinear, and area contracting map is made. Several dynamical features in the system such as tangent, period-doubling, pitchfork, and cusp bifurcations were found and discussed together with cascades of period-adding, period-doubling, and the Feigeinbaum scenario. The presence of spring and saddle-area structures allow us to conclude that cubic homoclinic tangencies are present in the system. A set of complex sets such as streets with the same periodicity and the period-adding of spring-areas are observed in the parameter space of the mapping.

Scaling and self-similarity for the dynamics of a particle confined to an asymmetric time-dependent potential well.

The dynamics of a classical point particle confined to an asymmetric time-dependent potential well is investigated under the framework of scaling. The potential corresponds to a reduced version of a particle moving along an infinitely periodic sequence of synchronously oscillating potential barriers. The dynamics of the model is described by a two-dimensional nonlinear and area preserving map in energy and phase variables. The asymmetric potential well is defined by two regions: Region I with fixed null potential and region II with an oscillating potential. The time-dependent potential of region II makes, for certain initial conditions, the particle to undergo a number of multiple reflections η at the border of the two regions and stay trapped in region I. Such trappings are described by histograms of multiple reflections η, obeying the power-law H(η)∝η^{-ν} with ν≈3, which are scale invariant with a scaling parameter depending of the control parameters of the mapping. We identify the location of the sets of initial conditions in phase space producing the multiple reflections and show that they generate well defined self-similar structures in density plots of trajectories in energy space. The self-similar structures can be enhanced by properly tuning the system parameters.

Transition from normal to ballistic diffusion in a one-dimensional impact system.

We characterize a transition from normal to ballistic diffusion in a bouncing ball dynamics. The system is composed of a particle, or an ensemble of noninteracting particles, experiencing elastic collisions with a heavy and periodically moving wall under the influence of a constant gravitational field. The dynamics lead to a mixed phase space where chaotic orbits have a free path to move along the velocity axis, presenting a normal diffusion behavior. Depending on the control parameter, one can observe the presence of featured resonances, known as accelerator modes, that lead to a ballistic growth of velocity. Through statistical and numerical analysis of the velocity of the particle, we are able to characterize a transition between the two regimes, where transport properties were used to characterize the scenario of the ballistic regime. Also, in an analysis of the probability of an orbit to reach an accelerator mode as a function of the velocity, we observe a competition between the normal and ballistic transport in the midrange velocity.

Transport of chaotic trajectories from regions distant from or near to structures of regular motion of the Fermi-Ulam model.

The chaotic portion of phase space of the simplified Fermi-Ulam model is studied under the context of transport of trajectories in two scenarios: (i) the trajectories are originated from a region distant from the islands of regular motion and are transported to a region located at a high portion of phase space and (ii) the trajectories are originated from chaotic regions around the islands of regular motion and are transported to other regions around islands of regular motion. The transport is investigated in terms of the observables histogram of transport and survival probability. We show that the histogram curves are scaling invariant and we organize the survival probability curves in four kinds of behavior, namely (a) transition from exponential decay to power law decay, (b) transition from exponential decay to stretched exponential decay, (c) transition from an initial fast exponential decay to a slower exponential decay, and (d) a single exponential decay. We show that, depending on choice of the regions of origin and destination, the transport process is weakly affected by the stickiness of trajectories around islands of regular motion.

Thermodynamics of a time-dependent and dissipative oval billiard: A heat transfer and billiard approach.

We study some statistical properties for the behavior of the average squared velocity-hence the temperature-for an ensemble of classical particles moving in a billiard whose boundary is time dependent. We assume the collisions of the particles with the boundary of the billiard are inelastic, leading the average squared velocity to reach a steady-state dynamics for large enough time. The description of the stationary state is made by using two different approaches: (i) heat transfer motivated by the Fourier law and (ii) billiard dynamics using either numerical simulations and theoretical description.

On the statistical and transport properties of a non-dissipative Fermi-Ulam model.

The transport and diffusion properties for the velocity of a Fermi-Ulam model were characterized using the decay rate of the survival probability. The system consists of an ensemble of non-interacting particles confined to move along and experience elastic collisions with two infinitely heavy walls. One is fixed, working as a returning mechanism of the colliding particles, while the other one moves periodically in time. The diffusion equation is solved, and the diffusion coefficient is numerically estimated by means of the averaged square velocity. Our results show remarkably good agreement of the theory and simulation for the chaotic sea below the first elliptic island in the phase space. From the decay rates of the survival probability, we obtained transport properties that can be extended to other nonlinear mappings, as well to billiard problems.

Global ballistic acceleration in a bouncing-ball model.

The ballistic increase for the velocity of a particle in a bouncing-ball model was investigated. The phenomenon is caused by accelerating structures in phase space known as accelerator modes. They lead to a regular and monotonic increase of the velocity. Here, both regular and ballistic Fermi acceleration coexist in the dynamics, leading the dynamics to two different growth regimes. We characterized deaccelerator modes in the dynamics, corresponding to unstable points in the antisymmetric position of the accelerator modes. In control parameter space, parameter sets for which these accelerations and deaccelerations constitute structures were obtained analytically. Since the mapping is not symplectic, we found fractal basins of influence for acceleration and deacceleration bounded by the stable and unstable manifolds, where the basins affect globally the average velocity of the system.

Dynamics of classical particles in oval or elliptic billiards with a dispersing mechanism.

Some dynamical properties for an oval billiard with a scatterer in its interior are studied. The dynamics consists of a classical particle colliding between an inner circle and an external boundary given by an oval, elliptical, or circle shapes, exploring for the first time some natural generalizations. The billiard is indeed a generalization of the annular billiard, which is of strong interest for understanding marginally unstable periodic orbits and their role in the boundary between regular and chaotic regions in both classical and quantum (including experimental) systems. For the oval billiard, which has a mixed phase space, the presence of an obstacle is an interesting addition. We demonstrate, with details, how to obtain the equations of the mapping, and the changes in the phase space are discussed. We study the linear stability of some fixed points and show both analytically and numerically the occurrence of direct and inverse parabolic bifurcations. Lyapunov exponents and generalized bifurcation diagrams are obtained. Moreover, histograms of the number of successive iterations for orbits that stay in a cusp are studied. These histograms are shown to be scaling invariant when changing the radius of the scatterer, and they have a power law slope around -3. The results here can be generalized to other kinds of external boundaries.