Escape of particles in a time-dependent potential well

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.

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.

Nonlinear dynamics of an elliptic vortex embedded in an oscillatory shear flow

The nonlinear dynamics of an elliptic vortex subjected to a time-periodic linear external shear flow is studied numerically. Making use of the ideas from the theory of nonlinear resonance overlaps, the study focuses on the appearance of chaotic regimes in the ellipse dynamics. When the superimposed flow is stationary, two general types of the steady-state phase portrait are considered: one that features a homoclinic separatrix delineating bounded and unbounded phase trajectories and one without a separatrix (all the phase trajectories are bounded in a periodic domain). When the external flow is time-periodic, the ensuing nonlinear dynamics differs significantly in both cases. For the case with a separatrix and two distinct types of phase trajectories: bounded and unbounded, the effect of the most influential nonlinear resonance with the winding number of 1:1 is analyzed in detail. Namely, the process of occupying the central stability region associated with the steady-state elliptic critical point by the stability region associated with the nonlinear resonance of 1:1 as the perturbation frequency gradually varies is investigated. A stark increase in the persistence of the central regular dynamics region against perturbation when the resonance of 1:1 associated stability region occupies the region associated with the steady-state elliptic critical point is observed. An analogous persistence of the regular motion occurs for higher perturbation frequencies when the corresponding stability islands reach the central stability region associated with the steady-state elliptic point. An analysis for the case with the resonance of 1:2 is presented. For the second case with only bounded phase trajectories and, therefore, no separatrix, the appearance of much bigger stability islands associated with nonlinear resonances compared with the case with a separatrix is reported.

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.

Analysis of resonant population transfer in time-dependent elliptical quantum billiards

A Fermi golden rule for population transfer between instantaneous eigenstates of elliptical quantum billiards with oscillating boundaries is derived. Thereby the occurrence of both the recently observed resonant population transfer between instantaneous eigenstates and the empirical criterion stating that these transitions occur when the driving frequency matches the mean difference of the latter [Lenz et al., New J. Phys. 13, 103019 (2011)] is explained. As a second main result a criterion judging which resonances are resolvable in a corresponding experiment of certain duration is provided. Our analysis is complemented by numerical simulations for three different driving laws. The corresponding resonance spectra are in agreement with the predictions of both criteria.

Scaling investigation for the dynamics of charged particles in an electric field accelerator

Some dynamical properties of an ensemble of trajectories of individual (non-interacting) classical particles of mass m and charge q interacting with a time-dependent electric field and suffering the action of a constant magnetic field are studied. Depending on both the amplitude of oscillation of the electric field and the intensity of the magnetic field, the phase space of the model can either exhibit: (i) regular behavior or (ii) a mixed structure, with periodic islands of regular motion, chaotic seas characterized by positive Lyapunov exponents, and invariant Kolmogorov-Arnold-Moser curves preventing the particle to reach unbounded energy. We define an escape window in the chaotic sea and study the transport properties for chaotic orbits along the phase space by the use of scaling formalism. Our results show that the escape distribution and the survival probability obey homogeneous functions characterized by critical exponents and present universal behavior under appropriate scaling transformations. We show the survival probability decays exponentially for small iterations changing to a slower power law decay for large time, therefore, characterizing clearly the effects of stickiness of the islands and invariant tori. For the range of parameters used, our results show that the crossover from fast to slow decay obeys a power law and the behavior of survival orbits is scaling invariant.

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 nonintegrable (ε≠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 εc, 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.