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

Abstract

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.

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

“…We see several structures, which, apparently, does not present a pattern. It differs from the results obtained for a time-dependent potential barrier in a previous work [ 25 ]. In that work, it was possible to show the structures had an auto-similar pattern, but here, the structures are more complex.…”

“…When increasing the value of i , we will check an increased number of structures, which contribute to the last scenario shown in Figure 9 a. It is important to mention that similar structures appear, for example, in the Fermi–Ulam model [ 28 ], in concave billiards [ 29 , 30 ] and in the time-dependent potential barrier [ 25 ], but more studies are necessary to understand completely the mechanism that produces these structures.…”

“…We consider a classical particle starting at , with initial energy and an initial time . Each time the particle reaches the position , we save the energy E and time t ; then, this position will be our Poincaré section [ 26 , 27 ], which is different from previous publications [ 25 ]. This change is necessary because some observables in this paper are better observed when using this specific Poincaré section.…”

“…The authors in Ref. [ 25 ] showed, for a time-dependent potential barrier, that there are some regions in the phase space presenting multiple reflections. Inspired by this work, in this paper, we study a potential well.…”