Phase space properties and chaotic transport for a particle moving in a time dependent step potential well

Abstract

Some dynamical properties for an ensemble of non-interacting classical particles along chaotic orbits and transport properties over the chaotic sea for the problem of a step and time dependent potential well are considered. The dynamics of each particle is described by a two-dimensional, nonlinear and area preserving mapping for the variables energy and time. The phase space is of mixed-type and contains periodic islands, a set of invariant KAM curves and chaotic seas. The chaotic orbits are characterized by the use of Lyapunov exponents. Transport over the chaotic sea is considered and scaling exponents are obtained. A sticky region around a chain of periodic islands produces local and temporarily trapping of the dynamics and discussions of the rearrangement of the phase space are made.

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

“…Similar wells have been studied in Refs. [12,14,20]; none of those works considered the investigation we are dealing with in this paper. Also, we have defined two regions in the well, labeled as I and II in Fig.…”

“…[1][2][3][4][5][6][7][8]. This class of systems can be described by the use of different procedures that may range from quantum approaches, where the Schrödinger equation is solved, to classical-chaos investigations, where chaotic seas are characterized by Lyapunov exponents, passing through the description of phase transitions with the variation of control parameters, among other approaches [9][10][11][12][13][14]. The effect of noise in the dynamics and other kinds of perturbations are also subjects of interest [12].…”

“…The effect of noise in the dynamics and other kinds of perturbations are also subjects of interest [12]. In the classical case and considering time perturbations to the potential, the particles may exhibit chaotic motion leading to scaling invariance under the variation of control parameters whenever also the escape of particles from specific regions of the phase space can be considered [12,14]. To cite an application of the quantum case, the Gaussian wave-packet dynamics and quantum tunneling in asymmetric double-well systems were reported in [15].…”

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

“…Escape processes are usually studied in the context where a system presents a natural leak or hole through which a particle or a wave leaves the system [1][2][3][4][5][6]. A leak or hole can be introduced in closed systems in order to study the properties of trajectories in phase space [7][8][9] or to simulate loss in confinement of plasma and measurement devices, e.g., antennas and sensors [10][11][12][13]. Transport is the process by which trajectories initially located at a region of origin evolves to a destination region.…”

“…Eventually, these chaotic trajectories come close enough to a KAM island and stick to the border of this regular region for long periods of time [27,28]. As a consequence, in a system with phase space composed by mixed structure, i.e., regions of chaotic motion coexisting with regular trajectories, the Poincaré recurrence times and survival probability present an initial regime of exponential decay followed by a slower decay described by power law [8,9] or stretched exponential [29]. These slower-than-exponential decays are attributed to the temporary trapping, or stick, of chaotic trajectories near KAM islands or other periodic regions.…”