Limit of small exits in open Hamiltonian systems

An investigation of escape and scaling properties of a billiard system

We investigate some statistical properties of escaping particles in a billiard system whose boundary is described by two control parameters with a hole on its boundary. Initially, we analyze the survival probability for different hole positions and sizes. We notice that the survival probability follows an exponential decay with a characteristic power-law tail when the hole is positioned partially or entirely over large stability islands in phase space. We find that the survival probability exhibits scaling invariance with respect to the hole size. In contrast, the survival probability for holes placed in predominantly chaotic regions deviates from the exponential decay. We introduce two holes simultaneously and investigate the complexity of the escape basins for different hole sizes and control parameters by means of the basin entropy and the basin boundary entropy. We find a non-trivial relation between these entropies and the system's parameters and show that the basin entropy exhibits scaling invariance for a specific control parameter interval.

Basin Entropy and Shearless Barrier Breakup in Open Non-Twist Hamiltonian Systems

We consider open non-twist Hamiltonian systems represented by an area-preserving two-dimensional map describing incompressible planar flows in the reference frame of a propagating wave, and possessing exits through which map orbits can escape. The corresponding escape basins have a fractal nature that can be revealed by the so-called basin entropy, a novel concept developed to quantify final-state uncertainty in dynamical systems. Since the map considered violates locally the twist condition, there is a shearless barrier that prevents global chaotic transport. In this paper, we show that it is possible to determine the shearless barrier breakup by considering the variation in the escape basin entropy with a tunable parameter.

Dependence of the escape from an axially symmetric galaxy on the energy

The escape of a particle from a dynamical system depends on the intersection between the ingoing and outgoing asymptotic trajectories to certain periodic orbits placed at the openings of the curves of zero velocity of the system. Although many efforts have been devoted to the analysis of the escape from potentials presenting multiple openings, there are still few studies on potentials with only one opening. In this article, we clarify the way in which the energy affects the escape in this type of systems, showing that, contrary to what one could expect, there are several bifurcations for certain values of the energy.

On the escape from potentials with two exit channels

The aim of this paper is to investigate the escape dynamics in a Hamiltonian system describing the motion of stars in a galaxy with two exit channels through the analysis of the successive intersections of the stable and unstable manifolds to the main unstable periodic orbits with an adequate surface of section. We describe in detail the origin of the spirals shapes of the windows through which stars escape.

Thin structure of the transit time distributions of open billiards

It is known that typical open billiards distribution of transit times is an exponentially decaying function, possibly with a power-law tail. In the paper we show that on small scales some of such distributions change their appearance. These distributions contain a quasiperiodic thin structure, which carries a significant amount of information about the system. Origin and properties of this structure are discussed.

Fractal structures in nonlinear plasma physics

Fractal structures appear in many situations related to the dynamics of conservative as well as dissipative dynamical systems, being a manifestation of chaotic behaviour. In open area-preserving discrete dynamical systems we can find fractal structures in the form of fractal boundaries, associated to escape basins, and even possessing the more general property of Wada. Such systems appear in certain applications in plasma physics, like the magnetic field line behaviour in tokamaks with ergodic limiters. The main purpose of this paper is to show how such fractal structures have observable consequences in terms of the transport properties in the plasma edge of tokamaks, some of which have been experimentally verified. We emphasize the role of the fractal structures in the understanding of mesoscale phenomena in plasmas, such as electromagnetic turbulence.

Poincaré recurrences and transient chaos in systems with leaks

In order to simulate observational and experimental situations, we consider a leak in the phase space of a chaotic dynamical system. We obtain an expression for the escape rate of the survival probability by applying the theory of transient chaos. This expression improves previous estimates based on the properties of the closed system and explains dependencies on the position and size of the leak and on the initial ensemble. With a subtle choice of the initial ensemble, we obtain an equivalence to the classical problem of Poincaré recurrences in closed systems, which is treated in the same framework. Finally, we show how our results apply to weakly chaotic systems and justify a split of the invariant saddle into hyperbolic and nonhyperbolic components, related, respectively, to the intermediate exponential and asymptotic power-law decays of the survival probability.

Fractional kinetic model for chaotic transport in nonintegrable Hamiltonian systems

We propose a kinetic model of transport in nonintegrable Hamiltonian systems, based on a fractional kinetic equation with spatially dependent diffusion coefficient. The diffusion coefficient is estimated from the remainder of the optimal normal form for the given region of the phase space. After partitioning the phase space into building blocks, a separate equation can be constructed for each block. Solving the kinetic equations approximately and estimating the diffusion time scales, we convolve the solutions to get the description of the macroscopic behavior. We show that, in the limit of infinitely many blocks, one can expect an approximate scaling relation between the Lyapunov time and the diffusion (or escape) time, which is either an exponential or a power law. We check our results numerically on over a dozen Hamiltonians and find a good agreement.

Crisis-induced intermittency in two coupled chaotic maps: towards understanding chaotic itinerancy

The present paper considers crisis-induced intermittency in a system composed of two coupled logistic maps. Its purpose is to clarify a bifurcation scenario generating such intermittent behaviors that can be regarded as a simple example of chaotic itinerancy. The intermittent dynamics appears immediately after an attractor-merging crisis of two off-diagonal chaotic attractors in a symmetrically coupled system. The scenario for the crisis is investigated through analyses of sequential bifurcations leading to the two chaotic attractors and successive changes in basin structures with variation of a system parameter. The successive changes of the basins are also characterized by variation of a dimension of a fractal basin boundary. A numerical analysis shows that simultaneous contacts between the attractors and the fractal basin boundary bring about the crisis and a snap-back repeller generated at the crisis produces the intermittent transitions. Furthermore, a modified scenario for intermittent behaviors in an asymmetrically coupled system is also discussed.

Quasiscarred resonances in a spiral-shaped microcavity

We study resonance patterns of a spiral-shaped dielectric microcavity with chaotic ray dynamics. Many resonance patterns of this microcavity, with refractive indices n=2 and 3, exhibit strong localization of simple geometric shape, and we call them quasiscarred resonances in the sense that there is, unlike conventional scarring, no underlying periodic orbits. It is shown that the formation of a quasiscarred pattern can be understood in terms of ray dynamical probability distributions and wave properties like uncertainty and interference.

Ray splitting in paraxial optical cavities

We present a numerical investigation of the ray dynamics in a paraxial optical cavity when a ray-splitting mechanism is present. The cavity is a conventional two-mirror stable resonator and the ray splitting is achieved by inserting an optical beam splitter perpendicular to the cavity axis. We show that depending on the position of the beam splitter the optical resonator can become unstable and the ray dynamics displays a positive Lyapunov exponent.