Chaotic escape from an open vase-shaped cavity. I. Numerical and experimental results

Abstract

We present part I in a two-part study of an open chaotic cavity shaped as a vase. The vase possesses an unstable periodic orbit in its neck. Trajectories passing through this orbit escape without return. For our analysis, we consider a family of trajectories launched from a point on the vase boundary. We imagine a vertical array of detectors past the unstable periodic orbit and, for each escaping trajectory, record the propagation time and the vertical detector position. We find that the escape time exhibits a complicated recursive structure. This recursive structure is explored in part I of our study. We present an approximation to the Helmholtz equation for waves escaping the vase. By choosing a set of detector points, we interpolate trajectories connecting the source to the different detector points. We use these interpolated classical trajectories to construct the solution to the wave equation at a detector point. Finally, we construct a plot of the detector position versus the escape time and compare this graph to the results of an experiment using classical ultrasound waves. We find that generally the classical trajectories organize the escaping ultrasound waves.

Topological analysis of chaotic transport through a ballistic atom pump

“…This discrepancy is caused by our choice of using only a small anount of initial topological information (three iterates of the fundamental segments of the unstable manifolds). Agreement between the two methods can be extended to higher iterates by using more initial topological information as the basis for the symbolic representation [47,48].…”

“…As mentioned earlier, an important difference between this problem and earlier ones [37][38][39][40][41][42][43][44][45][46][47][48][49][50] is that we are now examining a full scattering system, with particles approaching from and receding to infinite distance. (In previous work, we examined only "half-scattering," i.e., escape from a confined region.)…”

“…In a future complementary paper, we will present a thorough examination of ballistic atom pumps using classical, semiclassical, and quantum theories. In earlier work, we have studied three other models of chaotic transport: escape of excited electrons from atoms in strong electric and magnetic fields, escape of light or sound from a vase-shaped enclosure, and escape of points from a region of the phase plane in an abstract two-dimensional map [37][38][39][40][41][42][43][44][45][46][47][48][49]. In all of these cases, a classical description of the system leads to a homoclinic tangle, the interior of which we call a "complex," and the trajectories begin on some line in the tangle.…”

Experimental validation of phase space conduits of transition between potential wells

“…To name but a few, ionization of a hydrogen atom under electromagnetic field in atomic physics [1], transport of defects in solid state and semiconductor physics [2], isomerization of clusters [3], reaction rates in chemical physics [4,5], buckling modes in structural mechanics [6,7], ship motion and capsize [8][9][10], escape and recapture of comets and asteroids in celestial mechanics [11][12][13], and escape into inflation or re-collapse to singularity in cosmology [14]. The theoretical criteria of transition and its agreement with laboratory experiment have been shown for 1 degree of freedom (DOF) systems [15][16][17]. Detailed experimental validation of the geometrical framework for predicting transition in higher dimensional phase space ( 4, that is for 2 or more DOF systems) is still lacking.…”

Exact relations between homoclinic and periodic orbit actions in chaotic systems

“…The fundamental ingredients of chaos [3], exponential compression and stretching, as well as the folding and mixing of phase space volumes, are all characterized by stable and unstable manifolds and the complicated patterns they form, namely homoclinic tangles [4,5]. They govern various dynamical properties like phase space mixing [6,7], transport [8] or escape rates [9][10][11][12][13]. Intersections of stable and unstable manifolds give rise to homoclinic and heteroclinic orbits [1], which have fixed past and future asymptotes.…”