The chaotic set and the cross section for chaotic scattering in three degrees of freedom

Abstract

This article treats chaotic scattering with three degrees of freedom, where one of them is open and the other two are closed, as a first step toward a more general understanding of chaotic scattering in higher dimensions. Despite of the strong restrictions it breaks the essential simplicity implicit in any two-dimensional time-independent scattering problem. Introducing the third degree of freedom by breaking a continuous symmetry, we first explore the topological structure of the homoclinic/heteroclinic tangle and the structures in the scattering functions. Then we work out implications of these structures for the doubly differential cross section. The most prominent structures in the cross section are rainbow singularities. They form a fractal pattern which reflects the fractal structure of the chaotic invariant set. This allows to determine structures in the cross section from the invariant set and conversely, to obtain information about the topology of the invariant set from the cross section. The latter is a contribution to the inverse scattering problem for chaotic systems.

The vibrational dynamics of 3D HOCl above dissociation

“…Our analysis of Cl-HO scattering dynamics uses a potential energy surface (PES), constructed by Weiss et al [16], that realistically governs the dynamics of the bound HOCl molecule, and the Cl-HO scattering system for an interval of energies just above dissociation of Cl from HO. For the 3D Cl-HO scattering process, we find behavior similar to that observed by Jung et al [17], with the vibrational degree of freedom of HO playing the role of a weakly coupled third degree of freedom (at least for some parameter regimes). For a given initial energy of Cl and HO, the initial conditions of both objects can be specified in terms of two initial "phases" (which we define later).…”

Revealing roaming on the double Morse potential energy surface with Lagrangian descriptors

“…In both figures, some lines converge directly to constant values of the momentum in the asymptotic region. These lines correspond to the branches of stable and unstable manifolds that go directly to the asymptotic region, close to these branches there are the lobules where [35,37]. In figure 10 there are three disconnected regions: two unbounded regions on the extremes and one bounded region in the middle surrounded by a white region.…”

“…The chaotic scattering is a common phenomenon in the open Hamiltonian system. Detailed studies of chaotic scattering in Hamiltonian systems with two and three degrees of freedom are in the recent literature [34][35][36][37]. The tangle between the stable and unstable manifolds generates rich dynamics in the interaction region.…”

Escape of particles in a time-dependent potential well

“…This power law arises from combined effects of an infinite hierarchy of islands with a corresponding hierarchy of time scales. Recent investigations of resonance splittings and homoclinic tangles in the vicinity of islands may provide insight into these mechanisms [25][26][27][28].…”