A Computational Procedure to Detect a New Type of High Dimensional Chaotic Saddle and its Application to the 3-D Hill's Problem

Abstract

A computational procedure that allows the detection of a new type of high-dimensional chaotic saddle in Hamiltonian systems with three degrees of freedom is presented. The chaotic saddle is associated with a so-called normally hyperbolic invariant manifold (NHIM). The procedure allows to compute appropriate homoclinic orbits to the NHIM from which we can infer the existence a chaotic saddle. NHIMs control the phase space transport across an equilibrium point of saddle-centre-...-centre stability type, which is a fundamental mechanism for chemical reactions, capture and escape, scattering, and, more generally, ``transformation'' in many different areas of physics. Consequently, the presented methods and results are of broad interest. The procedure is illustrated for the spatial Hill's problem which is a well known model in celestial mechanics and which gained much interest e.g. in the study of the formation of binaries in the Kuiper belt.

12 pages, 6 figures, pdflatex, submitted to JPhysA

References in corpus (2)

• Formation of Kuiper Belt Binaries
• Homoclinic chaos in the dynamics of a general Bianchi IX model

Cited by in corpus (26)

• Wigner's Dynamical Transition State Theory in Phase Space: Classical and Quantum
• Microcanonical rates, gap times, and phase space dividing surfaces
• Nonstatistical dynamics on potentials exhibiting reaction path bifurcations and valley-ridge inflection points
• Nonstatistical dynamics on the caldera
• Basin boundary, edge of chaos, and edge state in a two-dimensional model
• Capture and escape in the elliptic restricted three-body problem
• Phase space geometry and reaction dynamics near index two saddles
• Geometrical Models of the Phase Space Structures Governing Reaction Dynamics
• Index k saddles and dividing surfaces in phase space, with applications to isomerization dynamics
• Chaotic Dynamics in Multidimensional Transition States
• The chaotic set and the cross section for chaotic scattering beyond in 3 degrees of freedom
• Reaction rate calculation with time-dependent invariant manifolds
• Transition State Theory for dissipative systems without a dividing surface
• Quantum Theory of Reactive Scattering in Phase Space
• Transition State Theory for solvated reactions beyond recrossing-free dividing surfaces
• Strands and braids in narrow planetary rings: A scattering system approach
• Phase space structure and dynamics for the Hamiltonian isokinetic thermostat
• Ground Control to Niels Bohr: Exploring Outer Space with Atomic Physics
• Orbit classification in the Hill problem: I. The classical case
• Correlating the escape dynamics and the role of the normally hyperbolic invariant manifolds in a binary system of dwarf spheroidal galaxies
• Elucidating the escape dynamics of the four hill potential
• Hořava-Lifshitz Bouncing Bianchi IX Universes: A Dynamical System Analysis
• General Bianchi IX dynamics in bouncing braneworld cosmology: homoclinic chaos and the BKL conjecture
• Phase space structures causing the reaction rate decrease in the collinear hydrogen exchange reaction
• Escaping from a degenerate version of the four hill potential
• Low-Dimensional Projection of Reactive Islands in Chemical Reaction Dynamics Using a Supervised Dimensionality Reduction Method