Transition criteria and phase space structures in a three degree of freedom system with dissipation

Abstract

Escape from a potential well through an index-1 saddle can be widely found in some important physical systems. Knowing the criteria and phase space geometry that govern escape events plays an important role in making use of such phenomenon, particularly when realistic frictional or dissipative forces are present. We aim to extend the study the escape dynamics around the saddle from two degrees of freedom to three degrees of freedom, presenting both a methodology and phase space structures. Both the ideal conservative system and a perturbed, dissipative system are considered. We define the five-dimensional transition region, ${\mathcal{T}}_h$, as the set of initial conditions of a given initial energy $h$ for which the trajectories will escape from one side of the saddle to another. Invariant manifold arguments demonstrate that in the six-dimensional phase space, the boundary of the transition region, $\partial {\mathcal{T}}_h$, is topologically a four-dimensional hyper-cylinder in the conservative system, and a four-dimensional hyper-sphere in the dissipative system. The transition region ${\mathcal{T}}_h$ can be constructed by a solid three-dimensional ellipsoid (solid three-dimensional cylinder) in the three-dimensional configuration space, where at each point, there is a cone of velocity -- the velocity directions leading to transition are given by cones, with velocity magnitude given by the initial energy and the direction by two spherical angles with given limits. To illustrate our analysis, we consider an example system which has two potential minima connected by an index 1 saddle.

arXiv comment26 pages, 6 figures

References in corpus (14)

• Distinguished trajectories in time dependent vector fields
• Phase Space Analysis of the Dynamics on a Potential Energy Surface with an Entrance Channel and Two Potential Wells
• Snap buckling of a confined thin elastic sheet
• A Tube Dynamics Perspective Governing Stability Transitions: An Example Based on Snap-through Buckling
• Exploring Isomerization Dynamics on a Potential Energy Surface with an Index-2 Saddle using Lagrangian Descriptors
• Finding NHIM in 2 and 3 degrees-of-freedom with Hénon-Heiles type potential
• Invariant Manifolds and Rate Constants in Driven Chemical Reactions
• Tilting and Squeezing: Phase space geometry of Hamiltonian saddle-node bifurcation and its influence on chemical reaction dynamics
• Experimental validation of phase space conduits of transition between potential wells
• Isomerization dynamics of a buckled nanobeam
• Geometry of escape and transition dynamics in the presence of dissipative and gyroscopic forces in two degree of freedom systems
• Reactive Islands for Three Degrees-of-Freedom Hamiltonian Systems
• The role of depth and flatness of a potential energy surface in chemical reaction dynamics
• Painting the Phase Space of Dissipative Systems with Lagrangian Descriptors