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

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.