Geometry of escape and transition dynamics in the presence of dissipative and gyroscopic forces in two degree of freedom systems

Abstract

Escape from a potential well can occur in different physical systems, such as capsize of ships, resonance transitions in celestial mechanics, and dynamic snap-through of arches and shells, as well as molecular reconfigurations in chemical reactions. The criteria and routes of escape in one-degree of freedom systems has been well studied theoretically with reasonable agreement with experiment. The trajectory can only transit from the hilltop of the one-dimensional potential energy surface. The situation becomes more complicated when the system has higher degrees of freedom since it has multiple routes to escape through an equilibrium of saddle-type, specifically, an index-1 saddle. This paper summarizes the geometry of escape across a saddle in some widely known physical systems with two degrees of freedom and establishes the criteria of escape providing both a methodology and results under the conceptual framework known as tube dynamics. These problems are classified into two categories based on whether the saddle projection and focus projection in the symplectic eigenspace are coupled or not when damping and/or gyroscopic effects are considered. For simplicity, only the linearized system around the saddle points is analyzed, but the results generalize to the nonlinear system. We define a transition region, T h , as the region of initial conditions of a given initial energy h which transit from one side of a saddle to the other. We find that in conservative systems, the boundary of the transition region, ∂T h , is a cylinder, while in dissipative systems, ∂T h is an ellipsoid.

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

“…As a result, the above relations together define a cone at the position (X 0 1 , X 0 2 , X 0 3 ) within the tube of transition. Here we refer to it as the cone of velocity, in analogy with the wedge of velocity discussed by Conley [36] (see also [11]). Notice the cone only exists inside of the tube of transition to ensure R > 0.…”

“…For conservative systems with higher degrees of freedom, the transition boundary for all possible escape trajectories is known to be the stable and unstable invariant manifolds of a normally hyperbolic invariant manifold (NHIM) [10] of a given energy. Recently, it was found that the corresponding transition boundary for dissipative systems is the stable invariant manifolds of an index-1 saddle connecting the potential wells [11,12]. Specifically, for an intermediate case of two degree of freedom, the NHIM is a collection of periodic orbits around the index-1 saddle, each periodic orbit corresponding to a given energy [13,4,14,15].…”

“…Specifically, for an intermediate case of two degree of freedom, the NHIM is a collection of periodic orbits around the index-1 saddle, each periodic orbit corresponding to a given energy [13,4,14,15]. The geometry of corresponding stable invariant manifolds of the NHIM and the index-1 saddle are pieces of tubes (or cylinders) [4,9,16] and ellipsoids [17,12,11] in the conservative and dissipative systems, respectively, sometimes referred to as transition tubes and transition ellipsoids in tube dynamics [18,19,20]. Recently, the theory of transition tubes in two degrees of freedom was experimentally verified [21].…”

“…Compared to the well understood escape dynamics in the conservative systems with higher degrees of freedom [34,35,16], we have little understanding on higher degree of freedom systems with dissipation. Here we aim to extend the study on the linearized dynamics in two degree of freedom systems [11] to three degree of freedom systems in the presence of dissipative forces.…”

Reactive Islands for Three Degrees-of-Freedom Hamiltonian Systems

“…The deterministic evolution of a system between two stable states through an intermediate unstable state is a fundamental setting for an important form of dynamical evolution that informs our way of thinking of "transition phenomena". This dynamical situation is common to many diverse fields in science and engineering, such as celestial mechanics [1,2], structural mechanics [3][4][5], and chemical reaction dynamics. This setting is natural for studies in the latter area and will be the focus of the applications that we consider.…”