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

•For the first time, both dissipative and gyroscopic forces are considered in the context of the geometric theory of tube dynamics for escape across an index-1 saddle.•In N degrees of freedom, the boundary of transit orbits starting at the same initial energy goes from a (2N-2)-dimensional hyper-cylinder in the conservative case to a (2N-2)-dimensional hyper-ellipsoid in the dissipative case.•Several two degree of freedom example systems are considered that illustrate escape from potential wells and they are classified into systems with coupled or uncoupled saddle-focus dynamics.•A gyroscopic system with any amount of damping, and an inertial system with unequal damping in the multiple degree of freedom case both have coupled dynamics of the saddle and focus projections in the symplectic eigenspace. 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 have 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 the system state 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 uncoupled 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, Th, 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, ∂Th, is a cylinder, while in dissipative systems, ∂Th is an ellipsoid.