A tube dynamics perspective governing stability transitions: An example based on snap-through buckling

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

“…The transition tube of the conservative system not only gives all the initial conditions for transit orbits in phase space, but also gives the boundary of their evolution, while the transition 'tube' for the dissipative system merely gives the boundary of the initial conditions of a specific initial energy for transit orbits on a specific Poincaré section and the evolution of the transit orbits with those initial conditions is not along an invariant energy manifold any longer. As for the global structure of the phase space in the dissipative system that governs the initial conditions of transit orbits, this was not addressed in [2]. In the current study, we continue this study and answer in more detail the concern of how the situation changes when dissipation is present, finding that the transition tube in the conservative system becomes a transition ellipsoid in the dissipative system.…”

“…A linear two degrees of freedom conservative system with a saddle-center type equilibrium point (i.e., index-1 or rank-1 saddle) [2][3][4][5]8] can be transformed via a canonical transformation to normal form coordinates (q 1 , q 2 , p 1 , p 2 ) such that the quadratic Hamiltonian, H 2 , can be written in the normal form,…”

“…Transition events are very common in both the natural world, daily life and even industrial applications. Examples of transition are the snap-through of plant leaves and engineering structures responding to stimuli [1,2], the flipping over of umbrellas on a windy day, reaction rates in chemical reaction dynamics [3], the escape and recapture of comets and spacecraft in celestial mechanics [4][5][6], and the capsize of ships [7,8].…”

“…Some researchers have studied escape in conservative gyroscopic systems (e.g., [6,25]). There exist transition tubes controlling the escape which are topologically the same as in an inertial system [2,8,26]. However, to the best knowledge of the authors, no study has been carried out to study the escape in systems with both dissipative and gyroscopic forces present.…”

“…which is an ellipse in the configuration space called the ellipse of transition, and is merely the configuration space projection of the transition ellipsoid (48), first found in [2]. The ellipse of the transition confines the existence of transit orbits of a given initial energy which means the transit orbits can just exist inside the ellipse.…”