Global invariant manifolds delineating transition and escape dynamics in dissipative systems

TL;DR

This paper extends the concept of invariant manifolds from conservative to dissipative systems to analyze transition and escape dynamics, using a two-mode model of buckling to demonstrate the differences in transition boundaries.

Contribution

It introduces a new computational framework for studying invariant manifolds in dissipative systems, highlighting the change from cylindrical to ellipsoidal transition boundaries.

Findings

In conservative systems, transition boundaries are cylindrical.

In dissipative systems, transition boundaries become ellipsoidal.

The developed algorithms effectively analyze escape dynamics.

Abstract

Invariant manifolds play an important role in organizing global dynamical behaviors. For example, it is found that in multi-well conservative systems where the potential energy wells are connected by index-1 saddles, the motion between potential wells is governed by the invariant manifolds of a periodic orbit around the saddle. In two degree of freedom systems, such invariant manifolds appear as cylindrical conduits which are referred to as transition tubes. In this study, we apply the concept of invariant manifolds to study the transition between potential wells in not only conservative systems, but more realistic dissipative systems, by solving respective proper boundary-value problems. The example system considered is a two mode model of the snap-through buckling of a shallow arch. We define the transition region, $\mathcal{T}_h$, as a set of initial conditions of a given initial Hamiltonian energy $h$ with which the trajectories can escape from one potential well to another, which in the example system corresponds to snap-through buckling of a structure. The numerical results reveal that in the conservative system the boundary of the transition region, $\partial \mathcal{T}_h$, is a cylinder, while in the dissipative system, $\partial \mathcal{T}_h$ is an ellipsoid. The algorithms developed in the current research from the perspective of invariant manifold provides a robust theoretical-computational framework to study escape and transition dynamics.