  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.
