Phase Space Structure and Transport in a Caldera Potential Energy Surface

Abstract

We study phase space transport in a 2D caldera potential energy surface (PES) using techniques from nonlinear dynamics. The caldera PES is characterized by a flat region or shallow minimum at its center surrounded by potential walls and multiple symmetry related index one saddle points that allow entrance and exit from this intermediate region. We have discovered four qualitatively distinct cases of the structure of the phase space that govern phase space transport. These cases are categorized according to the total energy and the stability of the periodic orbits associated with the family of the central minimum, the bifurcations of the same family, and the energetic accessibility of the index one saddles. In each case we have computed the invariant manifolds of the unstable periodic orbits of the central region of the potential and the invariant manifolds of the unstable periodic orbits of the families of periodic orbits associated with the index one saddles. The periodic orbits of the central region are, for the first case, the unstable periodic orbits with period 10 that are outside the stable region of the stable periodic orbits of the family of the central minimum. In addition, the periodic orbits of the central region are, for the second and third case, the unstable periodic orbits of the family of the central minimum and for the fourth case the unstable periodic orbits with period 2 of a period-doubling bifurcation of the family of the central minimum. We have found that there are three distinct mechanisms determined by the invariant manifold structure of the unstable periodic orbits govern the phase space transport. The first mechanism explains the nature of the entrance of the trajectories from the region of the low energy saddles into the caldera and how they may become trapped in the central region of the potential. The second mechanism describes the trapping of the trajectories that begin from the central region of the caldera, their transport to the regions of the saddles, and the nature of their exit from the caldera. The third mechanism describes the phase space geometry responsible for the dynamical matching of trajectories originally proposed by Carpenter and described in [Collins et al., 2014] for the two dimensional caldera PES that we consider.

Phase Space Analysis of the Nonexistence of Dynamical Matching in a Stretched Caldera Potential Energy Surface

“…Dynamical matching is the behaviour of trajectories having initial conditions on the periodic orbit dividing surfaces of the upper saddles, that go straight across the caldera and exit via the opposite lower saddle. We showed that this occurs when there is no interaction of the invariant manifolds of the unstable periodic orbits of the upper saddles with the central region of the caldera [Katsanikas and Wiggins , 2018]. This implies that there is no trapping of these trajectories in the symmetric caldera potential energy surface.…”

“…The caldera potential energy surface studied in that paper possessed a symmetry (to be described shortly), and the effect of asymmetry, or stretching of the potential, on trajectories was also considered. In [Katsanikas and Wiggins , 2018] an analysis of the phase space structures that determined the different behaviors of trajectories was given for the symmetric caldera potential. In particular, we investigated the mechanisms of trapping trajectories and of dynamical matching in the symmetrical caldera potential energy surface.…”

“…The second option is to be trapped by the invariant manifolds of the unstable periodic orbits of the central region until they are transported from the unstable invariant manifolds to the exit from the caldera through the four different regions of saddles. The trajectories that have initial conditions on the dividing surfaces of the unstable periodic orbits of the upper saddles are not trapped but, on the contrary, we have the phenomenon of dynamical matching [Katsanikas and Wiggins , 2018]. Dynamical matching is the behaviour of trajectories having initial conditions on the periodic orbit dividing surfaces of the upper saddles, that go straight across the caldera and exit via the opposite lower saddle.…”

The dynamical matching mechanism in phase space for caldera-type potential energy surfaces

“…In this section we describe the Caldera potential energy surface that we consider. This model has previously been studied in [9][10][11]. The key features of the PES are a central minimum surrounded by four index-1 saddles.…”

“…A detailed study of the trajectory behavior in a two degree-of-freedom (DoF) caldera PES was given in [9], where a more general discussion of caldera-like PESs in organic reactions is also presented. Further work elucidating the phenomena of dynamical matching and trapping in this caldera model was carried out in [10,11]. We will describe the results in these papers in more detail when we describe the Hamiltonian model in the next section.…”