# Phase Space Analysis of the Non-Existence of Dynamical Matching in a Stretched Caldera Potential Energy Surface

## TL;DR

This study investigates how symmetry influences phase space structures and trajectory behavior in a caldera potential energy surface, revealing a critical symmetry value affecting dynamical crossing and trapping.

## Contribution

It identifies a critical symmetry parameter that governs phase space structures and trajectory crossing behaviors in a parametrized caldera potential.

## Key findings

- Existence of a critical symmetry value affecting phase space dynamics.
- Phase space structures change at the critical symmetry, influencing trajectory crossing.
- Explanation of the critical value via Henon stability parameter behavior.

## Abstract

In this paper we continue our studies of the two dimensional caldera potential energy surface in a parametrized family that allows for a study of the effect of symmetry on the phase space structures that govern how trajectories enter, cross, and exit the region of the caldera. As a particular form of trajectory crossing, we are able to determine the effect of symmetry and phase space structure on dynamical matching. We show that there is a critical value of the symmetry parameter which controls the phase space structures responsible for the manner of crossing, interacting with the central region (including trapping in this region) and exiting the caldera. We provide an explanation for the existence of this critical value in terms of the behavior of the Henon stability parameter for the associated periodic orbits.