# Tilting and Squeezing: Phase space geometry of Hamiltonian saddle-node bifurcation and its influence on chemical reaction dynamics

## TL;DR

This paper investigates how a Hamiltonian saddle-node bifurcation alters phase space structures that govern chemical reaction dynamics, using Lagrangian descriptors to analyze invariant manifolds in high-dimensional systems.

## Contribution

It introduces a phase space analysis framework for Hamiltonian saddle-node bifurcations, extending from one to two degrees of freedom, and examines the impact on reaction dynamics.

## Key findings

- Identification of invariant manifolds via Lagrangian descriptors.
- Description of phase space changes due to bifurcation.
- Analysis of NHIMs and their manifolds in reaction dynamics.

## Abstract

In this article we present the influence of a Hamiltonian saddle-node bifurcation on the high-dimensional phase space structures that mediate reaction dynamics. To achieve this goal, we identify the phase space invariant manifolds using Lagrangian descriptors, which is a trajectory-based diagnostic suitable for the construction of a complete ``phase space tomography'' by means of analyzing dynamics on low-dimensional slices. First, we build a Hamiltonian system with one degree-of-freedom (DoF) that models reaction, and study the effect of adding a parameter to the potential energy function that controls the depth of the well. Then, we extend this framework to a saddle-node bifurcation for a two DoF Hamiltonian, constructed by coupling a harmonic oscillator, i.e. a bath mode, to the other reactive DoF in the system. For this problem, we describe the phase space structures associated with the rank-1 saddle equilibrium point in the bottleneck region, which is a Normally Hyperbolic Invariant Manifold (NHIM) and its stable and unstable manifolds. Finally, we address the qualitative changes in the reaction dynamics of the Hamiltonian system due to changes in the well depth of the potential energy surface that gives rise to the saddle-node bifurcation.