Detection of Dynamical Matching in a Caldera Hamiltonian System using Lagrangian Descriptors

Abstract

The goal of this paper is to apply the method of Lagrangian descriptors to reveal the phase space mechanism by which a Caldera-type potential energy surface (PES) exhibits the dynamical matching phenomenon. Using this technique, we can easily establish that the non-existence of dynamical matching is a consequence of heteroclinic connections between the unstable manifolds of the unstable periodic orbits (UPOs) of the upper index-1 saddles (entrance channels to the Caldera) and the stable manifolds of the family of UPOs of the central minimum of the Caldera, resulting in the temporary trapping of trajectories. Moreover, dynamical matching will occur when there is no heteroclinic connection, which allows trajectories to enter and exit the Caldera without interacting with the shallow region of the central minimum. Knowledge of this phase space mechanism is relevant because it allows us to effectively predict the existence, and non-existence, of dynamical matching. In this work we explore a stretched Caldera potential by means of Lagrangian descriptors, allowing us to accurately compute the critical value for the stretching parameter for which dynamical matching behavior occurs in the system. This approach is shown to provide a tremendous advantage for exploring this mechanism in comparison to other methods from nonlinear dynamics that use phase space dividing surfaces.

16 pages, 9 figures

References in corpus (4)

• A Theoretical Framework for Lagrangian Descriptors
• Nonstatistical dynamics on the caldera
• Finding NHIM: Identifying High Dimensional Phase Space Structures in Reaction Dynamics using Lagrangian Descriptors
• Invariant Manifolds and Rate Constants in Driven Chemical Reactions

Cited by in corpus (10)

• Phase Space Analysis of the Dynamics on a Potential Energy Surface with an Entrance Channel and Two Potential Wells
• The Nature of Reactive and Non-reactive Trajectories for a Three Dimensional Caldera Potential Energy Surface
• The bifurcations of the critical points and the role of the depth in a symmetric Caldera potential energy surface
• Quantifying chaos using Lagrangian descriptors
• The Influence of a Pitchfork Bifurcation of the Critical Points of a Symmetric Caldera Potential Energy Surface on Dynamical Matching
• Phase space transport in a symmetric Caldera potential with three index-1 saddles and no minima
• Bifurcation of Dividing Surfaces Constructed from Period-Doubling Bifurcations of Periodic Orbits in a Caldera Potential Energy Surface
• From Poincare Maps to Lagrangian Descriptors: The Case of the Valley Ridge Inflection Point Potential
• Using Lagrangian descriptors to calculate the Maslov index of periodic orbits
• Painting the Phase Space of Dissipative Systems with Lagrangian Descriptors