Finding NHIM: Identifying High Dimensional Phase Space Structures in Reaction Dynamics using Lagrangian Descriptors

Abstract

Phase space structures such as dividing surfaces, normally hyperbolic invariant manifolds, their stable and unstable manifolds have been an integral part of computing quantitative results such as transition fraction, stability erosion in multi-stable mechanical systems, and reaction rates in chemical reaction dynamics. Thus, methods that can reveal their geometry in high dimensional phase space (4 or more dimensions) need to be benchmarked by comparing with known results. In this study, we assess the capability of one such method called Lagrangian descriptor for revealing the types of high dimensional phase space structures associated with index-1 saddle in Hamiltonian systems. The Lagrangian descriptor based approach is applied to two and three degree-of-freedom quadratic Hamiltonian systems where the high dimensional phase space structures are known, that is as closed-form analytical expressions. This leads to a direct comparison of features in the Lagrangian descriptor plots and the phase space structures' intersection with an isoenergetic two-dimensional surface and hence provides a validation of the approach.

39 pages, 7 figures, Submitted to Communications in Nonlinear Science and Numerical Simulation

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• The Phase Space Mechanism for Selectivity in a Symmetric Potential Energy Surface with a Post-Transition-State Bifurcation
• Finding NHIM in 2 and 3 degrees-of-freedom with Hénon-Heiles type potential
• Exploring Isomerization Dynamics on a Potential Energy Surface with an Index-2 Saddle using Lagrangian Descriptors
• Tilting and Squeezing: Phase space geometry of Hamiltonian saddle-node bifurcation and its influence on chemical reaction dynamics
• The edge as a Lagrangian Coherent Structure in a high-dimensional state space
• Phase space structure and escape time dynamics in a Van der Waals model for exothermic reactions
• Visualizing the Phase Space of the HeI_2 van der Waals Complex using Lagrangian Descriptors
• Lagrangian Descriptors and the Action Integral of Classical Mechanics
• An Extension of Discrete Lagrangian Descriptors for Unbounded Maps
• Using Lagrangian descriptors to uncover invariant structures in Chesnavich's Isokinetic Model with application to roaming
• Geometric parametrisation of Lagrangian Descriptors for 1 degree-of-freedom systems
• Influence of low frequency modes on dynamical concertedness in double proton transfer dynamics
• Reactive Islands for Three Degrees-of-Freedom Hamiltonian Systems
• Bifurcation Study on a Degenerate Double van der Waals Cirque Potential Energy Surface using Lagrangian Descriptors
• Controlling reaction dynamics in chemical model systems through external driving
• Painting the Phase Space of Dissipative Systems with Lagrangian Descriptors
• Transport and roaming on the Double van der Waals Potential Energy Surface
• Revealing the phase space structure of Hamiltonian systems using the action
• Phase space integrity in neural network models of Hamiltonian dynamics: A Lagrangian descriptor approach