Finding NHIM: Identifying high dimensional phase space structures in reaction dynamics using Lagrangian descriptors

Naik, Shibabrat; García-Garrido, Víctor J.; Wiggins, Stephen

doi:10.1016/j.cnsns.2019.104907

Cited by 35 publications

(20 citation statements)

References 52 publications

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“…In order to reveal the phase space structures that are responsible for the mechanism that allows and prevents dynamical matching, we use in this work the method of Lagrangian descriptors (LDs), see e.g. [12][13][14]. Lagrangian descriptors is a trajectory-based scalar diagnostic that has been developed in the nonlinear dynamics literature to explore the geometrical template of phase space structures that characterizes qualitatively distinct dynamical behavior.…”

Section: The Phase Space Mechanism Governing Dynamical Matchingmentioning

confidence: 99%

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

Katsanikas

García-Garrido

Wiggins

2020

Chemical Physics Letters

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Dynamical matching occurs in a variety of important organic chemical reactions. It is observed to be a result of a potential energy surface (PES) having specific geometric features. In particular, a region of relative flatness where entrance and exit to this region is controlled by index-one saddles. Examples of potential energy surfaces having these features are the so-called caldera potential energy surfaces. We develop a predictive level of understanding of the phenomenon of dynamical matching in a caldera potential energy surface. We show that the phase space structure that governs dynamical matching is a particular type of heteroclinic trajectory which gives rise to trapping of trajectories in the central region of the caldera PES. When the heteroclinic trajectory is broken, as a result of parameter variations, then dynamical matching occurs.

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Section: The Phase Space Mechanism Governing Dynamical Matchingmentioning

confidence: 99%

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

Katsanikas

García-Garrido

Wiggins

2020

Chemical Physics Letters

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“…The method was originally developed in the context of Lagrangian transport studies in fluid dynamics [30][31][32], but the wide applicability of the method has recently been recognized in chemistry [33][34][35][36][37][38][39][40][41]. The method is straightforward to implement computationally [42], the interpretation is clear, and it provides a "high resolution" method for exploring high-dimensional phase space with low-dimensional slices [43,44].…”

Section: B Three Degrees-of-freedom Hamiltonian Systemsmentioning

confidence: 99%

Reactive Islands for Three Degrees-of-Freedom Hamiltonian Systems

Krajňák,

García-Garrido,

Wiggins

2021

Preprint

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We develop the geometrical, analytical, and computational framework for reactive island theory for three degrees-of-freedom time-independent Hamiltonian systems. In this setting, the dynamics occurs in a 5-dimensional energy surface in phase space and is governed by four-dimensional stable and unstable manifolds of a three-dimensional normally hyperbolic invariant sphere. The stable and unstable manifolds have the geometrical structure of spherinders and we provide the means to investigate the ways in which these spherinders and their intersections determine the dynamical evolution of trajectories. This geometrical picture is realized through the computational technique of Lagrangian descriptors. In a set of trajectories, Lagrangian descriptors allow us to identify the ones closest to a stable or unstable manifold. Using an approximation of the manifold on a surface of section we are able to calculate the flux between two regions of the energy surface.

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“…where the f i 's are the components of the velocity field f , and p ∈ (0, 1] and τ ∈ R + are two parameters. We focus on the definition (8) used in the literature [52,53], although other alternative definitions have been suggested. It is convenient to split Eq.…”

Section: B Lagrangian Descriptorsmentioning

confidence: 99%

Edge manifold as a Lagrangian coherent structure in a high-dimensional state space

Beneitez

Duguet

Schlatter

et al. 2020

Phys. Rev. Research

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Dissipative dynamical systems characterized by two basins of attraction are found in many physical systems, notably in hydrodynamics where laminar and turbulent regimes can coexist. The state space of such systems is structured around a dividing manifold called the edge, which separates trajectories attracted by the laminar state from those reaching the turbulent state. We apply here concepts and tools from Lagrangian data analysis to investigate this edge manifold. This approach is carried out in the state space of autonomous arbitrarily highdimensional dissipative systems, in which the edge manifold is reinterpreted as a Lagrangian coherent structure (LCS). Two different diagnostics, finite-time Lyapunov exponents and Lagrangian descriptors, are used and compared with respect to their ability to identify the edge and their scalability. Their properties are illustrated on several low-order models of subcritical transition of increasing dimension and complexity, as well on wellresolved simulations of the Navier-Stokes equations in the case of plane Couette flow. They allow for a mapping of the global structure of both the state space and the edge manifold based on quantitative information. Both diagnostics can also be used to generate efficient bisection algorithms to approach asymptotic edge states, which outperform classical edge tracking.

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Finding NHIM: Identifying high dimensional phase space structures in reaction dynamics using Lagrangian descriptors

Cited by 35 publications

References 52 publications

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

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

Reactive Islands for Three Degrees-of-Freedom Hamiltonian Systems

Edge manifold as a Lagrangian coherent structure in a high-dimensional state space

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