Finding normally hyperbolic invariant manifolds in two and three degrees of freedom with Hénon-Heiles-type potential

Naik, Shibabrat; Wiggins, Stephen

doi:10.1103/physreve.100.022204

Cited by 38 publications

(25 citation statements)

References 93 publications

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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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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

Self Cite

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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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“…This leads to convergence within 3 steps in the sense of the trajectory returning to the initial condition. Once a small amplitude UPO is obtained, numerical continuation increases the amplitude and, correspondingly, the total energy, while a combination of bracketing and bisection method computes the UPO at the desired energy for a specified tolerance (Koon, Lo, Marsden, & Ross, 2011;Naik & Wiggins, 2019).…”

Section: Differential Correctionmentioning

confidence: 99%