An Eulerian method for computing the coherent ergodic partition of continuous dynamical systems

You, Guoqiao; Leung, Shingyu

doi:10.1016/j.jcp.2014.01.034

Cited by 17 publications

(11 citation statements)

References 30 publications

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“…Longer windows have been rarely considered. An example is provided in You & Leung (2014) where the time window is , eight times longer than the conventional one. In this section, the purpose is to conduct topological colouring choosing a time window such that a comparison is possible with the ROC finite-time analysis.…”

Section: Resultsmentioning

confidence: 99%

Topological colouring of fluid particles unravels finite-time coherent sets

2021

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Section: Resultsmentioning

confidence: 99%

Topological colouring of fluid particles unravels finite-time coherent sets

2021

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“…In (a), one can observe four closed regions or islands in which the tracer does not penetrate. These islands adopt either a circular or an arrow-like shape; they are not static and rotate in each half domain suffering deformation in their shape during their displacement, keeping the same size due to incompressibility [57]. When the injection is performed in the left circular blank island of (a), as shown in (b), the tracer never invades the central part of the circle.…”

Section: Driven Double Gyrementioning

confidence: 99%

Topology of dynamical reconstructions from Lagrangian data

Charó

Artana

Sciamarella

2020

Physica D: Nonlinear Phenomena

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Branched Manifold analysis through Homologies (BraMAH) is a technique that computes the state-space topology of a dynamical reconstruction from scalar data. This work introduces the application of this technique to Lagrangian time series. The approach unveils the topological structure underlying the behavior of a fluid particle. When applied to a set of sparse particles, the results of the analysis can be used to classify them according to the dynamics they deploy during a given time window. Topological grids can be constructed to portray the spatial organization of the topological classes. The connection between the topological grids and the transport properties of the flow is examined using streaklines. Even if demonstrated here in the context of kinematic flow models, the generality of the method allows for its potential application to experimental or observational Lagrangian data satisfying the technical requirements for the analysis.

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“…One disadvantage about the previous Eulerian approach for forward flow map computation as discussed in Section 2.2 and [23,24,38] is that the level set equation…”

Section: A Forward Time Marching Pde Approach For Constructing the Fomentioning

confidence: 99%

“…We first introduce the definition of various closely related Lyapunov exponents including the finite time Lyapunov exponent (FTLE) [16,12,13,34,21], the finite size Lyapunov exponent (FSLE) [2,3,22] and also the infinitesimal size Lyapunov exponent (ISLE) [18]. Then we discuss typical Lagrangian approaches and summarize the original Eulerian approaches as in [23,24,38,39]. The discussions here, however, will definitely not be a complete survey.…”

Section: Introductionmentioning

confidence: 99%

Eulerian Methods for Visualizing Continuous Dynamical Systems using Lyapunov Exponents

You¹,

Wong²,

Leung³

2017

SIAM J. Sci. Comput.

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We propose a new Eulerian numerical approach for constructing the forward flow maps in continuous dynamical systems. The new algorithm improves the original formulation developed in [23,24] so that the associated partial differential equations (PDEs) are solved forward in time and, therefore, the forward flow map can now be determined on the fly. Due to the simplicity in the implementations, we are now able to efficiently compute the unstable coherent structures in the flow based on quantities like the finite time Lyapunov exponent (FTLE), the finite size Lyapunov exponent (FSLE) and also a related infinitesimal size Lyapunov exponent (ISLE). When applied to the ISLE computations, the Eulerian method is in particularly computational efficient. For each separation factor r in the definition of the ISLE, typical Lagrangian methods require to shoot and monitor an individual set of ray trajectories. If the scale factor in the definition is changed, these methods have to restart the whole computations all over again. The proposed Eulerian method, however, requires to extract only an isosurface of a volumetric data for an individual value of r which can be easily done using any well-developed efficient interpolation method or simply an isosurface extraction algorithm. Moreover, we provide a theoretical link between the FTLE and the ISLE fields which explains the similarity in these solutions observed in various applications.

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An Eulerian method for computing the coherent ergodic partition of continuous dynamical systems

Cited by 17 publications

References 30 publications

Topological colouring of fluid particles unravels finite-time coherent sets

Topological colouring of fluid particles unravels finite-time coherent sets

Topology of dynamical reconstructions from Lagrangian data

Eulerian Methods for Visualizing Continuous Dynamical Systems using Lyapunov Exponents

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