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Eulerian Methods for Visualizing Continuous Dynamical Systems using Lyapunov Exponents

Guoqiao You, Tony Wong, Shingyu Leung
2017 SIAM Journal on Scientific Computing  
We propose a new Eulerian numerical approach for constructing the forward flow maps in continuous dynamical systems.  ...  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  ...  Introduction It is an important task to visualize, understand and then extract useful information in complex continuous dynamical systems from many science and engineering fields including flight simu-lation  ... 
doi:10.1137/16m1066890 fatcat:za4vqsc7hvcq7dedkzwlqd6s7m

Lagrangian chaos, Eulerian chaos, and mixing enhancement in converging–diverging channel flows

Cristina H. Amon, Amador M. Guzmán, Benoit Morel
1996 Physics of Fluids  
Classical fluid dynamics representations and dynamical system techniques characterize Eulerian flows, whereas Lagrangian trajectories and finite-time Lagrangian Lyapunov exponents identify Lagrangian chaotic  ...  Last, Lagrangian and Eulerian Lyapunov exponents are used to demonstrate that the onset of Eulerian chaos coincides with the spreading of a strong Lagrangian chaotic regime from the vortex region to the  ...  We use the spectral element method [25] [26] [27] to solve the governing Navier-Stokes and continuity equations and to obtain the Eulerian velocity and pressure fields.  ... 
doi:10.1063/1.868910 fatcat:pdwu5mqot5go3lnwyryfjoghle

An Alternative Formulation of Lyapunov Exponents for Computing Lagrangian Coherent Structures

Allen R. Sanderson
2014 2014 IEEE Pacific Visualization Symposium  
One of the most widely used tools for revealing Lagrangian coherent structures has been to calculate the finite-time Lyapunov exponents, whose maximal values appear as ridgelines to reveal Lagrangian coherent  ...  In this paper we explore an alternative formulation of Lyapunov exponents for computing Lagrangian coherent structures.  ...  These distances are also used for the Lyapunov exponent calculations. This image differs greatly from the one based on the arc length shown in Figure 4 .  ... 
doi:10.1109/pacificvis.2014.27 dblp:conf/apvis/Sanderson14 fatcat:hnv7o2te3rew5oggs6fexndm54

Dynamical flow characterization of transitional and chaotic regimes in converging–diverging channels

A. M. Guzmán, C. H. Amon
1996 Journal of Fluid Mechanics  
Modern dynamical system techniques such as timedelay reconstructions of pseudophase spaces, autocorrelation functions, fractal dimensions and Eulerian Lyapunov exponents are used for the dynamical flow  ...  Eulerian Lyapunov exponents and fractal dimensions are quantitative parameters to establish the onset of chaos, the persistence of chaotic flow behaviour, and the long-term persistent unpredictability  ...  In 93, we present the numerical algorithms for evaluating dynamical system parameters, we analyse the Eulerian flow dynamical representation, and we describe the route to chaos using techniques derived  ... 
doi:10.1017/s002211209600763x fatcat:44xlm652bfbrfiwynocsuggzou

Comparison between Eulerian diagnostics and finite-size Lyapunov exponents computed from altimetry in the Algerian basin

Francesco d'Ovidio, Jordi Isern-Fontanet, Cristóbal López, Emilio Hernández-García, Emilio García-Ladona
2009 Deep Sea Research Part I: Oceanographic Research Papers  
Climatologies of Lyapunov exponents do not show any compact relation with other Eulerian diagnostics, unveiling a different structure even at the basin scale.  ...  Here we compare the ability of standard Eulerian diagnostics and the finite-size Lyapunov exponent in detecting instantaneaous and climatological transport and mixing properties.  ...  Several methods exist, one of the simplest being the Finite-Size Lyapunov Exponent (FSLE).The FSLE is a generalization of the concept of Lyapunov exponent to finite separations.The standard definition  ... 
doi:10.1016/j.dsr.2008.07.014 fatcat:7vhvmmfxqrbilfysw4fj4hsnfm

Lyapunov exponent and Wasserstein metric as validation tools for assessing short-time dynamics and quantitative model evaluation of large-eddy simulation [article]

Hao Wu, Peter C. Ma, Yu Lv, Matthias Ihme
2018 arXiv   pre-print
In this work, methods for the evaluation of LES-quality and LES-accuracy are presented, which include the Lyapunov exponent for the analysis of short-time predictability of LES-calculation and the Wasserstein  ...  The analysis of the turbulent simulation data using the concept of the Lyapunov exponent and the Wasserstein metric provides a more quantitative approach to assess the mesh dependency of the simulation  ...  By observing that an Eulerian field can be regarded as a trajectory in an extended dynamical system [21] , a practical method for obtaining the Lyapunov exponent is to perturb the initial field φ(t 0  ... 
arXiv:1801.03046v1 fatcat:75mbunjnpva3pd4f26fuzv6vxm

Introduction to chaos and diffusion [article]

G. Boffetta, G. Lacorata, A. Vulpiani
2004 arXiv   pre-print
Our intention is twofold: to give a self-consistent introduction to chaos and diffusion, and to offer a guide for the reading of the rest of this volume.  ...  This contribution is relative to the opening lectures of the ISSAOS 2001 summer school and it has the aim to provide the reader with some concepts and techniques concerning chaotic dynamics and transport  ...  Indeed, using the standard methods for data analysis [27] , from Lagrangian trajectories one extracts the total Lyapunov exponent λ T and not λ L or λ E . IV.  ... 
arXiv:nlin/0411023v1 fatcat:u7g7rxj36fahfn374cdzfoj54y

Microscopic and Macroscopic Rayleigh-Benard Flows : Continuum and Particle Simulations, Turbulence, Fluctuations, Time Reversibility, and Lyapunov Instability [article]

Wm. G. Hoover, Carol G. Hoover
2012 arXiv   pre-print
We discuss the irreversibility, nonlocality, and fluctuations, as well as the Lyapunov and hydrodynamic instabilities characterizing atomistic, smooth-particle, and finite-difference solutions of the two-dimensional  ...  The simplest derivation of these three laws uses an Eulerian coordinate system fixed in space.  ...  Particle Methods: Nonequilibrium Molecular Dynamics and SPAM Nonequilibrium molecular dynamics is a straightforward but limited method for studying Rayleigh-Bénard flows.  ... 
arXiv:1205.4633v1 fatcat:rmthwb5jhbcpzfocklgrqplfsa

Lagrangian structures and stirring in the Earth's mantle

Cinzia G. Farnetani, Henri Samuel
2003 Earth and Planetary Science Letters  
The finite-size Lyapunov exponents estimate the local mixing properties of the flow using the finite dispersion of particles, while the finite-time Lyapunov exponents have been often used to locate dynamically  ...  The Eulerian velocity field is computed using a finite element code for solid state convection. We focus on a thermochemical model with a chemically denser layer at the base of the Earth's mantle.  ...  Manga for their thorough reviews. We are grate-  ... 
doi:10.1016/s0012-821x(02)01085-3 fatcat:ylikc5fegbgovewspo43sg6mvi

Thermal heat reservoirs via Gauss' principle of least constraint; Dissipation, chaos, and phase‐space dimensionality loss in one‐dimensional chains

William G. Hoover, Harald A. Posch, Luke W. Campbell
1993 Chaos  
We use Gauss' principle of least constraint to impose different kinetic temperatures on the two halves of a periodic one-dimensional chain.  ...  We document the sensitivity of the Lyapunov spectrum and the underlying phase-space topology by varying the chain length and the size of the kinetic-temperature difference.  ...  The maximum Lyapunov exponent found using the naive ver sion of Method 2N in the full l6-dimensional Cartesian phase space is 0.19.  ... 
doi:10.1063/1.165941 pmid:12780041 fatcat:v7ybjmwtrbbalk3cpcdbkc7rda

Lagrangian coherent structures in the California Current System – sensitivities and limitations

Cheryl S. Harrison, Gary A. Glatzmaier
2012 Geophysical and Astrophysical Fluid Dynamics  
We report on numerical experiments to test the sensitivity of Lagrangian coherent structures (LCS), found by identifying ridges of the finite-time Lyapunov exponent (FTLE), to errors in two systems representing  ...  At this expected error level the FTLE method is reliable for locating boundaries of large eddies and strong jets.  ...  Mangel for their support. We would also like to thank A. Pascual for discussion of her results.  ... 
doi:10.1080/03091929.2010.532793 fatcat:frypijsl2fabrencr2jo742vgi

Time-Symmetry Breaking in Hamiltonian Mechanics. III. A Memoir for Douglas James Henderson [1934-2020] [article]

William Graham Hoover, Carol Griswold Hoover
2020 arXiv   pre-print
We model these processes using classical molecular dynamics and Eulerian fluid mechanics in two dimensions.  ...  The explanations for these results are an interesting combination of two (irreversible) instabilities, Lyapunov and Navier-Stokes.  ...  See Figure 12 for details. We verified that the two methods, steady-state and Eulerian, are consistent and turned to considering the detailed differences between shock and rarefaction waves.  ... 
arXiv:2011.11180v3 fatcat:l73egj64jfc23p4dbrmxdm7kym

LDDS: Python package for computing and visualizing Lagrangian Descriptors for Dynamical Systems [article]

Broncio Aguilar Sanjuan, Víctor J. García-Garrido, Vladimír Krajňák, Shibabrat Naik, Stephen Wiggins
2021 Zenodo  
The Python package, LDDS, serves this need for analyzing deterministic and stochastic, continuous and discrete high-dimensional nonlinear dynamical systems described either by an analytical vector field  ...  In most nonlinear systems, the invariant manifolds are computed using numerical techniques that rely on some form of linearization around equilibrium points followed by continuation and globalization.  ...  LDDS: Python package for computing and visualizing Lagrangian Descriptors for Dynamical Systems.  ... 
doi:10.5281/zenodo.5519579 fatcat:ulevxg77pzaxplrzmbnkfzn4ty

Lagrangian Reduced Order Modeling Using Finite Time Lyapunov Exponents

Xuping Xie, Peter J. Nolan, Shane D. Ross, Changhong Mou, Traian Iliescu
2020 Fluids  
Lyapunov exponent (FTLE)), but also Eulerian fields (e.g., the streamfunction).  ...  We emphasize that the new Lagrangian ROMs do not employ any closure modeling to model the effect of discarded modes (which is standard procedure for low-dimensional ROMs of complex nonlinear systems).  ...  time Lyapunov exponent (FTLE)), but also Eulerian fields (e.g., the streamfunction).  ... 
doi:10.3390/fluids5040189 fatcat:3cpddu6ea5h5jbbgmmba5bdnxi

Wall Shear Stress Topological Skeleton Analysis in Cardiovascular Flows: Methods and Applications

Valentina Mazzi, Umberto Morbiducci, Karol Calò, Giuseppe De Nisco, Maurizio Lodi Rizzini, Elena Torta, Giuseppe Carlo Alp Caridi, Claudio Chiastra, Diego Gallo
2021 Mathematics  
Here, the theoretical and methodological bases supporting the Lagrangian- and Eulerian-based methods currently used in the literature are reported and discussed, highlighting their application to cardiovascular  ...  Based on dynamical system theory, the WSS topological skeleton is composed of fixed points, i.e., focal points where WSS locally vanishes, and unstable/stable manifolds, consisting of contraction/expansion  ...  Steinman for providing the patient-specific computational hemodynamic model of carotid bifurcation. Conflicts of Interest: The authors declare no conflict of interest. Mathematics 2021, 9, 720  ... 
doi:10.3390/math9070720 fatcat:7uqkp5qp7rgpxom3qz43o46gqa
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