Asymptotic observability of low-dimensional powder chaos in a three-degrees-of-freedom scattering system

Drótos, Gábor; Montoya, Francisco Gonzalez; Jung, Christof; Tél, Tamás

doi:10.1103/physreve.90.022906

Cited by 19 publications

(14 citation statements)

References 25 publications

(37 reference statements)

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“…55 Recent developments in classical chaotic scattering include the investigation of the ray dynamics in optical metamaterials, 56 of escape in celestial mechanics 57,58 and in medically relevant fluid flows, 59,60 and a basic understanding of the structure of chaotic saddles underlying scattering in higher dimensions. [61][62][63][64] Snapshot chaotic saddles and attractors exist in aperiodically driven system, 65 and represent instantaneous states of ensembles of trajectories. A novel observation of recent years is that they are uniquely defined not only in noisy systems 66 but also in the presence of smooth driving that might even be a one-sided temporal shift of some parameters.…”

Section: Discussionmentioning

confidence: 99%

The joy of transient chaos

Tél

2015

Chaos: An Interdisciplinary Journal of Nonlinear Science

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We intend to show that transient chaos is a very appealing, but still not widely appreciated, subfield of nonlinear dynamics. Besides flashing its basic properties and giving a brief overview of the many applications, a few recent transient-chaos-related subjects are introduced in some detail. These include the dynamics of decision making, dispersion, and sedimentation of volcanic ash, doubly transient chaos of undriven autonomous mechanical systems, and a dynamical systems approach to energy absorption or explosion.

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Section: Discussionmentioning

confidence: 99%

The joy of transient chaos

Tél

2015

Chaos: An Interdisciplinary Journal of Nonlinear Science

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“…Even the definition of finite-volume resonance zones and their associated lobes is not straightforward for 4D maps (and indeed even for 3D maps) and in many cases such zones and lobes do not exist [5,21,56,57]. In other recent developments, Jung and collaborators [16,17,27] have explored the topological structure of fractal chaotic scattering functions for three-degree-of-freedom Hamiltonian systems. Other approaches to higher-dimensional symbolic dynamics are based on topological simplexes [28,29] and higherdimensional braids [26].…”

Section: The Current Work Addresses This Question In 3dmentioning

confidence: 99%

Using Invariant Manifolds to Construct Symbolic Dynamics for Three-Dimensional Volume-Preserving Maps

Maelfeyt¹,

Smith²,

Mitchell³

2017

SIAM J. Appl. Dyn. Syst.

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Topological techniques are powerful tools for characterizing the complexity of many dynamical systems, including the commonly studied area-preserving maps of the plane. However, the extension of many topological techniques to higher dimensions is filled with roadblocks preventing their application. This article shows how to extend the homotopic lobe dynamics (HLD) technique, previously developed for 2D maps, to volume-preserving maps of a three-dimensional phase space. Such maps are physically relevant to particle transport by incompressible fluid flows or by magnetic field lines. Specifically, this manuscript shows how to utilize two-dimensional stable and unstable invariant manifolds, intersecting in a heteroclinic tangle, to construct a symbolic representation of the topological dynamics of the map. This symbolic representation can be used to classify system trajectories and to compute topological entropy. We illustrate the salient ideas through a series of examples with increasing complexity. These examples highlight new features of the HLD technique in 3D. Ultimately, in the final example, our technique detects a difference between the 2D stretching rate of surfaces and the 1D stretching rate of curves, illustrating the truly 3D nature of our approach.

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“…This was first realized by Wiggins [17], followed by an explicit chemical example by Gililan and Ezra [18]. As an alternative approach, Jung, Montoya, and collaborators [19,20,21,22,23] have studied the topological structure of chaotic scattering functions for threedegree-of-freedom Hamiltonian systems. They have shown how symbolic dynamics can be extracted from the doubly-differential cross section and then related back to the fractal structure of the chaotic saddle itself.…”

Section: Introductionmentioning

confidence: 99%

Topological Dynamics of Volume-Preserving Maps Without an Equatorial Heteroclinic Curve

Arenson,

Mitchell

2020

Preprint

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Understanding the topological structure of phase space for dynamical systems in higher dimensions is critical for numerous applications, including the computation of chemical reaction rates and transport of objects in the solar system. Many topological techniques have been developed to study maps of two-dimensional (2D) phase spaces, but extending these techniques to higher dimensions is often a major challenge or even impossible. Previously, one such technique, homotopic lobe dynamics (HLD), was generalized to analyze the stable and unstable manifolds of hyperbolic fixed points for volume-preserving maps in three dimensions. This prior work assumed the existence of an equatorial heteroclinic intersection curve, which was the natural generalization of the 2D case. The present work extends the previous analysis to the case where no such equatorial curve exists, but where intersection curves, connecting fixed points may exist. In order to extend HLD to this case, we shift our perspective from the invariant manifolds of the fixed points to the invariant manifolds of the invariant circle formed by the fixed-point-to-fixed-point intersections. The output of the HLD technique is a symbolic description of the minimal underlying topology of the invariant manifolds. We demonstrate this approach through a series of examples.

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Asymptotic observability of low-dimensional powder chaos in a three-degrees-of-freedom scattering system

Cited by 19 publications

References 25 publications

The joy of transient chaos

The joy of transient chaos

Using Invariant Manifolds to Construct Symbolic Dynamics for Three-Dimensional Volume-Preserving Maps

Topological Dynamics of Volume-Preserving Maps Without an Equatorial Heteroclinic Curve

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