The Conley Index and Rigorous Numerics for Attracting Periodic Orbits

Mrożek, Marian; Pilarczyk, Paweł

doi:10.1007/978-1-4612-0081-9_5

Cited by 4 publications

(7 citation statements)

References 14 publications

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“…For other parameter values, the Lorenz system may contain attracting periodic trajectories (a computer‐assisted proof of their existence has been given by [MP02]). One of these parameter settings is (σ,β,ρ) = (10,8/3,350).…”

Section: Resultsmentioning

confidence: 99%

“… Largest nearly recurrent component for the Lorenz system with σ= 10, β= 8/3 and ρ= 350 (magenta). A tubular neighborhood (of diameter equal to the grid size) of the periodic trajectory proven to exist in [MP02], approximated numerically, is shown in red. Note that it closely follows the loop formed by the nearly recurrent component.…”

Section: Resultsmentioning

confidence: 99%

“…This is an artifact of the approximation: it alters the stability of the stationary points and makes them separate nearly recurrent components. contain attracting periodic trajectories (a computer-assisted proof of their existence has been given by [MP02]). One of these parameter settings is (σ, β, ρ) = (10, 8/3, 350).…”

Section: Analytical Vector Fieldsmentioning

confidence: 99%

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Nearly Recurrent Components in 3D Piecewise Constant Vector Fields

Szymczak

Brunhart-Lupo

2012

Computer Graphics Forum

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We present an algorithm for computing nearly recurrent components, that represent areas of close to circulating or stagnant flow, for 3D piecewise constant (PC) vector fields defined on regular grids. Using a number of analytical and simulated data sets, we demonstrate that nearly recurrent components can provide interesting insight into the topological structure of 3D vector fields. Our approach is based on prior work on Morse decompositions for PC vector fields on surfaces and extends concepts previously developed with this goal in mind to the case of 3D vector fields defined on regular grids. Our contributions include a description of trajectories of 3D piecewise constant vector fields and an extension of the transition graph, a finite directed graph that represents all trajectories, to the 3D case. Nearly recurrent components are defined by strongly connected components of the transition graph.

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Section: Resultsmentioning

confidence: 99%

Section: Resultsmentioning

confidence: 99%

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Nearly Recurrent Components in 3D Piecewise Constant Vector Fields

Szymczak

Brunhart-Lupo

2012

Computer Graphics Forum

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“…One possible approach to represent a dynamical system in a finite way is based on partitioning, that is, subdividing the phase space into a finite family of compact sets with nonempty and disjoint interiors. This approach goes back several decades and has proved extremely successful for describing certain classes of systems such as uniformly hyperbolic diffeomorphisms and flows [7,9,41] and has been recently further developed and used in various applications (see [2,28,32,37,42] for some examples). However, working with this kind of discretization in more general systems may not well reflect the actual topology of the phase space.…”

Section: Abstract Theorymentioning

confidence: 99%

Finite Resolution Dynamics

Luzzatto

Pilarczyk

2011

Found Comput Math

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We develop a new mathematical model for describing a dynamical system at limited resolution (or finite scale), and we give precise meaning to the notion of a dynamical system having some property at all resolutions coarser than a given number. Open covers are used to approximate the topology of the phase space in a finite way, and the dynamical system is represented by means of a combinatorial multivalued map. We formulate notions of transitivity and mixing in the finite resolution setting in a computable and consistent way. Moreover, we formulate equivalent conditions for these properties in terms of graphs, and provide effective algorithms for their verification. As an application we show that the Henon attractor is mixing at all resolutions coarser than 10^-5.Comment: 25 pages. Final version. To appear in Foundations of Computational Mathematic

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“…A more challenging example arises when f is the translation map of a continuous dynamical system induced by an ODE. In this setting one can use the method introduced in [12], [23], [33] (an implementation is available at [2]), as, for instance, was done in [22], [25], [26]. With this set of examples as justification our approach for the remainder of this paper is to assume that an appropriate combinatorial representation has been found.…”

Section: Introductionmentioning

confidence: 99%