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Foundations of Computational Mathematics

, Volume 11, Issue 2, pp 211–239 | Cite as

Finite Resolution Dynamics

Article

Abstract

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 Hénon attractor is mixing at all resolutions coarser than 10−5.

Keywords

Dynamical system Finite resolution Open cover Combinatorial dynamics Rigorous numerics Directed graph Transitivity Mixing Algorithm 

Mathematics Subject Classification (2000)

37M99 65P20 65G20 

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Copyright information

© SFoCM 2011

Authors and Affiliations

  1. 1.Mathematics DepartmentImperial CollegeLondonUK
  2. 2.Abdus Salam, International Centre for Theoretical PhysicsTriesteItaly
  3. 3.Centro de MatemáticaUniversidade do MinhoBragaPortugal

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