Finite Resolution Dynamics

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 Henon attractor is mixing at all resolutions coarser than 10^-5.

25 pages. Final version. To appear in Foundations of Computational Mathematics

References in corpus (3)

• The Lorenz attractor is mixing
• An existence of simple choreographies for N-body problem - a computer assisted proof
• Computable conditions for the occurrence of non-uniform hyperbolicity in families of one-dimensional maps

Cited by in corpus (5)

• Local stability implies global stability for the 2-dimensional Ricker map
• Dynamical properties of spatial discretizations of a generic homeomorphism
• Rigorous Enclosures of a Slow Manifold
• Bistability and chaos in the discrete two-gene Andrecut-Kauffman model
• Rigorous computation of expansion in one-dimensional dynamics