Combinatorial approach to detection of fixed points, periodic orbits, and symbolic dynamics
| Autor: | Yitzchak Shmalo, Marian Gidea |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2017 |
| Předmět: |
Dynamical systems theory
Computer science Computation Structure (category theory) Symbolic dynamics Dynamical Systems (math.DS) 02 engineering and technology Fixed point Sperner's lemma 01 natural sciences 0103 physical sciences FOS: Mathematics Algebraic Topology (math.AT) Discrete Mathematics and Combinatorics Mathematics - Combinatorics Mathematics - Algebraic Topology Algebraic number Mathematics - Dynamical Systems 010302 applied physics Discrete mathematics Lemma (mathematics) Applied Mathematics 021001 nanoscience & nanotechnology Combinatorics (math.CO) 0210 nano-technology Analysis |
| Popis: | We present a combinatorial approach to rigorously show the existence of fixed points, periodic orbits, and symbolic dynamics in discrete-time dynamical systems, as well as to find numerical approximations of such objects. Our approach relies on the method of 'correctly aligned windows'. We subdivide 'windows' into cubical complexes, and we assign to the vertices of the cubes labels determined by the dynamics. In this way, we encode the information on the dynamics into combinatorial structure. We use a version of Sperner's Lemma to infer that, if the labeling satisfies certain conditions, then there exist fixed points/periodic orbits/orbits with prescribed itineraries. The method developed here does not require the computation of algebraic topology-type invariants, as only combinatorial information is needed; our arguments are elementary. |
| Databáze: | OpenAIRE |
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