An algorithmic approach to chain recurrence

W. D. Kalies, K. Mischaikow, R. C.A.M. VanderVorst

Research output: Contribution to journalArticle

57 Citations (Scopus)

Abstract

In this paper we give a new definition of the chain recurrent set of a continuous map using finite spatial discretizations. This approach allows for an algorithmic construction of isolating blocks for the components of Morse decompositions which approximate the chain recurrent set arbitrarily closely as well as discrete approximations of Conley's Lyapunov function. This is a natural framework in which to develop computational techniques for the analysis of qualitative dynamics including rigorous computer-assisted proofs.

Original languageEnglish (US)
Pages (from-to)409-449
Number of pages41
JournalFoundations of Computational Mathematics
Volume5
Issue number4
DOIs
StatePublished - Nov 1 2005

Fingerprint

Chain Recurrence
Lyapunov functions
Morse Decomposition
Computer-assisted Proof
Decomposition
Discrete Approximation
Computational Techniques
Continuous Map
Lyapunov Function
Discretization

All Science Journal Classification (ASJC) codes

  • Analysis
  • Computational Mathematics
  • Computational Theory and Mathematics
  • Applied Mathematics

Keywords

  • Algorithms
  • Chain recurrence
  • Combinatorial dynamics
  • Computation
  • Conley's decomposition theorem
  • Lyapunov function

Cite this

Kalies, W. D. ; Mischaikow, K. ; VanderVorst, R. C.A.M. / An algorithmic approach to chain recurrence. In: Foundations of Computational Mathematics. 2005 ; Vol. 5, No. 4. pp. 409-449.
@article{db296c04fd274feea192437ca51586a4,
title = "An algorithmic approach to chain recurrence",
abstract = "In this paper we give a new definition of the chain recurrent set of a continuous map using finite spatial discretizations. This approach allows for an algorithmic construction of isolating blocks for the components of Morse decompositions which approximate the chain recurrent set arbitrarily closely as well as discrete approximations of Conley's Lyapunov function. This is a natural framework in which to develop computational techniques for the analysis of qualitative dynamics including rigorous computer-assisted proofs.",
keywords = "Algorithms, Chain recurrence, Combinatorial dynamics, Computation, Conley's decomposition theorem, Lyapunov function",
author = "Kalies, {W. D.} and K. Mischaikow and VanderVorst, {R. C.A.M.}",
year = "2005",
month = "11",
day = "1",
doi = "10.1007/s10208-004-0163-9",
language = "English (US)",
volume = "5",
pages = "409--449",
journal = "Foundations of Computational Mathematics",
issn = "1615-3375",
publisher = "Springer New York",
number = "4",

}

An algorithmic approach to chain recurrence. / Kalies, W. D.; Mischaikow, K.; VanderVorst, R. C.A.M.

In: Foundations of Computational Mathematics, Vol. 5, No. 4, 01.11.2005, p. 409-449.

Research output: Contribution to journalArticle

TY - JOUR

T1 - An algorithmic approach to chain recurrence

AU - Kalies, W. D.

AU - Mischaikow, K.

AU - VanderVorst, R. C.A.M.

PY - 2005/11/1

Y1 - 2005/11/1

N2 - In this paper we give a new definition of the chain recurrent set of a continuous map using finite spatial discretizations. This approach allows for an algorithmic construction of isolating blocks for the components of Morse decompositions which approximate the chain recurrent set arbitrarily closely as well as discrete approximations of Conley's Lyapunov function. This is a natural framework in which to develop computational techniques for the analysis of qualitative dynamics including rigorous computer-assisted proofs.

AB - In this paper we give a new definition of the chain recurrent set of a continuous map using finite spatial discretizations. This approach allows for an algorithmic construction of isolating blocks for the components of Morse decompositions which approximate the chain recurrent set arbitrarily closely as well as discrete approximations of Conley's Lyapunov function. This is a natural framework in which to develop computational techniques for the analysis of qualitative dynamics including rigorous computer-assisted proofs.

KW - Algorithms

KW - Chain recurrence

KW - Combinatorial dynamics

KW - Computation

KW - Conley's decomposition theorem

KW - Lyapunov function

UR - http://www.scopus.com/inward/record.url?scp=31944446545&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=31944446545&partnerID=8YFLogxK

U2 - 10.1007/s10208-004-0163-9

DO - 10.1007/s10208-004-0163-9

M3 - Article

AN - SCOPUS:31944446545

VL - 5

SP - 409

EP - 449

JO - Foundations of Computational Mathematics

JF - Foundations of Computational Mathematics

SN - 1615-3375

IS - 4

ER -