A Singularity Analysis of Integrability and Chaos in Dynamical Systems

Tassos C. Bountis

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

The analysis of movable singularities in the complex time plane of n first order, ordinary differential equations has led to a better understanding of the real time behavior of the solutions of dynamical systems. The requirement that these singularities be poles, with n-1 free constants, i.e. the so-called Painlevé property, has identified many new completely integrable dynamical systems having, of course, no chaotic behavior whatsoever. On the other hand, the violation of the Painleve property with the introduction of logarithmic terms at higher orders in the series expansions has identified many dynamical systems with only "weakly chaotic" behavior. In this paper, I review these results and discuss the methods of singularity analysis with the aid of illustrative examples.

Original languageEnglish
Pages (from-to)353-373
Number of pages21
JournalNorth-Holland Mathematics Studies
Volume103
Issue numberC
DOIs
Publication statusPublished - 1985
Externally publishedYes

Fingerprint

Singularity Analysis
Integrability
Chaos
Dynamical system
Chaotic Behavior
Singularity
Painlevé
Series Expansion
Pole
Logarithmic
Ordinary differential equation
Higher Order
First-order
Requirements
Term

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

A Singularity Analysis of Integrability and Chaos in Dynamical Systems. / Bountis, Tassos C.

In: North-Holland Mathematics Studies, Vol. 103, No. C, 1985, p. 353-373.

Research output: Contribution to journalArticle

@article{8101ab40d8494b54b927618d51f110f3,
title = "A Singularity Analysis of Integrability and Chaos in Dynamical Systems",
abstract = "The analysis of movable singularities in the complex time plane of n first order, ordinary differential equations has led to a better understanding of the real time behavior of the solutions of dynamical systems. The requirement that these singularities be poles, with n-1 free constants, i.e. the so-called Painlev{\'e} property, has identified many new completely integrable dynamical systems having, of course, no chaotic behavior whatsoever. On the other hand, the violation of the Painleve property with the introduction of logarithmic terms at higher orders in the series expansions has identified many dynamical systems with only {"}weakly chaotic{"} behavior. In this paper, I review these results and discuss the methods of singularity analysis with the aid of illustrative examples.",
author = "Bountis, {Tassos C.}",
year = "1985",
doi = "10.1016/S0304-0208(08)72136-4",
language = "English",
volume = "103",
pages = "353--373",
journal = "North-Holland Mathematics Studies",
issn = "0304-0208",
publisher = "Elsevier",
number = "C",

}

TY - JOUR

T1 - A Singularity Analysis of Integrability and Chaos in Dynamical Systems

AU - Bountis, Tassos C.

PY - 1985

Y1 - 1985

N2 - The analysis of movable singularities in the complex time plane of n first order, ordinary differential equations has led to a better understanding of the real time behavior of the solutions of dynamical systems. The requirement that these singularities be poles, with n-1 free constants, i.e. the so-called Painlevé property, has identified many new completely integrable dynamical systems having, of course, no chaotic behavior whatsoever. On the other hand, the violation of the Painleve property with the introduction of logarithmic terms at higher orders in the series expansions has identified many dynamical systems with only "weakly chaotic" behavior. In this paper, I review these results and discuss the methods of singularity analysis with the aid of illustrative examples.

AB - The analysis of movable singularities in the complex time plane of n first order, ordinary differential equations has led to a better understanding of the real time behavior of the solutions of dynamical systems. The requirement that these singularities be poles, with n-1 free constants, i.e. the so-called Painlevé property, has identified many new completely integrable dynamical systems having, of course, no chaotic behavior whatsoever. On the other hand, the violation of the Painleve property with the introduction of logarithmic terms at higher orders in the series expansions has identified many dynamical systems with only "weakly chaotic" behavior. In this paper, I review these results and discuss the methods of singularity analysis with the aid of illustrative examples.

UR - http://www.scopus.com/inward/record.url?scp=77956934584&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77956934584&partnerID=8YFLogxK

U2 - 10.1016/S0304-0208(08)72136-4

DO - 10.1016/S0304-0208(08)72136-4

M3 - Article

VL - 103

SP - 353

EP - 373

JO - North-Holland Mathematics Studies

JF - North-Holland Mathematics Studies

SN - 0304-0208

IS - C

ER -