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Isochronous dynamical systems.

Authors
  • Calogero, F
Type
Published Article
Journal
Philosophical transactions. Series A, Mathematical, physical, and engineering sciences
Publication Date
Mar 28, 2011
Volume
369
Issue
1939
Pages
1118–1136
Identifiers
DOI: 10.1098/rsta.2010.0250
PMID: 21320909
Source
Medline
License
Unknown

Abstract

This is a terse review of recent results on isochronous dynamical systems, namely systems of (first-order, generally nonlinear) ordinary differential equations (ODEs) featuring an open set of initial data (which might coincide with the entire set of all initial data), from which emerge solutions all of which are completely periodic (i.e. periodic in all their components) with a fixed period (independent of the initial data, provided they are within the isochrony region). A leitmotif of this presentation is that 'isochronous systems are not rare'. Indeed, it is shown how any (autonomous) dynamical system can be modified or extended so that the new (also autonomous) system thereby obtained is isochronous with an arbitrarily assigned period T, while its dynamics, over time intervals much shorter than the period T, mimics closely that of the original system, or even, over an arbitrarily large fraction of its period T, coincides exactly with that of the original system. It is pointed out that this fact raises the issue of developing criteria providing, for a dynamical system, some kind of measure associated with a finite time scale of the complexity of its behaviour (while the current, standard definitions of integrable versus chaotic dynamical systems are related to the behaviour of a system over infinite time).

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