Obstruction to deterministic modeling of chaotic systems with an invariant subspace

Ying-Cheng Lai, Celso Grebogi

Research output: Contribution to journalArticle

11 Citations (Scopus)

Abstract

A natural process can be deterministically modeled if solutions from its mathematical model stay close to the ones produced by nature. The mathematical model, however, is not exact due to imperfections of the natural system. We describe, in this paper, that there exists a class of models of chaotic processes, for which severe obstruction to deterministic modeling may arise. In particular, such obstruction may occur when unstable periodic orbits embedded in the chaotic invariant set have a distinct number of unstable directions, a type of nonhyperbolicity called unstable-dimension variability. We make these ideas concrete by investigating a class of deterministic models: chaotic systems with an invariant subspace such as systems of coupled chaotic oscillators. We show that unstable-dimension variability can occur in wide parameter regimes of these systems. The implications of our results to scientific modeling are discussed.

Original languageEnglish (US)
Pages (from-to)683-693
Number of pages11
JournalInternational Journal of Bifurcation and Chaos in Applied Sciences and Engineering
Volume10
Issue number3
StatePublished - Mar 2000

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Chaotic systems
Invariant Subspace
Obstruction
Chaotic System
Unstable
Mathematical models
Modeling
Orbits
Concretes
Mathematical Model
Chaotic Oscillator
Defects
Deterministic Model
Imperfections
Coupled Oscillators
Invariant Set
Periodic Orbits
Distinct
Class
Model

ASJC Scopus subject areas

  • General
  • Applied Mathematics

Cite this

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