A route to computational chaos revisited

Noninvertibility and the breakup of an invariant circle

Christos E. Frouzakis, Yannis Kevrekidis, Bruce B. Peckham

Research output: Contribution to journalArticle

20 Citations (Scopus)

Abstract

In a one-parameter study of a noninvertible family of maps of the plane arising in the context of a numerical integration scheme, Lorenz studied a sequence of transitions from an attracting fixed point to "computational chaos". As part of the transition sequence, he proposed the following as a possible scenario for the breakup of an invariant circle (IC): the IC develops regions of increasingly sharper curvature until at a critical parameter value it develops cusps; beyond this parameter value, the IC fails to persist, and the system exhibits chaotic behavior on an invariant set with loops [Computational chaos - a prelude to computational instability, Physica D 35 (1989) 299]. We investigate this problem in more detail and show that the IC is actually destroyed in a global bifurcation before it has a chance to develop cusps. Instead, the global unstable manifolds of saddle-type periodic points are the objects which develop cusps and subsequently "loops" or "antennae". The one-parameter study is better understood when embedded in the full two-parameter space and viewed in the context of the two-parameter Arnold horn structure. Certain elements of the interplay of noninvertibility with this structure, the associated ICs, periodic points and global bifurcations are examined.

Original languageEnglish (US)
Pages (from-to)101-121
Number of pages21
JournalPhysica D: Nonlinear Phenomena
Volume177
Issue number1-4
DOIs
StatePublished - Mar 15 2003

Fingerprint

Breakup
Chaos theory
chaos
Chaos
Circle
Cusp
routes
Invariant
Global Bifurcation
Chaotic systems
Periodic Points
cusps
Two Parameters
Antennas
Unstable Manifold
Saddle
Invariant Set
Chaotic Behavior
Numerical integration
Parameter Space

All Science Journal Classification (ASJC) codes

  • Applied Mathematics
  • Statistical and Nonlinear Physics

Cite this

Frouzakis, Christos E. ; Kevrekidis, Yannis ; Peckham, Bruce B. / A route to computational chaos revisited : Noninvertibility and the breakup of an invariant circle. In: Physica D: Nonlinear Phenomena. 2003 ; Vol. 177, No. 1-4. pp. 101-121.
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A route to computational chaos revisited : Noninvertibility and the breakup of an invariant circle. / Frouzakis, Christos E.; Kevrekidis, Yannis; Peckham, Bruce B.

In: Physica D: Nonlinear Phenomena, Vol. 177, No. 1-4, 15.03.2003, p. 101-121.

Research output: Contribution to journalArticle

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