A route to computational chaos revisited: Noninvertibility and the breakup of an invariant circle

Christos E. Frouzakis, Ioanni G. Kevrekidis, Bruce B. Peckham

Research output: Contribution to journalArticlepeer-review

33 Scopus citations


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
Issue number1-4
StatePublished - Mar 15 2003

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • Condensed Matter Physics
  • Applied Mathematics


  • Bifurcation
  • Chaos
  • Integration
  • Invariant circles
  • Noninvertible maps


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