How often are chaotic saddles nonhyperbolic?

Ying-Cheng Lai, C. Grebogi, J. A. Yorke, I. Kan

Research output: Contribution to journalArticle

83 Citations (Scopus)

Abstract

The authors numerically investigate the fraction of nonhyperbolic parameter values in chaotic dynamical systems. By a nonhyperbolic parameter value they mean a parameter value at which there are tangencies between some stable and unstable manifolds. The nonhyperbolic parameter values are important because the dynamics in such cases is especially pathological. For example, near each such parameter value, there is another parameter value at which there are infinitely many coexisting attractors. In particular, Newhouse and Robinson (1983) proved that the existence of one nonhyperbolic parameter value typically implies the existence of an interval ('a Newhouse interval') of nonhyperbolic parameter values. They numerically compute the fraction of nonhyperbolic parameter values for the Henon map in the parameter range where there exist only chaotic saddles (i.e., nonattracting invariant chaotic sets). They discuss a theoretical model which predicts the fraction of nonhyperbolic parameter values for small Jacobians.

Original languageEnglish (US)
Article number007
Pages (from-to)779-797
Number of pages19
JournalNonlinearity
Volume6
Issue number5
DOIs
StatePublished - 1993
Externally publishedYes

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saddles
Saddle
Dynamical systems
intervals
Hénon Map
Chaotic Dynamical Systems
Stable and Unstable Manifolds
Interval
Mean Value
dynamical systems
Theoretical Model
Attractor
Imply
Predict

ASJC Scopus subject areas

  • Mathematical Physics
  • Statistical and Nonlinear Physics
  • Applied Mathematics
  • Mathematics(all)

Cite this

Lai, Y-C., Grebogi, C., Yorke, J. A., & Kan, I. (1993). How often are chaotic saddles nonhyperbolic? Nonlinearity, 6(5), 779-797. [007]. https://doi.org/10.1088/0951-7715/6/5/007

How often are chaotic saddles nonhyperbolic? / Lai, Ying-Cheng; Grebogi, C.; Yorke, J. A.; Kan, I.

In: Nonlinearity, Vol. 6, No. 5, 007, 1993, p. 779-797.

Research output: Contribution to journalArticle

Lai, Y-C, Grebogi, C, Yorke, JA & Kan, I 1993, 'How often are chaotic saddles nonhyperbolic?', Nonlinearity, vol. 6, no. 5, 007, pp. 779-797. https://doi.org/10.1088/0951-7715/6/5/007
Lai, Ying-Cheng ; Grebogi, C. ; Yorke, J. A. ; Kan, I. / How often are chaotic saddles nonhyperbolic?. In: Nonlinearity. 1993 ; Vol. 6, No. 5. pp. 779-797.
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