HOW OFTEN ARE CHAOTIC SADDLES NONHYPERBOLIC

Y C LAI, C GREBOGI, J A YORKE, I KAN, Ying-Cheng Lai

Research output: Contribution to journalArticle

80 Citations (Scopus)

Abstract

In this paper, we numerically investigate the fraction of nonhyperbolic parameter values in chaotic dynamical systems. By a nonhyperbolic parameter value we 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 proved that the existence of one nonhyperbolic parameter value typically implies the existence of an interval ('a Newhouse interval') of nonhyperbolic parameter values. We 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). We discuss a theoretical model which predicts the fraction of nonhyperbolic parameter values for small Jacobians. Two-dimensional diffeomorphisms with similar chaotic saddles may arise in the study of Poincare return map for physical systems. Our results suggest that (1) nonhyperbolic chaotic saddles are common in chaotic dynamical systems; and (2) Newhouse intervals can be quite large in the parameter space.

Original languageEnglish
Pages (from-to)779-797
Number of pages19
JournalNonlinearity
Volume6
Issue number5
Publication statusPublished - Sep 1993

Keywords

  • TRAJECTORIES
  • DIMENSIONS
  • DYNAMICS
  • ORBITS

Cite this

LAI, Y. C., GREBOGI, C., YORKE, J. A., KAN, I., & Lai, Y-C. (1993). HOW OFTEN ARE CHAOTIC SADDLES NONHYPERBOLIC. Nonlinearity, 6(5), 779-797.

HOW OFTEN ARE CHAOTIC SADDLES NONHYPERBOLIC. / LAI, Y C ; GREBOGI, C ; YORKE, J A ; KAN, I ; Lai, Ying-Cheng.

In: Nonlinearity, Vol. 6, No. 5, 09.1993, p. 779-797.

Research output: Contribution to journalArticle

LAI, YC, GREBOGI, C, YORKE, JA, KAN, I & Lai, Y-C 1993, 'HOW OFTEN ARE CHAOTIC SADDLES NONHYPERBOLIC' Nonlinearity, vol. 6, no. 5, pp. 779-797.
LAI YC, GREBOGI C, YORKE JA, KAN I, Lai Y-C. HOW OFTEN ARE CHAOTIC SADDLES NONHYPERBOLIC. Nonlinearity. 1993 Sep;6(5):779-797.
LAI, Y C ; GREBOGI, C ; YORKE, J A ; KAN, I ; Lai, Ying-Cheng. / HOW OFTEN ARE CHAOTIC SADDLES NONHYPERBOLIC. In: Nonlinearity. 1993 ; Vol. 6, No. 5. pp. 779-797.
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N2 - In this paper, we numerically investigate the fraction of nonhyperbolic parameter values in chaotic dynamical systems. By a nonhyperbolic parameter value we 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 proved that the existence of one nonhyperbolic parameter value typically implies the existence of an interval ('a Newhouse interval') of nonhyperbolic parameter values. We 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). We discuss a theoretical model which predicts the fraction of nonhyperbolic parameter values for small Jacobians. Two-dimensional diffeomorphisms with similar chaotic saddles may arise in the study of Poincare return map for physical systems. Our results suggest that (1) nonhyperbolic chaotic saddles are common in chaotic dynamical systems; and (2) Newhouse intervals can be quite large in the parameter space.

AB - In this paper, we numerically investigate the fraction of nonhyperbolic parameter values in chaotic dynamical systems. By a nonhyperbolic parameter value we 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 proved that the existence of one nonhyperbolic parameter value typically implies the existence of an interval ('a Newhouse interval') of nonhyperbolic parameter values. We 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). We discuss a theoretical model which predicts the fraction of nonhyperbolic parameter values for small Jacobians. Two-dimensional diffeomorphisms with similar chaotic saddles may arise in the study of Poincare return map for physical systems. Our results suggest that (1) nonhyperbolic chaotic saddles are common in chaotic dynamical systems; and (2) Newhouse intervals can be quite large in the parameter space.

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