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Journal of Dynamics and Differential Equations

, Volume 22, Issue 3, pp 381–398 | Cite as

Lyapunov Exponents and Sensitive Dependence

  • Hüseyin Koçak
  • Kenneth J. Palmer
Article

Abstract

Precise relationships between Lyapunov exponents and the instability/stability of orbits of scalar discrete dynamical systems are investigated. It is known that positive Lyapunov exponent does not, in general, imply instability, or, equivalently, sensitive dependence on initial conditions. A notion of strong Lyapunov exponent is introduced and it is proved that for continuously differentiable maps on an interval, a positive strong Lyapunov exponent implies sensitive dependence on initial conditions. However, to have a strong Lyapunov exponent an orbit must stay away from critical points. Nevertheless, it is shown for a restricted class of maps that a positive Lyapunov exponent implies sensitive dependence even for orbits which go arbitrarily close to critical points. Finally, it is proved that for twice differentiable maps on an interval negative Lyapunov exponent does imply exponential stability of orbits.

Keywords

Lyapunov exponent Strong Lyapunov exponent Sensitive dependence Chaotic orbits Chaos 

Mathematics Subject Classification (2000)

58F22 58F13 65H10 

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Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Departments of Computer Science and MathematicsUniversity of MiamiCoral GablesUSA
  2. 2.Department of MathematicsNational Taiwan UniversityTaipeiTaiwan

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