Lyapunov Exponents and Sensitive Dependence
Resource
Journal of Dynamics and Differential Equations, 22(3), 381-398
Journal
Journal of Dynamics and Differential Equations
Pages
381-398
Date Issued
2010
Date
2010
Author(s)
Kocak, Hueseyin
Palmer, Kenneth J.
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. © 2010 Springer Science+Business Media, LLC.
Type
journal article
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