Lyapunov exponents and sensitive dependence
type: Item
instance of: scholarly article
title: Lyapunov exponents and sensitive dependence
zbMATH Open document ID: 1205.37024
published in: Journal of Dynamics and Differential Equations
publication date: 2010-11-22T00:00:00Z
full work available at URL: 106.pdf
review text: 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.
reviewed by: Ilia V. Boikov
Mathematics Subject Classification ID: 37B25
Mathematics Subject Classification ID: 65H10
zbMATH DE Number: 5818069
zbMATH Keywords: Lyapunov exponent
zbMATH Keywords: strong Lyapunov exponent
zbMATH Keywords: sensitive dependence
zbMATH Keywords: chaotic orbits
zbMATH Keywords: chaos
author: Hüseyin Koçak
author: Kenneth Palmer
MaRDI profile type: Publication
OpenAlex ID: W2075953208
cites work: On Lyapunov exponent and sensitivity.
cites work: Chaos
cites work: Q5842342
cites work: Q5848416
cites work: Q5533446
cites work: Quantitative hyperbolicity estimates in one-dimensional dynamics
cites work: A note on positive Lyapunov exponent and sensitive dependence on initial conditions
cites work: Sensitive dependence to initial conditions for one dimensional maps
cites work: Q5580482
cites work: Q4132178
cites work: A study of synchronous asymptotic stability
cites work: Stability by Liapunov's direct method. With applications
cites work: Q3293556
cites work: TIME-VARYING LINEARIZATION AND THE PERRON EFFECTS
cites work: Q4003408
cites work: Q4793015
DOI: 10.1007/S10884-010-9169-Y
description: scientific article; zbMATH DE number 5818069