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.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 381-398 |
| Number of pages | 18 |
| Journal | Journal of Dynamics and Differential Equations |
| Volume | 22 |
| Issue number | 3 |
| DOIs | |
| State | Published - 2010 |
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Keywords
- Chaos
- Chaotic orbits
- Lyapunov exponent
- Sensitive dependence
- Strong Lyapunov exponent
ASJC Scopus subject areas
- Analysis
Cite this
Lyapunov Exponents and Sensitive Dependence. / Kocak, Huseyin; Palmer, Kenneth J.
In: Journal of Dynamics and Differential Equations, Vol. 22, No. 3, 2010, p. 381-398.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Lyapunov Exponents and Sensitive Dependence
AU - Kocak, Huseyin
AU - Palmer, Kenneth J.
PY - 2010
Y1 - 2010
N2 - 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.
AB - 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.
KW - Chaos
KW - Chaotic orbits
KW - Lyapunov exponent
KW - Sensitive dependence
KW - Strong Lyapunov exponent
UR - http://www.scopus.com/inward/record.url?scp=77956617010&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=77956617010&partnerID=8YFLogxK
U2 - 10.1007/s10884-010-9169-y
DO - 10.1007/s10884-010-9169-y
M3 - Article
AN - SCOPUS:77956617010
VL - 22
SP - 381
EP - 398
JO - Journal of Dynamics and Differential Equations
JF - Journal of Dynamics and Differential Equations
SN - 1040-7294
IS - 3
ER -