Entropy, Lyapunov exponents, and Hausdorff dimension in differentiable dynamical systems

1983 ◽  
Vol 30 (8) ◽  
pp. 599-607 ◽  
Author(s):  
Lai-Sang Young
Keyword(s):  
Dynamical Systems ◽  
Hausdorff Dimension ◽  
Lyapunov Exponents

scholarly journals A computational framework for finding parameter sets associated with chaotic dynamics

2021 ◽  
pp. 1-11
Author(s):  
S. Koshy-Chenthittayil ◽  
E. Dimitrova ◽  
E.W. Jenkins ◽  
B.C. Dean
Keyword(s):  
Experimental Data ◽  
Dynamical Systems ◽  
Lyapunov Exponents ◽  
Parameter Space ◽  
Chaotic Dynamics ◽  
Chaotic Behavior ◽  
Computational Framework ◽  
Independent Variables ◽  
Systems Model ◽  
Small Support

Many biological ecosystems exhibit chaotic behavior, demonstrated either analytically using parameter choices in an associated dynamical systems model or empirically through analysis of experimental data. In this paper, we use existing software tools (COPASI, R) to explore dynamical systems and uncover regions with positive Lyapunov exponents where thus chaos exists. We evaluate the ability of the software’s optimization algorithms to find these positive values with several dynamical systems used to model biological populations. The algorithms have been able to identify parameter sets which lead to positive Lyapunov exponents, even when those exponents lie in regions with small support. For one of the examined systems, we observed that positive Lyapunov exponents were not uncovered when executing a search over the parameter space with small spacings between values of the independent variables.

Invariant tensor fields of dynamical systems with pinched Lyapunov exponents and rigidity of geodesic flows

1989 ◽  
Vol 9 (3) ◽  
pp. 427-432 ◽  
Author(s):  
Renato Feres ◽  
Anatoly Katok
Keyword(s):  
Dynamical Systems ◽  
Lyapunov Exponents ◽  
Riemannian Manifolds ◽  
Negative Curvature ◽  
Geodesic Flows ◽  
Tensor Fields ◽  
Invariant Tensor ◽  
Affine Connections

AbstractWe consider in this note smooth dynamical systems equipped with smooth invariant affine connections and show that, under a pinching condition on the Lyapunov exponents, certain invariant tensor fields are parallel. We then apply this result to a problem of rigidity of geodesic flows for Riemannian manifolds with negative curvature.

scholarly journals The Lyapunov spectrum of some parabolic systems

2009 ◽  
Vol 29 (3) ◽  
pp. 919-940 ◽  
Author(s):  
KATRIN GELFERT ◽  
MICHAŁ RAMS
Keyword(s):  
Lyapunov Exponent ◽  
Hausdorff Dimension ◽  
Lyapunov Exponents ◽  
Topological Entropy ◽  
Level Set ◽  
Level Sets ◽  
Parabolic Systems ◽  
Lyapunov Spectrum ◽  
Interval Maps ◽  
Set Of Points

AbstractWe study the Hausdorff dimension for Lyapunov exponents for a class of interval maps which includes several non-hyperbolic situations. We also analyze the level sets of points with given lower and upper Lyapunov exponents and, in particular, with zero lower Lyapunov exponent. We prove that the level set of points with zero exponent has full Hausdorff dimension, but carries no topological entropy.

DYNAMICAL ASPECTS OF INTERACTION NETWORKS

2005 ◽  
Vol 15 (11) ◽  
pp. 3467-3480 ◽  
Author(s):  
G. NICOLIS ◽  
A. GARCÍA CANTÚ ◽  
C. NICOLIS
Keyword(s):  
Phase Space ◽  
Dynamical Systems ◽  
Stochastic Processes ◽  
Lyapunov Exponents ◽  
Network Theory ◽  
Finite Size ◽  
Interaction Networks ◽  
Deterministic Systems ◽  
Connectivity Patterns ◽  
Probabilistic Process

A connection between dynamical systems and network theory is outlined based on a mapping of the dynamics into a discrete probabilistic process, whereby the phase space is partitioned into finite size cells. It is found that the connectivity patterns of networks generated by deterministic systems can be related to the indicators of the dynamics such as local Lyapunov exponents. The procedure is extended to networks generated by stochastic processes.

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