9 Entropy Production, Information Gain and Lyapunov Exponents of Random Hyperbolic Dynamical Systems

2003 ◽  
pp. 215-252
Author(s):  
Da-Quan Jiang ◽  
Min Qian ◽  
Min-Ping Qian
Keyword(s):  
Dynamical Systems ◽  
Entropy Production ◽  
Lyapunov Exponents ◽  
Information Gain ◽  
Hyperbolic Dynamical Systems

Analytic regularity of solutions of Livsic's cohomology equation and some applications to analytic conjugacy of hyperbolic dynamical systems

1997 ◽  
Vol 17 (3) ◽  
pp. 649-662 ◽  
Author(s):  
R. DE LA LLAVE
Keyword(s):  
Vector Field ◽  
Dynamical Systems ◽  
Lyapunov Exponents ◽  
Continuous Solution ◽  
Periodic Points ◽  
Regularity Of Solutions ◽  
Hyperbolic Dynamical Systems ◽  
Anosov Systems ◽  
Low Dimensional ◽  
Analytic Regularity

We study Livsic's problem of finding $\phi$ satisfying $X\phi=\eta$, where $\eta$ is a given function and $X$ is a given Anosov vector field. We show that, if $\phi$ is a continuous solution and $X,\eta$ are analytic, then $\phi$ is analytic. We use the previous result to show that if two low-dimensional Anosov systems are topologically conjugate and the Lyapunov exponents at corresponding periodic points agree, the conjugacy is analytic. Analogous results hold for diffeomorphisms.

scholarly journals Weak specification properties and large deviations for non-additive potentials

2013 ◽  
Vol 35 (3) ◽  
pp. 968-993 ◽  
Author(s):  
PAULO VARANDAS ◽  
YUN ZHAO
Keyword(s):  
Dynamical Systems ◽  
Large Deviations ◽  
Lyapunov Exponents ◽  
Large Deviation Principle ◽  
Large Deviation ◽  
Deviation Principle ◽  
Hyperbolic Dynamical Systems

AbstractWe obtain large deviation bounds for the measure of deviation sets associated with asymptotically additive and sub-additive potentials under some weak specification properties. In particular, a large deviation principle is obtained in the case of uniformly hyperbolic dynamical systems. Some applications to the study of the convergence of Lyapunov exponents are given.

scholarly journals Almost all $S$-integer dynamical systems have many periodic points

1998 ◽  
Vol 18 (2) ◽  
pp. 471-486 ◽  
Author(s):  
T. B. WARD
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Cellular Automata ◽  
Zeta Function ◽  
Periodic Points ◽  
Radius Of Convergence ◽  
Dynamical Zeta Function ◽  
Hyperbolic Dynamical Systems ◽  
Almost All

We show that for almost every ergodic $S$-integer dynamical system the radius of convergence of the dynamical zeta function is no larger than $\exp(-\frac{1}{2}h_{\rm top})<1$. In the arithmetic case almost every zeta function is irrational.We conjecture that for almost every ergodic $S$-integer dynamical system the radius of convergence of the zeta function is exactly $\exp(-h_{\rm top})<1$ and the zeta function is irrational.In an important geometric case (the $S$-integer systems corresponding to isometric extensions of the full $p$-shift or, more generally, linear algebraic cellular automata on the full $p$-shift) we show that the conjecture holds with the possible exception of at most two primes $p$.Finally, we explicitly describe the structure of $S$-integer dynamical systems as isometric extensions of (quasi-)hyperbolic dynamical systems.

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.

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