Generalized Lyapunov exponents in high-dimensional chaotic dynamics and products of large random matrices

1988 ◽  
Vol 53 (3-4) ◽  
pp. 583-601 ◽  
Author(s):  
Andrea Crisanti ◽  
Giovanni Paladin ◽  
Angelo Vulpiani
Keyword(s):  
Lyapunov Exponents ◽  
Random Matrices ◽  
Chaotic Dynamics ◽  
High Dimensional ◽  
Generalized Lyapunov Exponents

CHAOS AND COMPLEXITY

2001 ◽  
Vol 11 (01) ◽  
pp. 19-26 ◽  
Author(s):  
RAY BROWN ◽  
ROBERT BEREZDIVIN ◽  
LEON O. CHUA
Keyword(s):  
Dynamical Systems ◽  
Lyapunov Exponents ◽  
Chaotic Dynamics ◽  
High Dimensional ◽  
Topological Conjugacy ◽  
Integer Lattices ◽  
Homoclinic Tangles ◽  
Dimensional Complexity ◽  
Finite Integer ◽  
The Relationship

In this paper we show how to relate a form of high-dimensional complexity to chaotic and other types of dynamical systems. The derivation shows how "near-chaotic" complexity can arise without the presence of homoclinic tangles or positive Lyapunov exponents. The relationship we derive follows from the observation that the elements of invariant finite integer lattices of high-dimensional dynamical systems can, themselves, be viewed as single integers rather than coordinates of a point in n-space. From this observation it is possible to construct high-dimensional dynamical systems which have properties of shifts but for which there is no conventional topological conjugacy to a shift. The particular manner in which the shift appears in high-dimensional dynamical systems suggests that some forms of complexity arise from the presence of chaotic dynamics which are obscured by the large dimensionality of the system domain.

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.

scholarly journals A New Test for Chaotic Dynamics Using Lyapunov Exponents

2003 ◽  
Author(s):  
Fernando Fernández Rodríguez ◽  
Simón Javier Sosvilla Rivero ◽  
Julián Andrada Félix
Keyword(s):  
Lyapunov Exponents ◽  
Chaotic Dynamics

scholarly journals Recurrent flows: the clockwork behind turbulence

2013 ◽  
Vol 726 ◽  
pp. 1-4 ◽  
Author(s):  
Predrag Cvitanović
Keyword(s):  
Fluid Dynamics ◽  
Reynolds Number ◽  
Chaotic Dynamics ◽  
High Dimensional ◽  
Governing Equations ◽  
Infinite Dimensional ◽  
Long Run ◽  
Moderate Reynolds Number ◽  
Dimensional State Space ◽  
Exhaustive Study

AbstractThe understanding of chaotic dynamics in high-dimensional systems that has emerged in the last decade offers a promising dynamical framework to study turbulence. Here turbulence is viewed as a walk through a forest of exact solutions in the infinite-dimensional state space of the governing equations. Recently, Chandler & Kerswell (J. Fluid Mech., vol. 722, 2013, pp. 554–595) carry out the most exhaustive study of this programme undertaken so far in fluid dynamics, a feat that requires every tool in the dynamicist’s toolbox: numerical searches for recurrent flows, computation of their stability, their symmetry classification, and estimating from these solutions statistical averages over the turbulent flow. In the long run this research promises to develop a quantitative, predictive description of moderate-Reynolds-number turbulence, and to use this description to control flows and explain their statistics.

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