scholarly journals Dynamical study of Lyapunov exponents for Hide’s coupled dynamo model

2021 ◽  
Vol 54 (1) ◽  
pp. 189-195
Author(s):  
Teflah Alresheedi ◽  
Ali Allahem
Keyword(s):  
Dynamical Systems ◽  
Numerical Solution ◽  
Lyapunov Exponents ◽  
Dynamo Model ◽  
Periodic Flow ◽  
Solution Behavior ◽  
Dynamical Study ◽  
Regular Flow

Abstract In this paper, we introduced the Lyapunov exponents (LEs) as a significant tool that is used to study the numerical solution behavior of the dynamical systems. Moreover, Hide’s coupled dynamo model presents a valuable dynamical study. We simulate the convergence of the LEs of the model in three cases by means of periodic flow, regular flow, and chaos flow. In addition, we compared these cases in logic connections and proved them in a mathematical way.

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.

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.

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.

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