Beyond the Li-Yorke Definition of Chaos

Kloeden, Peter E.; Li, Zhong

doi:10.1007/3-540-32502-6_1

Cited by 3 publications

(3 citation statements)

References 27 publications

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“…All the conditions of a theorem in Ref. [15] which generalizes Marotto's snap-back repeller result to saddle points are thus satisfied, so the twisted horseshoe mapping behaves chaotically in a slight generalization of the sense of Li and Yorke (see [18]). In fact, the twisted-horseshoe mapping also satisfies the Sharkovsky cycle coexistence ordering, due to a generalization in Ref.…”

Section: Twisted Horseshoe Mappingmentioning

confidence: 93%

Semi-hyperbolicity and bi-shadowing in nonautonomous difference equations with Lipschitz mappings

Diamond

Kloeden

Kozyakin

2008

Journal of Difference Equations and Applications

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Dedicated to Gerry Ladas on his 70th birthdayIt is shown how known results for autonomous difference equations can be adapted to definitions of semi-hyperbolicity and bi-shadowing generalized to nonautonomous difference equations with Lipschitz continuous mappings. In particular, invertibility and smoothness of the mappings are not required and, for greater applicability, the mappings are allowed to act between possibly different Banach spaces.

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Section: Twisted Horseshoe Mappingmentioning

confidence: 93%

Semi-hyperbolicity and bi-shadowing in nonautonomous difference equations with Lipschitz mappings

Diamond

Kloeden

Kozyakin

2008

Journal of Difference Equations and Applications

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“…We observe that theorems in [Kloeden, 1981;Kloeden & Li, 2006a, 2006b] require some conditions which include "covering relations". On the contrary, there are different approaches that involve topological tools which are about "crossing relations".…”

Section: Theoretical Background On Chaotic Dynamicsmentioning

confidence: 99%

“…In [Kloeden, 1981;Kloeden & Li, 2006a, 2006b], the authors used a theorem developed by Kloeden on chaotic dynamics, which generalizes the result obtained by Li and Yorke to the case of first order N -dimensional difference equations:…”

Section: Introductionmentioning

confidence: 99%

About Chaotic Dynamics in the Twisted Horseshoe Map

Sovrano

2016

Int. J. Bifurcation Chaos

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The twisted horseshoe map was developed in order to study a class of density dependent Leslie population models with two age classes. From the beginning, scientists have tried to prove that this map presents chaotic dynamics. Some demonstrations that have appeared in mathematical literature present some difficulties or delicate issues. In this paper, we give a simple and rigorous proof based on a different approach. We also highlight the possibility of getting chaotic dynamics for a broader class of maps.

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Chaotifying delayed recurrent neural networks via impulsive effects

Şaylı

Yılmaz

2016

Chaos: An Interdisciplinary Journal of Nonlinear Science

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In this paper, chaotification of delayed recurrent neural networks via chaotically changing moments of impulsive actions is considered. Sufficient conditions for the presence of Li-Yorke chaos with its ingredients proximality, frequent separation, and existence of infinitely many periodic solutions are theoretically proved. Finally, effectiveness of our theoretical results is illustrated by an example with numerical simulations.

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Beyond the Li-Yorke Definition of Chaos

Cited by 3 publications

References 27 publications

Semi-hyperbolicity and bi-shadowing in nonautonomous difference equations with Lipschitz mappings

Semi-hyperbolicity and bi-shadowing in nonautonomous difference equations with Lipschitz mappings

About Chaotic Dynamics in the Twisted Horseshoe Map

Chaotifying delayed recurrent neural networks via impulsive effects

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