ROBUSTNESS OF SYMBOLIC DYNAMICS AND SYNCHRONIZATION PROPERTIES

In this paper we introduce the method for investigation of coupled chaotic systems using topological methods. We show that if the coupling is small then there exists independent symbolic dynamics for every coupled subsystem and in consequence the systems are not synchronized. As an example we consider coupled Hénon maps. Using computer interval arithmetic we find parameter mismatch and perturbation range for which the symbolic dynamics in the Hénon system is sustained. For coupled Hénon maps we compute the value of coupling strength for which the symbolic dynamics in every subsystem survives.

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Design of PDC Controllers by Matrix Reversibility for Synchronization ofYinandYangChaotic Takagi-Sugeno Fuzzy Henon Maps

Abstract and Applied Analysis ◽

10.1155/2012/358201 ◽

2012 ◽

Vol 2012 ◽

pp. 1-11 ◽

Cited By ~ 1

Author(s):

Chun-Yen Ho ◽

Hsien-Keng Chen ◽

Zheng-Ming Ge

Keyword(s):

Chaos Synchronization ◽

Matrix Theory ◽

Chinese Philosophy ◽

Fuzzy Model ◽

Invertible Matrix ◽

Henon Maps ◽

Hénon Maps ◽

Feminine Principle ◽

Simulation Results ◽

Takagi Sugeno

This paper investigates the synchronization ofYinandYangchaotic T-S fuzzy Henon maps via PDC controllers. Based on the Chinese philosophy,Yinis the decreasing, negative, historical, or feminine principle in nature, whileYangis the increasing, positive, contemporary, or masculine principle in nature.YinandYangare two fundamental opposites in Chinese philosophy. The Henon map is an invertible map; so the Henon maps with increasing and decreasing argument can be called theYangandYinHenon maps, respectively. Chaos synchronization ofYinandYangT-S fuzzy Henon maps is achieved by PDC controllers. The design of PDC controllers is based on the linear invertible matrix theory. The T-S fuzzy model ofYinandYangHenon maps and the design of PDC controllers are novel, and the simulation results show that the approach is effective.

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Reversible Complex Hénon Maps

Experimental Mathematics ◽

10.1080/10586458.2002.10504479 ◽

2002 ◽

Vol 11 (3) ◽

pp. 339-347 ◽

Cited By ~ 4

Author(s):

C. R. Jordan ◽

D. A. Jordan ◽

J. H. Jordan

Keyword(s):

Henon Maps ◽

Hénon Maps

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Generalized Hénon maps: the cubic diffeomorphisms of the plane

Physica D Nonlinear Phenomena ◽

10.1016/s0167-2789(00)00105-6 ◽

2000 ◽

Vol 143 (1-4) ◽

pp. 262-289 ◽

Cited By ~ 28

Author(s):

H.R. Dullin ◽

J.D. Meiss

Keyword(s):

Henon Maps ◽

Hénon Maps

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Stabilization of Higher Periodic Orbits of the Lozi and Hénon Maps using Meta-evolutionary Approaches

2021 IEEE Congress on Evolutionary Computation (CEC) ◽

10.1109/cec45853.2021.9504798 ◽

2021 ◽

Author(s):

Radomil Matousek ◽

Rene Pierre Lozi ◽

Tomas Hulka

Keyword(s):

Periodic Orbits ◽

Henon Maps ◽

Hénon Maps

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Generalized Henon Maps as Difference Equations

New Trends in Difference Equations ◽

10.1201/9781482265156-93 ◽

2002 ◽

pp. 211-212

Keyword(s):

Difference Equations ◽

Henon Maps ◽

Hénon Maps

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ADAPTIVE OBSERVER BASED SYNCHRONIZATION OF A CLASS OF UNCERTAIN CHAOTIC SYSTEMS

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127408021798 ◽

2008 ◽

Vol 18 (08) ◽

pp. 2425-2435 ◽

Cited By ~ 4

Author(s):

SAMUEL BOWONG ◽

RENÉ YAMAPI

Keyword(s):

Chaotic Systems ◽

Lyapunov Stability Theory ◽

Adaptive Observer ◽

Adaptive Synchronization ◽

External Disturbances ◽

Parameter Mismatch ◽

Lipschitz Constants ◽

Response Systems ◽

Uncertain Chaotic Systems ◽

Robust Adaptive

This study addresses the adaptive synchronization of a class of uncertain chaotic systems in the drive-response framework. For a class of uncertain chaotic systems with parameter mismatch and external disturbances, a robust adaptive observer based on the response system is constructed to practically synchronize the uncertain drive chaotic system. Lyapunov stability theory ensures the practical synchronization between the drive and response systems even if Lipschitz constants on function matrices and bounds on uncertainties are unknown. Numerical simulation of two illustrative examples are given to verify the effectiveness of the proposed method.

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Topological classification of periodic orbits in the Kuramoto–Sivashinsky equation

Modern Physics Letters B ◽

10.1142/s0217984918501555 ◽

2018 ◽

Vol 32 (15) ◽

pp. 1850155 ◽

Cited By ~ 6

Author(s):

Chengwei Dong

Keyword(s):

Nonlinear System ◽

Variational Method ◽

Periodic Orbits ◽

Symbolic Dynamics ◽

Chaotic Systems ◽

Building Blocks ◽

Topological Classification ◽

One Dimensional ◽

Sivashinsky Equation

In this paper, we systematically research periodic orbits of the Kuramoto–Sivashinsky equation (KSe). In order to overcome the difficulties in the establishment of one-dimensional symbolic dynamics in the nonlinear system, two basic periodic orbits can be used as basic building blocks to initialize cycle searching, and we use the variational method to numerically determine all the periodic orbits under parameter [Formula: see text] = 0.02991. The symbolic dynamics based on trajectory topology are very successful for classifying all short periodic orbits in the KSe. The current research can be conveniently adapted to the identification and classification of periodic orbits in other chaotic systems.

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