COMPUTER ASSISTED VERIFICATION OF CHAOS IN THREE-NEURON CELLULAR NEURAL NETWORKS

2007 ◽  
Vol 17 (12) ◽  
pp. 4381-4386 ◽  
Author(s):  
QUAN YUAN ◽  
XIAO-SONG YANG
Keyword(s):  
Neural Networks ◽  
Topological Entropy ◽  
Chaotic Behavior ◽  
Cellular Neural Networks ◽  
Computer Assisted ◽  
Topological Horseshoe ◽  
Poincaré Maps ◽  
New Class ◽  
Numerical Studies ◽  
Connection Matrices

In this paper, we study a new class of simple three-neuron chaotic cellular neural networks with very simple connection matrices. To study the chaotic behavior in these cellular neural networks demonstrated in numerical studies, we resort to Poincaré section and Poincaré map technique and present a rigorous verification of the existence of horseshoe chaos using topological horseshoe theory and the estimate of topological entropy in derived Poincaré maps.

CHAOS IN SIMPLE CELLULAR NEURAL NETWORKS WITH CONNECTION MATRICES SATISFYING DALE'S RULE

2007 ◽  
Vol 17 (02) ◽  
pp. 583-587 ◽  
Author(s):  
XIAO-SONG YANG ◽  
QINGDU LI
Keyword(s):  
Neural Networks ◽  
Three Dimensional ◽  
Cellular Neural Networks ◽  
Computer Assisted ◽  
Topological Horseshoe ◽  
Connection Matrices

In this paper, it is shown that chaos can take place in simple three-dimensional cellular neural networks with connection matrices satisfying Dale's rule. In addition, a rigorous computer-assisted verification of chaoticity in these cellular neural networks is given by virtue of topological horseshoe theory.

CHAOS AND HYPERCHAOS IN A CLASS OF SIMPLE CELLULAR NEURAL NETWORKS MODELED BY O.D.E.

2006 ◽  
Vol 16 (09) ◽  
pp. 2729-2736 ◽  
Author(s):  
XIAO-SONG YANG ◽  
YAN HUANG
Keyword(s):  
Neural Networks ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Lyapunov Exponents ◽  
Cellular Neural Networks ◽  
New Class ◽  
Connection Matrices ◽  
Low Dimensional

This paper presents a new class of chaotic and hyperchaotic low dimensional cellular neural networks modeled by ordinary differential equations with some simple connection matrices. The chaoticity of these neural networks is indicated by positive Lyapunov exponents calculated by a computer.

CHAOS, BIFURCATION AND ROBUSTNESS OF A CLASS OF HOPFIELD NEURAL NETWORKS

2011 ◽  
Vol 21 (03) ◽  
pp. 885-895 ◽  
Author(s):  
WEN-ZHI HUANG ◽  
YAN HUANG
Keyword(s):  
Neural Networks ◽  
Dynamical Systems ◽  
Chaotic Behavior ◽  
Robustness Analysis ◽  
Original System ◽  
Hopfield Neural Networks ◽  
Phase Portraits ◽  
Chaotic Attractors ◽  
Computer Assisted ◽  
New Class

Chaos, bifurcation and robustness of a new class of Hopfield neural networks are investigated. Numerical simulations show that the simple Hopfield neural networks can display chaotic attractors and limit cycles for different parameters. The Lyapunov exponents are calculated, the bifurcation plot and several important phase portraits are presented as well. By virtue of horseshoes theory in dynamical systems, rigorous computer-assisted verifications for chaotic behavior of the system with certain parameters are given, and here also presents a discussion on the robustness of the original system. Besides this, quantitative descriptions of the complexity of these systems are also given, and a robustness analysis of the system is presented too.

HORSESHOE CHAOS IN CELLULAR NEURAL NETWORKS

2006 ◽  
Vol 16 (01) ◽  
pp. 157-161 ◽  
Author(s):  
XIAO-SONG YANG ◽  
QINGDU LI
Keyword(s):  
Neural Networks ◽  
Cross Section ◽  
Cellular Neural Networks ◽  
Computer Assisted ◽  
Poincaré Maps ◽  
Poincare Maps ◽  
Computer Assisted Proof ◽  
Low Dimensional

In this paper, we demonstrate chaos in low dimensional cellular neural networks for some weight matrices. To verify chaoticity of the dynamics in these cellular neural networks, we consider a cross-section properly chosen for the attractors obtained and study the dynamics of the corresponding Poincaré maps, and rigorously verify the existence of horseshoe in the manner of computer-assisted proof arguments.

CHAOS AND TWO-TORI IN A NEW FAMILY OF 4-CNNS

2007 ◽  
Vol 17 (03) ◽  
pp. 953-963 ◽  
Author(s):  
XIAO-SONG YANG ◽  
YAN HUANG
Keyword(s):  
Neural Networks ◽  
Limit Cycle ◽  
Limit Cycles ◽  
Cellular Neural Networks ◽  
Lyapunov Spectrum ◽  
Regular Array ◽  
Poincaré Maps ◽  
One Dimensional ◽  
Poincare Maps ◽  
New Family

In this paper we demonstrate chaos, two-tori and limit cycles in a new family of Cellular Neural Networks which is a one-dimensional regular array of four cells. The Lyapunov spectrum is calculated in a range of parameters, the bifurcation plots are presented as well. Furthermore, we confirm the nature of limit cycle, chaos and two-tori by studying Poincaré maps.

COMPUTER ASSISTED VERIFICATION OF CHAOS IN THE SMOOTH CHUA'S EQUATION

2008 ◽  
Vol 18 (08) ◽  
pp. 2391-2396 ◽  
Author(s):  
QUAN YUAN ◽  
XIAO-SONG YANG
Keyword(s):  
Poincaré Map ◽  
Chaotic Behavior ◽  
Computer Assisted ◽  
Poincaré Section ◽  
Poincare Map ◽  
Poincare Section ◽  
Numerical Studies ◽  
Topological Horseshoes

In this paper, chaos in the smooth Chua's equation is revisited. To confirm the chaotic behavior in the smooth Chua's equation demonstrated in numerical studies, we resort to Poincaré section and Poincaré map technique and present a computer assisted verification of existence of horseshoe chaos by virtue of topological horseshoes theory.

scholarly journals Dynamical Analysis of the Lorenz-84 Atmospheric Circulation Model

2014 ◽  
Vol 2014 ◽  
pp. 1-15 ◽  
Author(s):  
Hu Wang ◽  
Yongguang Yu ◽  
Guoguang Wen
Keyword(s):  
Atmospheric Circulation ◽  
Chaotic Behavior ◽  
Circulation Model ◽  
Dynamical Analysis ◽  
Computer Assisted ◽  
Topological Horseshoe ◽  
Normal Form Theory ◽  
Computer Assisted Proof ◽  
Parameter Interval ◽  
The Stability

The dynamical behaviors of the Lorenz-84 atmospheric circulation model are investigated based on qualitative theory and numerical simulations. The stability and local bifurcation conditions of the Lorenz-84 atmospheric circulation model are obtained. It is also shown that when the bifurcation parameter exceeds a critical value, the Hopf bifurcation occurs in this model. Then, the conditions of the supercritical and subcritical bifurcation are derived through the normal form theory. Finally, the chaotic behavior of the model is also discussed, the bifurcation diagrams and Lyapunov exponents spectrum for the corresponding parameter are obtained, and the parameter interval ranges of limit cycle and chaotic attractor are calculated in further. Especially, a computer-assisted proof of the chaoticity of the model is presented by a topological horseshoe theory.

A NEW CLASS OF CHAOTIC SIMPLE THREE-NEURON CELLULAR NEURAL NETWORKS

2006 ◽  
Vol 16 (04) ◽  
pp. 1019-1021 ◽  
Author(s):  
XIAO-SONG YANG ◽  
YAN HUANG
Keyword(s):  
Neural Networks ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Three Dimensional ◽  
Cellular Neural Networks ◽  
New Class

In this letter we report a new class of chaotic three-neuron cellular neural networks that are described by special three-dimensional autonomous ordinary differential equations.

BIFURCATIONS AND CHAOS IN CELLULAR NEURAL NETWORKS

2003 ◽  
Vol 12 (04) ◽  
pp. 417-433 ◽  
Author(s):  
M. BIEY ◽  
P. CHECCO ◽  
M. GILLI
Keyword(s):  
Neural Networks ◽  
Bifurcation Analysis ◽  
Complex Dynamics ◽  
Chaotic Behavior ◽  
Period Doubling ◽  
Cellular Neural Networks ◽  
Two Dimensional ◽  
First Order ◽  
Stable Periodic ◽  
Torus Breakdown

The dynamic behavior of first-order autonomous space invariant cellular neural networks (CNNs) is investigated. It is shown that complex dynamics may occur in very simple CNN structures, described by two-dimensional templates that present only vertical and horizontal couplings. The bifurcation processes are analyzed through the computation of the limit cycle Floquet's multipliers, the evaluation of the Lyapunov exponents and of the signal spectra. As a main result a detailed and accurate two-dimensional bifurcation diagram is reported. The diagram allows one to distinguish several regions in the parameter space of a single CNN. They correspond to stable, periodic, quasi-periodic, and chaotic behavior, respectively. In particular it is shown that chaotic regions can be reached through two different routes: period doubling and torus breakdown. We remark that most practical CNN implementations exploit first order cells and space-invariant templates: so far only a few examples of complex dynamics and no complete bifurcation analysis have been presented for such networks.

Bifurcation and Chaos in Noninteger Order Cellular Neural Networks

1998 ◽  
Vol 08 (07) ◽  
pp. 1527-1539 ◽  
Author(s):  
P. Arena ◽  
R. Caponetto ◽  
L. Fortuna ◽  
D. Porto
Keyword(s):  
Neural Networks ◽  
Equilibrium Points ◽  
Cellular Neural Networks ◽  
Cell System ◽  
Bifurcation And Chaos ◽  
Chaotic Motions ◽  
First Order ◽  
New Class ◽  
Onset Of Chaos ◽  
Coupling Parameters

In this paper a new class of Cellular Neural Networks (CNNs) is introduced. The peculiarity of the new CNN model consists in replacing the traditional first order cell with a noninteger order one. The introduction of fractional order cells, with a suitable choice of the coupling parameters, leads to the onset of chaos in a two-cell system of a total order of less than three. A theoretical approach, based on the interaction between equilibrium points and limit cycles, is used to discover chaotic motions in fractional CNNs.

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