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 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 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 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.

Symmetry-Breaking and Chaos in Oscillatory Neural Networks

2001 ◽  
Author(s):  
Yonghong Chen ◽  
Jianxue Xu ◽  
Tong Fang
Keyword(s):  
Neural Networks ◽  
Hopf Bifurcation ◽  
Normal Form ◽  
Symmetry Breaking ◽  
Chaotic Attractor ◽  
Phase Locking ◽  
Dynamical Behavior ◽  
Second Order ◽  
Poincaré Maps ◽  
Poincare Maps

Abstract Complex dynamical behavior of neural networks may lead to new methodology of information processing. In this paper the dynamics of a neural network designed by the normal form for Hopf bifurcation is studied. The secondary Hopf bifurcation of the network is discussed and a two-torus is observed. Examining the phase-locking motions on the two-torus, we present the conditions of symmetry-breaking occurring in the system. If the ratio of the two frequencies of the codimension two Hopf bifurcation is represented by an irreducible fraction, then the symmetry-breaking will occur when either the numerator or the denominator of the fraction is an even number. Chaotic attractors may be created with the sigmoid nonlinearities added to the right hand side of the normal form equations. The phase trajectory and the second order Poincaré maps of the chaotic attractor are given. The chaotic attractor looks like a butterfly on some of the second order Poincaré maps. This is a marvelous example for chaos to mimic nature.

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