SNAP-BACK REPELLERS AND CHAOTIC TRAVELING WAVES IN ONE-DIMENSIONAL CELLULAR NEURAL NETWORKS

2007 ◽  
Vol 17 (06) ◽  
pp. 1969-1983 ◽  
Author(s):  
YA-WEN CHANG ◽  
JONQ JUANG ◽  
CHIN-LUNG LI
Keyword(s):  
Neural Networks ◽  
Fixed Point ◽  
Discrete Time ◽  
Traveling Waves ◽  
Sufficient Conditions ◽  
Cellular Neural Networks ◽  
One Dimensional ◽  
Time Analogy

In 1998, Chen et al. [1998] found an error in Marotto's paper [1978]. It was pointed out by them that the existence of an expanding fixed point z of a map F in Br( z ), the ball of radius r with center at z does not necessarily imply that F is expanding in Br( z ). Subsequent efforts (see e.g. [Chen et al., 1998; Lin et al., 2002; Li & Chen, 2003]) in fixing the problems have some discrepancies since they only give conditions for which F is expanding "locally". In this paper, we give sufficient conditions so that F is "globally" expanding. This, in turn, gives more satisfying definitions of a snap-back repeller. We then use those results to show the existence of chaotic backward traveling waves in a discrete time analogy of one-dimensional Cellular Neural Networks (CNNs). Some computer evidence of chaotic traveling waves is also given.

EXISTENCE ANALYSIS OF MOSAIC SOLUTIONS FOR ONE-DIMENSIONAL CELLULAR NEURAL NETWORKS

2006 ◽  
Vol 16 (12) ◽  
pp. 3669-3677 ◽  
Author(s):  
YUN-QUAN KE ◽  
FENG-YAN ZHOU
Keyword(s):  
Neural Networks ◽  
Sufficient Conditions ◽  
Cellular Neural Networks ◽  
One Dimensional

In this letter, the mosaic solutions of one-dimensional Cellular Neural Networks system (CNNs) are investigated. Three types of parameters, the synaptic weights, the input terms and the threshold are properly chosen in terms of Chua's driving-point plot. Moreover, we give sufficient conditions for the existence of the mosaic solutions.

scholarly journals Pseudo Almost Periodic Sequence Solutions of Discrete Time Cellular Neural Networks

2009 ◽  
Vol 14 (3) ◽  
pp. 283-301 ◽  
Author(s):  
S. Abbas
Keyword(s):  
Neural Network ◽  
Neural Networks ◽  
Discrete Time ◽  
Existence And Uniqueness ◽  
Global Attractivity ◽  
Sufficient Conditions ◽  
Cellular Neural Networks ◽  
Periodic Sequence ◽  
Almost Periodic ◽  
Pseudo Almost Periodic

In this paper we discuss the existence and uniqueness of a k-pseudo almost periodic sequence solutions of a discrete time neural network. We give several sufficient conditions for the exponential and global attractivity of the solution.

scholarly journals Fixed Point and Asymptotic Analysis of Cellular Neural Networks

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
Xianghong Lai ◽  
Yutian Zhang
Keyword(s):  
Neural Networks ◽  
Fixed Point ◽  
Existence And Uniqueness ◽  
Fixed Point Theory ◽  
Sufficient Conditions ◽  
Point Theory ◽  
Cellular Neural Networks ◽  
Time Varying ◽  
Trivial Equilibrium ◽  
The Stability

We firstly employ the fixed point theory to study the stability of cellular neural networks without delays and with time-varying delays. Some novel and concise sufficient conditions are given to ensure the existence and uniqueness of solution and the asymptotic stability of trivial equilibrium at the same time. Moreover, these conditions are easily checked and do not require the differentiability of delays.

scholarly journals Traveling Waves for Delayed Cellular Neural Networks with Nonmonotonic Output Functions

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Zhi-Xian Yu ◽  
Rong Yuan ◽  
Cheng-Hsiung Hsu ◽  
Ming-Shu Peng
Keyword(s):  
Neural Networks ◽  
Traveling Waves ◽  
Signal Transmission ◽  
Distributed Delay ◽  
Traveling Wave Solutions ◽  
Cellular Neural Networks ◽  
Switching Speed ◽  
One Dimensional ◽  
The One ◽  
Nondecreasing Functions

This work investigates traveling waves for a class of delayed cellular neural networks with nonmonotonic output functions on the one-dimensional integer latticeZ. The dynamics of each given cell depends on itself and its nearestmleft orlright neighborhood cells with distributed delay due to, for example, finite switching speed and finite velocity of signal transmission. Our technique is to construct two appropriate nondecreasing functions to squeeze the nonmonotonic output functions. Then we construct a suitable wave profiles set and derive the existence of traveling wave solutions by using Schauder's fixed point theorem.

TOPOLOGICAL STRUCTURE OF CHAOS IN DISCRETE-TIME NEURAL NETWORKS WITH GENERALIZED INPUT–OUTPUT FUNCTION

2001 ◽  
Vol 11 (07) ◽  
pp. 1835-1851 ◽  
Author(s):  
JINLIANG WANG ◽  
ZHUJUN JING
Keyword(s):  
Neural Networks ◽  
Fixed Point ◽  
Discrete Time ◽  
Sufficient Conditions ◽  
Sigmoid Function ◽  
Output Function ◽  
Input Output ◽  
Attracting Set ◽  
The Neural Networks ◽  
Searching Ability

By analogue of [Chen & Aihara, 1995, 1997, 1999], we theoretically investigate the topologically chaotic structure, attracting set and global searching ability of discrete-time recurrent neural networks with the form of [Formula: see text] where the input–output function is defined as a generalized sigmoid function, such as vi = tanh (μiui), [Formula: see text] and [Formula: see text], etc. We first derive sufficient conditions of existence for a fixed point, and then prove that this fixed point eventually evolves into a snap-back repeller which generates chaotic structure when certain conditions are satisfied. Furthermore we prove that there exists an attracting set which includes not only the homoclinic orbit but also the globally unstable set of fixed points, thereby ensuring the neural networks to have global searching ability. Numerical simulations are also provided to demonstrate the theoretical results. The results indicated in this paper can be viewed as an extension of the works of [Chen & Aihara, 1997, 1999] and others [Gopalsamy & He, 1994; Wang & Smith, 1998].

scholarly journals State Estimation for Discrete-Time Fuzzy Cellular Neural Networks with Mixed Time Delays

2014 ◽  
Vol 2014 ◽  
pp. 1-13 ◽  
Author(s):  
Lijie Geng ◽  
Haiying Li ◽  
Bingchen Zhao ◽  
Guang Su
Keyword(s):  
Neural Networks ◽  
State Estimation ◽  
Discrete Time ◽  
Time Delays ◽  
Estimation Error ◽  
Sufficient Conditions ◽  
Cellular Neural Networks ◽  
Linear Matrix ◽  
Fuzzy Cellular Neural Networks ◽  
Mixed Time Delays

This paper is concerned with the exponential state estimation problem for a class of discrete-time fuzzy cellular neural networks with mixed time delays. The main purpose is to estimate the neuron states through available output measurements such that the dynamics of the estimation error is globally exponentially stable. By constructing a novel Lyapunov-Krasovskii functional which contains a triple summation term, some sufficient conditions are derived to guarantee the existence of the state estimator. The linear matrix inequality approach is employed for the first time to deal with the fuzzy cellular neural networks in the discrete-time case. Compared with the present conditions in the form ofM-matrix, the results obtained in this paper are less conservative and can be checked readily by the MATLAB toolbox. Finally, some numerical examples are given to demonstrate the effectiveness of the proposed results.

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