DEFORMATION OF TRAVELING WAVES IN DELAYED CELLULAR NEURAL NETWORKS

2003 ◽  
Vol 13 (04) ◽  
pp. 797-813 ◽  
Author(s):  
PEIXUAN WENG ◽  
JIANHONG WU
Keyword(s):  
Neural Networks ◽  
Traveling Waves ◽  
Delay Differential Equations ◽  
Wave Speed ◽  
Signal Transmission ◽  
Cellular Neural Networks ◽  
Profile Equation ◽  
Damped Oscillation ◽  
Wave Profiles ◽  
Delay Differential

In this paper, we establish the existence and describe the global structure of traveling waves for a class of lattice delay differential equations describing cellular neural networks with distributed delayed signal transmission. We describe the transition of wave profiles from monotonicity, damped oscillation, periodicity, unboundedness and back to monotonicity as the wave speed is varied. We also describe an interval of the wave speed where the structure of the wave solution is unknown since the corresponding profile equation involves distributed argument of both advanced and retarded types, and we present some preliminary numerical simulation to illustrate the complexity.

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.

scholarly journals New Conditions for the Oscillation of Second-Order Differential Equations with Sublinear Neutral Terms

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1159
Author(s):  
Shyam Sundar Santra ◽  
Omar Bazighifan ◽  
Mihai Postolache
Keyword(s):  
Fluid Dynamics ◽  
Neural Networks ◽  
Differential Equations ◽  
Delay Differential Equations ◽  
Sufficient Conditions ◽  
Second Order ◽  
Qualitative Behavior ◽  
Neutral Differential Equations ◽  
Second Order Differential Equations ◽  
Delay Differential

In continuous applications in electrodynamics, neural networks, quantum mechanics, electromagnetism, and the field of time symmetric, fluid dynamics, neutral differential equations appear when modeling many problems and phenomena. Therefore, it is interesting to study the qualitative behavior of solutions of such equations. In this study, we obtained some new sufficient conditions for oscillations to the solutions of a second-order delay differential equations with sub-linear neutral terms. The results obtained improve and complement the relevant results in the literature. Finally, we show an example to validate the main results, and an open problem is included.

DIVERSITY OF TRAVELING WAVE SOLUTIONS IN DELAYED CELLULAR NEURAL NETWORKS

2008 ◽  
Vol 18 (12) ◽  
pp. 3515-3550 ◽  
Author(s):  
CHENG-HSIUNG HSU ◽  
CHUN-HSIEN LI ◽  
SUH-YUH YANG
Keyword(s):  
Neural Networks ◽  
Traveling Wave ◽  
Wave Speed ◽  
Distributed Delay ◽  
Traveling Wave Solutions ◽  
Cellular Neural Networks ◽  
Monotonicity Condition ◽  
Switching Speed ◽  
Wave Solutions ◽  
Special Cases

This work investigates the diversity of traveling wave solutions for a class of delayed cellular neural networks on the one-dimensional integer lattice ℤ1. The dynamics of a given cell is characterized by instantaneous self-feedback and neighborhood interaction with distributed delay due to, for example, finite switching speed and finite velocity of signal transmission. Applying the monotone iteration scheme, we can deduce the existence of monotonic traveling wave solutions provided the templates satisfy the so-called quasi-monotonicity condition. We then consider two special cases of the delayed cellular neural network in which each cell interacts only with either the nearest m left neighbors or the nearest m right neighbors. For the former case, we can directly figure out the analytic solution in an explicit form by the method of step with the help of the characteristic function and then prove that, in addition to the existence of monotonic traveling wave solutions, for certain templates there exist nonmonotonic traveling wave solutions such as camel-like waves with many critical points. For the latter case, employing the comparison arguments repeatedly, we can clarify the deformation of traveling wave solutions with respect to the wave speed. More specifically, we can describe the transition of profiles from monotonicity, damped oscillation, periodicity, unboundedness and back to monotonicity as the wave speed is varied. Some numerical results are also given to demonstrate the theoretical analysis.

scholarly journals Stability and Hopf-Bifurcating Periodic Solution for Delayed Hopfield Neural Networks withnNeuron

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Haji Mohammad Mohammadinejad ◽  
Mohammad Hadi Moslehi
Keyword(s):  
Neural Networks ◽  
Delay Differential Equations ◽  
Sufficient Conditions ◽  
Neural System ◽  
Hopfield Neural Networks ◽  
Local Asymptotic Stability ◽  
Trivial Equilibrium ◽  
Delay Differential ◽  
Bifurcating Periodic Solution ◽  
Theoretical Results

We consider a system of delay differential equations which represents the general model of a Hopfield neural networks type. We construct some new sufficient conditions for local asymptotic stability about the trivial equilibrium based on the connection weights and delays of the neural system. We also investigate the occurrence of an Andronov-Hopf bifurcation about the trivial equilibrium. Finally, the simulating results demonstrate the validity and feasibility of our theoretical results.

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.

TRAVELING WAVES IN CELLULAR NEURAL NETWORKS

1999 ◽  
Vol 09 (07) ◽  
pp. 1307-1319 ◽  
Author(s):  
CHENG-HSIUNG HSU ◽  
SONG-SUN LIN ◽  
WENXIAN SHEN
Keyword(s):  
Neural Networks ◽  
Traveling Waves ◽  
Traveling Wave ◽  
Shooting Method ◽  
Wave Train ◽  
Traveling Wave Solutions ◽  
Periodic Wave ◽  
Cellular Neural Networks ◽  
Periodic Wave Solutions ◽  
Wave Solutions

In this paper, we study the structure of traveling wave solutions of Cellular Neural Networks of the advanced type. We show the existence of monotone traveling wave, oscillating wave and eventually periodic wave solutions by using shooting method and comparison principle. In addition, we obtain the existence of periodic wave train solutions.

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