EXISTENCE OF MONOTONIC TRAVELING WAVES IN LATTICE DYNAMICAL SYSTEMS

2005 ◽  
Vol 15 (08) ◽  
pp. 2375-2394 ◽  
Author(s):  
CHENG-HSIUNG HSU ◽  
SUH-YUH YANG
Keyword(s):  
Neural Networks ◽  
Dynamical Systems ◽  
Standing Wave ◽  
Upper And Lower Solutions ◽  
Traveling Wave Solutions ◽  
Cellular Neural Networks ◽  
Iteration Scheme ◽  
Monotone Iteration ◽  
Wave Solutions ◽  
Monotone Iteration Scheme

This paper deals with the existence of monotonic traveling and standing wave solutions for a certain class of lattice differential equations. Employing the techniques of monotone iteration coupled with the concept of upper and lower solutions in the theory of monotone dynamical systems, we can classify the monotonic traveling wave solutions with various asymptotic boundary conditions. For the case of zero wave speed, a novel discrete monotone iteration scheme is established for proving the existence of monotonic standing wave solutions. Applications are made to several models including cellular neural networks, original and modified RTD-based cellular neural networks. Numerical simulations of the monotone iteration schemes are also given.

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.

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.

TRAVELING WAVE SOLUTIONS FOR DISCRETE-TIME MODEL OF DELAYED CELLULAR NEURAL NETWORKS

2013 ◽  
Vol 23 (06) ◽  
pp. 1350107 ◽  
Author(s):  
CHENG-HSIUNG HSU ◽  
JIAN-JHONG LIN
Keyword(s):  
Neural Networks ◽  
Discrete Time ◽  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
Cellular Neural Networks ◽  
Time Model ◽  
Discrete Time Model ◽  
Characteristic Roots ◽  
Wave Solutions ◽  
Existence And Stability

The aim of this work is to study the existence and stability of traveling wave solutions for discrete-time model of delayed cellular neural networks distributed in the one-dimensional integer lattice ℤ1. Since the dynamics of each given cell depends on its left and right neighboring cells, it is not easy to construct the traveling wave solutions. Using the method of step along with positive characteristic roots of the equations, we successfully prove the existence of traveling wave solutions. Moreover, we show that all the traveling wave solutions are unstable. We also provide some numerical results to support our results, and point out the different structures of traveling wave solutions between the continuous-time and discrete-time models.

scholarly journals Traveling Wave Solutions in a Reaction-Diffusion Epidemic Model

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Sheng Wang ◽  
Wenbin Liu ◽  
Zhengguang Guo ◽  
Weiming Wang
Keyword(s):  
Traveling Wave ◽  
Epidemic Model ◽  
Wave Speed ◽  
Upper And Lower Solutions ◽  
Traveling Wave Solutions ◽  
Reaction Diffusion ◽  
Monotone Iteration ◽  
Wave Solutions ◽  
Minimal Wave Speed

We investigate the traveling wave solutions in a reaction-diffusion epidemic model. The existence of the wave solutions is derived through monotone iteration of a pair of classical upper and lower solutions. The traveling wave solutions are shown to be unique and strictly monotonic. Furthermore, we determine the critical minimal wave speed.

scholarly journals Minimal Wave Speed in a Predator-Prey System with Distributed Time Delay

2018 ◽  
Vol 2018 ◽  
pp. 1-8
Author(s):  
Fuzhen Wu ◽  
Dongfeng Li
Keyword(s):  
Time Delay ◽  
Traveling Wave ◽  
Wave Speed ◽  
Upper And Lower Solutions ◽  
Traveling Wave Solutions ◽  
Predator Prey ◽  
Wave Solutions ◽  
Prey System ◽  
Minimal Wave Speed ◽  
Distributed Time Delay

This paper is concerned with the minimal wave speed of traveling wave solutions in a predator-prey system with distributed time delay, which does not satisfy comparison principle due to delayed intraspecific terms. By constructing upper and lower solutions, we obtain the existence of traveling wave solutions when the wave speed is the minimal wave speed. Our results complete the known conclusions and show the precisely asymptotic behavior of traveling wave solutions.

scholarly journals Instability of standing waves for non-linear Schrödinger-type equations

1988 ◽  
Vol 8 (8) ◽  
pp. 119-138 ◽  
Keyword(s):  
Dynamical Systems ◽  
Standing Wave ◽  
Optical Waveguides ◽  
Standing Waves ◽  
Positive Eigenvalue ◽  
Shooting Argument ◽  
Wave Solutions ◽  
Non Linear ◽  
Linearized Operator

AbstractA theorem is proved giving a condition under which certain standing wave solutions of non-linear Schrödinger-type equations are linearly unstable. The eigenvalue equations for the linearized operator at the standing wave can be analysed by dynamical systems methods. A positive eigenvalue is then shown to exist by means of a shooting argument in the space of Lagrangian planes. The theorem is applied to a situation arising in optical waveguides.

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