Advances in Difference Equations
Volume 2009 (2009), Article ID 243245, 29 pages
Research Article

Doubly Periodic Traveling Waves in a Cellular Neural Network with Linear Reaction

Department of Mathematics, Tsing Hua University, Hsinchu 30043, Taiwan

Received 4 June 2009; Accepted 13 October 2009

Academic Editor: Roderick Melnik

Copyright © 2009 Jian Jhong Lin and Sui Sun Cheng. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Szekeley observed that the dynamic pattern of the locomotion of salamanders can be explained by periodic vector sequences generated by logical neural networks. Such sequences can mathematically be described by “doubly periodic traveling waves” and therefore it is of interest to propose dynamic models that may produce such waves. One such dynamic network model is built here based on reaction-diffusion principles and a complete discussion is given for the existence of doubly periodic waves as outputs. Since there are 2 parameters in our model and 4 a priori unknown parameters involved in our search of solutions, our results are nontrivial. The reaction term in our model is a linear function and hence our results can also be interpreted as existence criteria for solutions of a nontrivial linear problem depending on 6 parameters.