Stability of Traveling Waves Solutions for Nonlinear Cellular Neural Networks with Distributed Delays

• Published in: Journal of systems science and complexity Vol. 35; no. 1; pp. 18 - 31
• Main Authors: Guo, Yingxin, Ge, Shuzhi Sam, Arbi, Adnène
• Format: Journal Article
• Language: English
• Published: Beijing Academy of Mathematics and Systems Science, Chinese Academy of Sciences 01-02-2022; Springer Nature B.V
• Subjects: Cellular communication; Complex Systems; Control; Differential calculus; Energy methods; Mathematics; Mathematics and Statistics; Mathematics of Computing; Neural networks; Operations Research/Decision Theory; Perturbation; Stability; Statistics; Systems Theory; Traveling waves; Cellular delayed neural networks; stability analysis; traveling waves solutions; comparison principle
• ISSN: 1009-6124; 1559-7067
• DOI: 10.1007/s11424-021-0180-7

This paper investigates the exponential stability of traveling wave solutions for nonlinear delayed cellular neural networks. As a continuity of the past work (Wu and Niu, 2016; Yu, et al., 2011) on the existence and uniqueness of the traveling wave solutions, it is very reasonable and interesting to consider the exponential stability of the traveling wave solutions. By the weighted energy method, comparison principle and the first integral mean value theorem, this paper proves that, for all monotone traveling waves with the wave speed
c
<
c
1
∗
<
0
or
c
>
c
2
∗
>
0
, the solutions converge time-exponentially to the corresponding traveling waves, when the initial perturbations decay at some fields.