Diversity of Traveling Wave Solutions in Delayed Cellular Neural Networks

Abstract

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 for nonlinear cellular neural networks with distributed delays

“…Thus, Theorems 1.1 and 1.4 improve and cover Theorems 3.1 and 3.2 in [14]. (4) Authors in [6] investigated the existence of traveling waves of (1.3) with a piecewise-linear output…”

“…This approach has been used for some special output functions (e.g. a piecewise-linear output function (1.2) (see, [7,8,11,6,10,21]) and nonlinear nondecreasing output functions (1.5)-(1.6) (see, [14])). In the present paper, our approach is to use Schauder's fixed point theorem combining upper-lower solutions, which is different from the monotone iteration technique.…”

“…The real constant coefficients a i , α and β i of the output function f constitute the so-called space-invariant template that measures the synaptic weights of self-feedback and with its nearest m left-right neighbors. We refer to the readers to the surveys [1][2][3][6][7][8] for more details about the model (1.1). For a piecewise-linear output function given by (1.2) authors in [8] obtained the existence of monotone traveling waves for (1.1) with m = 1, a 1 = 0, α > 0 and β 1 > 0 by using the monotone iteration scheme.…”

“…In models of electronic neural networks, the dynamics of each given cell depends on itself and its nearest m left-right neighbors where delays exist in leftright neighborhood interactions due to, for example, finite switching speed and finite velocity of signal transmission (see [1]). More precisely, authors in [6] investigated the CNN model with a piecewiselinear output function (1.2) and distributed delays present in left-right neighborhood interactions (for short, DCNN model):…”

“…It is well known that authors in [6,9,21] only investigated CNN models with finite distributed delays and piecewise-linear output functions. In [14], authors further considered CNN model (1.4) with finite distributed delays and a nonlinear nondecreasing output function.…”