Vol. 4, No. 1 Modern Applied Science 2 Some Sufficient Conditions for Oscillation of Symmetric Cellular Neural Networks with Delay Sarhan M. Musa College of Engineering, Prairie View A&M University Prairie View, TX 77447, USA Tel: 936-261-9860 E-mail: smmusa@pvamu.edu Abstract In this paper, some sufficient conditions for oscillation of symmetric cellular neural networks with delay (DCNNs) are introduced. These conditions impose constrains on the system of dynamic state equations when the cells are working in linear region 1 () () ( ) 0 xt Hx t Axt , where the coefficients H and 1 A are real 2 2 matrix 22 R , R , and 21 ( ), ( ) xt xt R are the state vector and its derivative at time t . Keywords: Cellular neural networks with delay (DCNNs), Oscillation 1. Introduction Cellular neural networks (CNNs) were introduced by (Chua, L O et al, 1988, p. 1257-1272; Chua, L O et al, 1988, 1273-1290); they found important applications in pattern recognition and signal processing, especially in statistic image treatment. The stability of CCNs has been investigated in (Chua, L O et al, 1990, p. 1520-1527; Zou, F et al, 1991, p. 38:675-677; Savaci, F A et al, 1992, p. 240 – 245). Processing of moving images requires the introduction of a delay in the signal transmitted among the cells, which led to introducing delayed cellular neural networks (DCNNs) by (Chua, L O et al, 1990, p. 12-25; Chua, L O, 1992, p. 449-459) and the consideration of their dynamic behavior in (Roska, T et al, 1992, p. 487-490). The most important uses of DCNNs (signal and image processing, non-linear/transcendental equation solving and so on) rely on stability. The stability of DCCNs has been investigated in many papers (Liao, T L et al, 1999, p. 1347-1349; Arik, S et al, 2000, p. 571-574; Liao, T L et al; 2000, p. 1481-1484; Takahshi, N N, 2000, p. 793-799; Arik, S, 2002, p. 1211-1214; Arik, S, 2003, p. 156-160). However, this work presents some sufficient conditions for oscillation of symmetric cellular neural networks with delay (DCNNs) for the dynamic state equations. 2. Cellular neural networks with delay A CNN is a massively parallel computing architecture made of simple processing elements (cells) which are locally connected. A cell is the basic circuit unit containing linear and nonlinear circuit elements, which are linear resistors, linear capacitors, linear and nonlinear controlled sources and independent sources. Any cell in a CNN is connected only to its neighbor cells. Due to the propagations effects on continuous time dynamics of CCNs, the cells that not directly connected to each other can affect each other indirectly (Chua, L O et al, 1988, p. 985- 988). These properties of the CCNs are similar to the cellular automata. In this paper we refer to Civalleri and Gilli study on some stability properties of DCNNs with consideration to reciprocal and non-reciprocal of the networks. They showed that a symmetric DCNN can become unstable if the delay is suitably chosen and they present a sufficient condition to assure the complete stability (Civalleri, P P et al, 1992, p. 94 -99; Civalleri, P P, 1993, p. 157-165). This paper gives new sufficient conditions for oscillation of symmetric cellular neural networks with a delay . These conditions impose constrain on the system of dynamic state equations, () x t , when the cells are working in linear region. Now, suppose the cells of the DCNNs are working in linear region, i.e.: (in matrix notation) 1 () () ( ) 0 xt Hx t Axt , (1)