Piecewise Two-Dimensional Maps and Applications to Cellular Neural Networks

Here a0 < 0, a1, a−1 > 1, b > 0, and c1 ∈ R is a biased term. The graph of F is given in Fig. 1. The motivation for studying such a map is, in part, due to the form of the map is a generalized version of Lozi map [Lozi, 1978]. More importantly, the map arises in the study of complexity of a set of bounded stable stationary solutions of one-dimensional Cellular Neural Networks (CNNs) (see e.g. [Chua, 1998; Chua & Yang, 1998a, 1998b]). In this paper, we first prove a theorem which states that a semiconjugate condition for T implies the existence of Smale horseshoe. Second, we apply the theorem to show the spatial chaos of one-dimensional Cellular Neural Networks. Such CNNs are of the form (e.g. [Ban et al., 2002, 2001; Hsu, 2000]).