Snap-back repellers and chaotic traveling waves in one-dimensional cellular neural networks

Abstract

In 1998, Chen et al. [1998] found an error in Marotto's paper [1978]. It was pointed out by them that the existence of an expanding fixed point z of a map F in Br(z), the ball of radius r with center at z does not necessarily imply that F is expanding in Br( z). Subsequent efforts (see e. g. [Chen et al., 1998; Lin et al., 2002; Li & Chen, 2003]) in. xing the problems have some discrepancies since they only give conditions for which F is expanding "locally". In this paper, we give suficient conditions so that F is "globally" expanding. This, in turn, gives more satisfying definitions of a snap-back repeller. We then use those results to show the existence of chaotic backward traveling waves in a discrete time analogy of one-dimensional Cellular Neural Networks (CNNs). Some computer evidence of chaotic traveling waves is also given.