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

Ya Wen Chang*, Juang Jonq, Chin Lung Li

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

1 Scopus citations


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 fixing the problems have some discrepancies since they only give conditions for which F is expanding "locally". In this paper, we give sufficient 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.

Original languageEnglish
Pages (from-to)1969-1983
Number of pages15
JournalInternational Journal of Bifurcation and Chaos
Issue number6
StatePublished - 1 Jan 2007


  • Cellular neural networks
  • Snap-back repellers
  • Traveling waves


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