TY - JOUR

T1 - Snap-back repellers and chaotic traveling waves in one-dimensional cellular neural networks

AU - Chang, Ya Wen

AU - Jonq, Juang

AU - Li, Chin Lung

PY - 2007/6

Y1 - 2007/6

N2 - 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.

AB - 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.

KW - Cellular neural networks

KW - Snap-back repellers

KW - Traveling waves

UR - http://www.scopus.com/inward/record.url?scp=34548405263&partnerID=8YFLogxK

U2 - 10.1142/S0218127407018154

DO - 10.1142/S0218127407018154

M3 - Article

AN - SCOPUS:34548405263

SN - 0218-1274

VL - 17

SP - 1969

EP - 1983

JO - International Journal of Bifurcation and Chaos

JF - International Journal of Bifurcation and Chaos

IS - 6

ER -