Geodesic
Periodic traveling-wave-type solutions in circular chains of unidirectionally coupled equations
Teoretičeskaâ i matematičeskaâ fizika, Tome 175 (2013) no. 1, pp. 62-83

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We consider special systems of ordinary differential equations, so-called circular unidirectional chains. For this class of systems, we develop a new method for studying the existence and stability problems for periodic solutions. A feature of the proposed approach is that some auxiliary systems with delay are used in both seeking cycles and analyzing their stability properties. We illustrate the relevance of this approach with a concrete example of a circular Hopfield neural network.
Keywords: chain of unidirectionally coupled equations, system with delay, Hopfield neural network, periodic solution, asymptotic behavior, stability. 
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     author = {S. D. Glyzin and A. Yu. Kolesov and N. Kh. Rozov},
     title = {Periodic traveling-wave-type solutions in circular chains of unidirectionally coupled equations},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
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     publisher = {mathdoc},
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     number = {1},
     year = {2013},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2013_175_1_a4/}
}
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S. D. Glyzin; A. Yu. Kolesov; N. Kh. Rozov. Periodic traveling-wave-type solutions in circular chains of unidirectionally coupled equations. Teoretičeskaâ i matematičeskaâ fizika, Tome 175 (2013) no. 1, pp. 62-83. http://geodesic.mathdoc.fr/item/TMF_2013_175_1_a4/