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TMF, 2013, Volume 175, Number 1, Pages 62–83 (Mi tmf8441)  

This article is cited in 14 scientific papers (total in 14 papers)

Periodic traveling-wave-type solutions in circular chains of unidirectionally coupled equations

S. D. Glyzina, A. Yu. Kolesova, N. Kh. Rozovb

a Yaroslavl State University, Yaroslavl, Russia
b Lomonosov Moscow State University, Moscow, Russia

Abstract: 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

DOI: https://doi.org/10.4213/tmf8441

Full text: PDF file (536 kB)
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English version:
Theoretical and Mathematical Physics, 2013, 175:1, 499–517

Bibliographic databases:

Received: 14.11.2012
Revised: 23.11.2012

Citation: S. D. Glyzin, A. Yu. Kolesov, N. Kh. Rozov, “Periodic traveling-wave-type solutions in circular chains of unidirectionally coupled equations”, TMF, 175:1 (2013), 62–83; Theoret. and Math. Phys., 175:1 (2013), 499–517

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. S. D. Glyzin, A. Yu. Kolesov, N. Kh. Rozov, “Autowave processes in continual chains of unidirectionally coupled oscillators”, Proc. Steklov Inst. Math., 285 (2014), 81–98  mathnet  crossref  crossref  isi  elib  elib
    2. S. D. Glyzin, A. Yu. Kolesov, N. Kh. Rozov, “Buffering effect in continuous chains of unidirectionally coupled generators”, Theoret. and Math. Phys., 181:2 (2014), 1349–1366  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib  elib
    3. S. D. Glyzin, A. Yu. Kolesov, N. Kh. Rozov, “Buffer phenomenon and chaos in circular chains of unidirectionally coupled oscillators”, Dokl. Math., 90:1 (2014), 509–512  crossref  mathscinet  zmath  isi  scopus
    4. S. D. Glyzin, A. Yu. Kolesov, N. Kh. Rozov, “Avtovolnovye protsessy v koltsevoi neironnoi tsepi s odnonapravlennoi svyazyu”, Model. i analiz inform. sistem, 22:3 (2015), 404–419  mathnet  crossref  mathscinet  elib
    5. A. Yu. Kolesov, N. Kh. Rozov, V. A. Sadovnichii, “Periodic solutions of travelling-wave type in circular gene networks”, Izv. Math., 80:3 (2016), 523–548  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    6. S. D. Glyzin, A. Yu. Kolesov, N. Kh. Rozov, “Buffering in cyclic gene networks”, Theoret. and Math. Phys., 187:3 (2016), 935–951  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    7. S. D. Glyzin, A. Yu. Kolesov, N. Kh. Rozov, “Traveling-wave solutions in continuous chains of unidirectionally coupled oscillators”, VI International Conference Problems of Mathematical Physics and Mathematical Modelling, Journal of Physics Conference Series, 937, IOP Publishing Ltd, 2017, UNSP 012015  crossref  isi  scopus
    8. S. D. Glyzin, A. Yu. Kolesov, N. Kh. Rozov, “Stable relaxation cycle in a bilocal neuron model”, Differ. Equ., 54:10 (2018), 1285–1309  crossref  crossref  mathscinet  zmath  isi  elib  scopus
    9. S. D. Glyzin, A. Yu. Kolesov, N. Kh. Rozov, “Quasi-stable structures in circular gene networks”, Comput. Math. Math. Phys., 58:5 (2018), 659–679  mathnet  crossref  crossref  isi  elib
    10. S. D. Glyzin, A. Yu. Kolesov, N. Kh. Rozov, “A self-symmetric cycle in a system of two diffusely connected Hutchinson's equations”, Sb. Math., 210:2 (2019), 184–233  mathnet  crossref  crossref  adsnasa  isi  elib
    11. A. Yu. Kolesov, N. Kh. Rozov, V. A. Sadovnichii, “On a Mathematical Model of Biological Self-Organization”, Proc. Steklov Inst. Math., 304 (2019), 160–189  mathnet  crossref  crossref  isi  elib
    12. Glyzin S.D., Kolesov A.Yu., Rozov N.Kh., International Conference on Computer Simulation in Physics and Beyond, Journal of Physics Conference Series, 1163, ed. Shchur L., IOP Publishing Ltd, 2019  crossref  isi
    13. Preobrazhenskaia M.M., International Conference on Computer Simulation in Physics and Beyond, Journal of Physics Conference Series, 1163, ed. Shchur L., IOP Publishing Ltd, 2019  crossref  isi
    14. S. D. Glyzin, A. Yu. Kolesov, N. Kh. Rozov, “Novyi podkhod k modelirovaniyu gennykh setei”, Model. i analiz inform. sistem, 26:3 (2019), 365–404  mathnet  crossref
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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