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Zh. Vychisl. Mat. Mat. Fiz., 2009, Volume 49, Number 1, Pages 76–89 (Mi zvmmf53)  

This article is cited in 11 scientific papers (total in 11 papers)

Extremal dynamics of the generalized Hutchinson equation

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

a Faculty of Mathematics, Yaroslavl State University, ul. Sovetskaya 14, Yaroslavl, 150000, Russia
b Faculty of Mechanics and Mathematics, Moscow State University, Moscow, 119992, Russia

Abstract: A scalar nonlinear differential-difference equation with two delays that generalizes Hutchinson's equation is considered. The bifurcation of self-oscillations of this equation from the zero equilibrium is studied in the extremal situation when one delay is asymptotically large while the other parameters are on the order of unity. Analytical methods combined with numerical techniques are used to show that the well-known buffer phenomenon occurs in the equation in this case. This means that an arbitrary finite number of different attractors coexist in the phase space of the equation with suitably chosen parameters.

Key words: differential-difference equation, bifurcation, quasi-normal form, buffer phenomenon.

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English version:
Computational Mathematics and Mathematical Physics, 2009, 49:1, 71–83

Bibliographic databases:

UDC: 519.624.2
Received: 13.02.2008
Revised: 20.03.2008

Citation: S. D. Glyzin, A. Yu. Kolesov, N. Kh. Rozov, “Extremal dynamics of the generalized Hutchinson equation”, Zh. Vychisl. Mat. Mat. Fiz., 49:1 (2009), 76–89; Comput. Math. Math. Phys., 49:1 (2009), 71–83

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Glyzin S.D., Kolesov A.Yu., Rozov N.Kh., “Buffer phenomenon in neurodynamics”, Dokl. Math., 85:2 (2012), 297–300  crossref  mathscinet  zmath  isi  elib  elib  scopus
    2. S. D. Glyzin, A. Yu. Kolesov, N. Kh. Rozov, “Modeling the Bursting Effect in Neuron Systems”, Math. Notes, 93:5 (2013), 676–690  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    3. S. A. Kaschenko, “Relaksatsionnye kolebaniya v sisteme s zapazdyvaniyami, modeliruyuschei zadachu «khischnik–zhertva»”, Model. i analiz inform. sistem, 20:1 (2013), 52–98  mathnet
    4. A. S. Bobok, S. D. Glyzin, A. Yu. Kolesov, “Ekstremalnaya dinamika sistemy trekh odnonapravlenno svyazannykh singulyarno vozmuschennykh uravnenii iz neirodinamiki”, Model. i analiz inform. sistem, 20:5 (2013), 158–167  mathnet
    5. S. D. Glyzin, E. A. Marushkina, “Relaksatsionnye tsikly v obobschennoi neironnoi modeli s dvumya zapazdyvaniyami”, Model. i analiz inform. sistem, 20:6 (2013), 179–199  mathnet
    6. S. V. Aleshin, S. A. Kaschenko, “Lokalnaya dinamika logisticheskogo uravneniya, soderzhaschego zapazdyvanie”, Model. i analiz inform. sistem, 21:1 (2014), 73–88  mathnet
    7. M. M. Preobrazhenskaya, “Primenenie metoda kvazinormalnykh form k matematicheskoi modeli otdelnogo neirona”, Model. i analiz inform. sistem, 21:5 (2014), 38–48  mathnet
    8. S. D. Glyzin, A. Yu. Kolesov, N. Kh. Rozov, “Self-excited relaxation oscillations in networks of impulse neurons”, Russian Math. Surveys, 70:3 (2015), 383–452  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    9. Kaschenko S.A., “Relaxation Oscillations in a System With Delays Modeling the Predator-Prey Problem”, Autom. Control Comp. Sci., 49:7 (2015), 547–581  crossref  isi  elib  scopus
    10. Glyzin S.D. Kolesov A.Yu. Preobrazhenskaia M.M., “Existence and Stability of Periodic Solutions of Quasi-Linear Korteweg - de Vries Equation”, V International Conference on Problems of Mathematical and Theoretical Physics and Mathematical Modelling, Journal of Physics Conference Series, 788, IOP Publishing Ltd, 2017, UNSP 012016  crossref  isi  scopus
    11. Glyzin S.D., Kashchenko S.A., Preobrazhenskaia M.M., Computer Simulations in Physics and Beyond (Csp2017), Journal of Physics Conference Series, 955, IOP Publishing Ltd, 2018  crossref  isi  scopus
  • Журнал вычислительной математики и математической физики Computational Mathematics and Mathematical Physics
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