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Zh. Vychisl. Mat. Mat. Fiz., 2008, Volume 48, Number 12, Pages 2141–2150 (Mi zvmmf69)  

This article is cited in 27 scientific papers (total in 27 papers)

Local dynamics of equations with large delay

I. S. Kashchenko

Yaroslavl State University, ul. Sovetskaya 14, Yaroslavl, 150000, Russia

Abstract: The local dynamics of a differential equation with large delay is analyzed using the normal forms technique. It is shown that, in the critical cases, families of parabolic equations play the role of infinite-dimensional normal forms. It is demonstrated analytically that even a very simple first-order delay equation can have a complicated dynamical behavior. Methods for constructing classes of stable modes for such equations are described. The proposed methods are extended for the case of secondorder equations.

Key words: dynamics of ordinary differential equations with large delay, normal forms technique, parabolic equations, roots of quasi-polynomials.

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English version:
Computational Mathematics and Mathematical Physics, 2008, 48:12, 2172–2181

Bibliographic databases:

UDC: 519.62
Received: 03.12.2007

Citation: I. S. Kashchenko, “Local dynamics of equations with large delay”, Zh. Vychisl. Mat. Mat. Fiz., 48:12 (2008), 2141–2150; Comput. Math. Math. Phys., 48:12 (2008), 2172–2181

Citation in format AMSBIB
\by I.~S.~Kashchenko
\paper Local dynamics of equations with large delay
\jour Zh. Vychisl. Mat. Mat. Fiz.
\yr 2008
\vol 48
\issue 12
\pages 2141--2150
\jour Comput. Math. Math. Phys.
\yr 2008
\vol 48
\issue 12
\pages 2172--2181

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    This publication is cited in the following articles:
    1. I. S. Kaschenko, “Normalizatsiya uravneniya s lineino raspredelennym zapazdyvaniem”, Model. i analiz inform. sistem, 16:4 (2009), 109–116  mathnet
    2. E. V. Grigoreva, I. S. Kaschenko, S. A. Kaschenko, “Multistabilnost v modeli lazera s bolshim zapazdyvaniem”, Model. i analiz inform. sistem, 17:2 (2010), 17–27  mathnet
    3. Kashchenko I.S., Kashchenko S.A., “Rapidly oscillating spatially inhomogeneous structures in coherent nonlinear optical systems”, Dokl. Math., 82:3 (2010), 850–853  crossref  mathscinet  zmath  isi  elib  elib  scopus
    4. Kashchenko I.S., “Multistability in Nonlinear Parabolic Systems with Low Diffusion”, Dokl. Math., 82:3 (2010), 878–881  crossref  mathscinet  zmath  isi  elib  scopus
    5. I. S. Kaschenko, “Normalizatsiya v sisteme s dvumya blizkimi bolshimi zapazdyvaniyami”, Nelineinaya dinam., 6:1 (2010), 169–180  mathnet
    6. Kashchenko I.S., “Dynamics of an equation with a large coefficient of delay control”, Dokl. Math., 83:2 (2011), 258–261  crossref  mathscinet  zmath  isi  elib  elib  scopus
    7. Grigoreva E.V., Kaschenko I.S., Kaschenko S.A., “Gipermultistabilnost v modelyakh lazerov s bolshim zapazdyvaniem”, Izvestiya vysshikh uchebnykh zavedenii. Prikladnaya nelineinaya dinamika, 19:3 (2011), 1–15  zmath  elib
    8. S. A. Kaschenko, “Dinamika kvazilineinoi kraevoi zadachi, obobschayuschei uravnenie s bolshim zapazdyvaniem”, Model. i analiz inform. sistem, 18:1 (2011), 28–31  mathnet
    9. I. S. Kaschenko, “Dinamicheskie svoistva odnoi modeli passivnogo zakhvata mod”, Model. i analiz inform. sistem, 18:1 (2011), 32–36  mathnet
    10. I. S. Kaschenko, “Lokalnaya dinamika uravneniya s dlitelnym eksponentsialno raspredelennym zapazdyvaniem”, Model. i analiz inform. sistem, 18:3 (2011), 42–49  mathnet
    11. Kashchenko I.S., “Asymptotic study of the corporate dynamics of systems of equations coupled by delay control”, Dokl. Math., 85:2 (2012), 163–166  crossref  mathscinet  mathscinet  zmath  isi  elib  elib  scopus
    12. Gnoenskii L.S., Shishkin E.A., “On the operation accuracy of linear stationary delay control systems”, Journal of Computer and Systems Sciences International, 51:1 (2012), 1–21  crossref  mathscinet  zmath  isi  elib  scopus
    13. E. V. Grigoreva, I. S. Kaschenko, S. A. Kaschenko, “Kvazinormalnye formy dlya uravnenii Lenga–Kobayashi s bolshim koeffitsientom upravleniya”, Model. i analiz inform. sistem, 20:1 (2013), 18–29  mathnet
    14. I. S. Kashchenko, S. A. Kashchenko, “Local dynamics of an equation with large delay and distributed deviation of the space variable”, Siberian Math. J., 55:2 (2014), 254–261  mathnet  crossref  mathscinet  isi
    15. Kashchenko I., “Normalization of a System With Two Large Delays”, Int. J. Bifurcation Chaos, 24:8 (2014), 1440021  crossref  mathscinet  zmath  isi  scopus
    16. Kashchenko I.S., “Local Dynamics of An Equation With Distributed Delay”, Differ. Equ., 50:1 (2014), 15–24  crossref  mathscinet  zmath  isi  elib  scopus
    17. Kashchenko I.S., Kashchenko S.A., “Local Dynamics of An Equation With a Large State-Dependent Delay”, Dokl. Math., 92:2 (2015), 581–584  crossref  mathscinet  zmath  isi  elib  scopus
    18. Kashchenko I., Kaschenko S., “Normal and Quasinormal Forms For Systems of Difference and Differential-Difference Equations”, Commun. Nonlinear Sci. Numer. Simul., 38 (2016), 243–256  crossref  mathscinet  isi  elib  scopus
    19. Sergey A. Kashchenko, “The Dynamics of Second-order Equations with Delayed Feedback and a Large Coefficient of Delayed Control”, Regul. Chaotic Dyn., 21:7-8 (2016), 811–820  mathnet  crossref
    20. Kashchenko I.S., “Local dynamics of an equation with two large different-order delays”, Dokl. Math., 94:2 (2016), 578–582  crossref  zmath  isi  elib  scopus
    21. Kashchenko S.A., “Dynamics of the logistic equation with two delays”, Differ. Equ., 52:5 (2016), 538–548  crossref  mathscinet  zmath  isi  elib  scopus
    22. I. S. Kashchenko, “Local Dynamics of a Second-Order Differential-Difference Equation with Large Delay at the First Derivative”, Math. Notes, 101:2 (2017), 379–381  mathnet  crossref  crossref  mathscinet  isi  elib
    23. S. A. Kaschenko, “O bifurkatsiyakh pri malykh vozmuscheniyakh v logisticheskom uravnenii s zapazdyvaniem”, Model. i analiz inform. sistem, 24:2 (2017), 168–185  mathnet  crossref  elib
    24. M. I. Balakin, N. M. Ryskin, “Bifurkatsionnyi mekhanizm formirovaniya razvitoi multistabilnosti v ostsillyatore van der Polya s zapazdyvayuschei obratnoi svyazyu”, Nelineinaya dinam., 13:2 (2017), 151–164  mathnet  crossref  elib
    25. I. S. Kashchenko, S. A. Kashchenko, “Analysis of local dynamics of difference and close to them differential-difference equations”, Russian Math. (Iz. VUZ), 62:9 (2018), 24–34  mathnet  crossref  isi
    26. E. V. Grigoreva, S. A. Kaschenko, D. V. Glazkov, “Osobennosti lokalnoi dinamiki modeli optiko-elektronnogo ostsillyatora s zapazdyvaniem”, Model. i analiz inform. sistem, 25:1 (2018), 71–82  mathnet  crossref  elib
    27. Grigorieva E.V., Kashchenko S.A., Glazkov D.V., “Local Dynamics of a Model of An Opto-Electronic Oscillator With Delay”, Autom. Control Comp. Sci., 52:7 (2018), 700–707  crossref  mathscinet  isi  scopus
  • Журнал вычислительной математики и математической физики Computational Mathematics and Mathematical Physics
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