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Zh. Vychisl. Mat. Mat. Fiz., 2010, Volume 50, Number 12, Pages 2099–2112 (Mi zvmmf4976)  

This article is cited in 26 scientific papers (total in 27 papers)

A modification of Hutchinson's equation

A. Yu. Kolesova, E. F. Mishchenkob, N. Kh. Rozovc

a Faculty of Mathematics, Yaroslavl State University, Sovetskaya ul. 14, Yaroslavl, 150000 Russia
b Steklov Mathematical Institute, Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119333 Russia
c Faculty of Mechanics and Mathematics, Moscow State University, Moscow, 119991 Russia

Abstract: A new mathematical object is introduced, namely, a scalar nonlinear delay differential-difference equation is considered that is a modification of Hutchinson's equation, which is well known in ecology. The existence and stability of its relaxation self-oscillations are analyzed.

Key words: differential-difference equation, relaxation cycle, asymptotic behavior, stability.

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English version:
Computational Mathematics and Mathematical Physics, 2010, 50:12, 1990–2002

Bibliographic databases:

Document Type: Article
UDC: 519. 624.2
Received: 26.04.2010

Citation: A. Yu. Kolesov, E. F. Mishchenko, N. Kh. Rozov, “A modification of Hutchinson's equation”, Zh. Vychisl. Mat. Mat. Fiz., 50:12 (2010), 2099–2112; Comput. Math. Math. Phys., 50:12 (2010), 1990–2002

Citation in format AMSBIB
\by A.~Yu.~Kolesov, E.~F.~Mishchenko, N.~Kh.~Rozov
\paper A modification of Hutchinson's equation
\jour Zh. Vychisl. Mat. Mat. Fiz.
\yr 2010
\vol 50
\issue 12
\pages 2099--2112
\jour Comput. Math. Math. Phys.
\yr 2010
\vol 50
\issue 12
\pages 1990--2002

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    This publication is cited in the following articles:
    1. D. V. Anosov, S. M. Aseev, R. V. Gamkrelidze, S. P. Konovalov, M. S. Nikol'skii, N. Kh. Rozov, “Evgenii Frolovich Mishchenko (on the 90th anniversary of his birth)”, Russian Math. Surveys, 67:2 (2012), 385–402  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    2. A. Yu. Kolesov, N. Kh. Rozov, “Discrete autowaves in systems of delay differential–difference equations in ecology”, Proc. Steklov Inst. Math., 277 (2012), 94–136  mathnet  crossref  mathscinet  isi
    3. S. D. Glyzin, A. Yu. Kolesov, N. Kh. Rozov, “Discrete autowaves in neural systems”, Comput. Math. Math. Phys., 52:5 (2012), 702–719  mathnet  crossref  mathscinet  isi  elib  elib
    4. S. D. Glyzin, A. Yu. Kolesov, N. Kh. Rozov, “Relaxation self-oscillations in Hopfield networks with delay”, Izv. Math., 77:2 (2013), 271–312  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    5. 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
    6. S. D. Glyzin, A. Yu. Kolesov, N. Kh. Rozov, “Periodic traveling-wave-type solutions in circular chains of unidirectionally coupled equations”, Theoret. and Math. Phys., 175:1 (2013), 499–517  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    7. Kolesov A.Yu. Rozov N.Kh., “New Methods for the Investigation of Periodic Solutions in Ring Systems of Unidirectionally Coupled Oscillators”, Ukr. Math. J., 65:1 (2013), 91–113  crossref  mathscinet  zmath  isi  elib  scopus
    8. Glyzin S.D. Kolesov A.Yu. Rozov N.Kh., “On a Method for Mathematical Modeling of Chemical Synapses”, Differ. Equ., 49:10 (2013), 1193–1210  crossref  mathscinet  zmath  isi  elib  scopus
    9. 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
    10. S. D. Glyzin, A. Yu. Kolesov, N. Kh. Rozov, “The buffer phenomenon in ring-like chains of unidirectionally connected generators”, Izv. Math., 78:4 (2014), 708–743  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    11. Yu. V. Bogomolov, S. D. Glyzin, A. Yu. Kolesov, “O chisle sosuschestvuyuschikh avtovolnovykh reshenii tsepochki diffuzionno svyazannykh ostsillyatorov neironnogo tipa”, Model. i analiz inform. sistem, 21:5 (2014), 162–180  mathnet
    12. 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
    13. Wu G.-Ch., Baleanu D., “Discrete Chaos in Fractional Delayed Logistic Maps”, Nonlinear Dyn., 80:4, SI (2015), 1697–1703  crossref  mathscinet  isi  scopus
    14. 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
    15. 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
    16. Glyzin S.D. Kolesov A.Yu. Rozov N.Kh., “Periodic Two-Cluster Synchronization Modes in Completely Connected Genetic Networks”, Differ. Equ., 52:2 (2016), 157–176  crossref  mathscinet  zmath  isi  elib  scopus
    17. Glyzin S.D., Kolesov A.Yu., Rozov N.Kh., “Self-Sustained Relaxation Oscillations in Time-Delay Neural Systems”, Murphys-Hsfs-2014: 7Th International Workshop on Multi-Rate Processes & Hysteresis (Murphys) & the 2Nd International Workshop on Hysteresis and Slow-Fast Systems (Hsfs), Journal of Physics Conference Series, 727, eds. Klein O., Dimian M., Gurevich P., Knees D., Rachinskii D., Tikhomirov S., IOP Publishing Ltd, 2016, UNSP 012004  crossref  isi  scopus
    18. Azhmyakov V., Ahmed A., Verriest E.I., “On the Optimal Control of Systems Evolving With State Suprema”, 2016 IEEE 55Th Conference on Decision and Control (Cdc), IEEE Conference on Decision and Control, IEEE, 2016, 3617–3623  isi
    19. S. D. Glyzin, A. Yu. Kolesov, E. A. Marushkina, “Relaksatsionnye avtokolebaniya v sisteme iz dvukh sinapticheski svyazannykh impulsnykh neironov”, Model. i analiz inform. sistem, 24:1 (2017), 82–93  mathnet  crossref  mathscinet  elib
    20. M. M. Preobrazhenskaya, “Relaksatsionnye tsikly v modeli sinapticheski vzaimodeistvuyuschikh ostsillyatorov”, Model. i analiz inform. sistem, 24:2 (2017), 186–204  mathnet  crossref  elib
    21. A. Yu. Perevaryukha, “Razrushenie relaksatsionnykh kolebanii v novoi modeli ekstremalnoi dinamiki chislennosti populyatsii”, Vestn. Volgogr. gos. un-ta. Ser. 1, Mat. Fiz., 2017, no. 1(38), 55–65  mathnet  crossref
    22. M. M. Preobrazhenskaya, “Impulsno-refrakternyi rezhim v koltsevoi tsepi sinapticheski svyazannykh ostsillyatorov neironnogo tipa”, Model. i analiz inform. sistem, 24:5 (2017), 550–566  mathnet  crossref  elib
    23. A. Yu. Perevaryukha, “Stsenarii nevynuzhdennoi destruktsii populyatsii v modifikatsii uravneniya Khatchinsona”, Vladikavk. matem. zhurn., 19:4 (2017), 58–69  mathnet
    24. Verriest E.I., Azhmyakov V., “Advances in Optimal Control of Differential Systems With the State Suprema”, 2017 IEEE 56Th Annual Conference on Decision and Control (Cdc), IEEE Conference on Decision and Control, IEEE, 2017  isi
    25. Glyzin S.D. Kolesov A.Yu. Marushkina E.A., “Relaxation Oscillations in a System of Two Pulsed Synaptically Coupled Neurons”, Autom. Control Comp. Sci., 51:7 (2017), 658–665  crossref  isi  scopus
    26. Preobrazhenskaia M.M., “Relaxation Cycles in a Model of Synaptically Interacting Oscillators”, Autom. Control Comp. Sci., 51:7 (2017), 783–797  crossref  mathscinet  isi  scopus
    27. S. D. Glyzin, A. Yu. Kolesov, N. Kh. Rozov, “Samosimmetrichnyi tsikl v sisteme dvukh diffuzionno svyazannykh uravnenii Khatchinsona”, Matem. sb., 210:2 (2019), 24–74  mathnet  crossref  elib
  • Журнал вычислительной математики и математической физики Computational Mathematics and Mathematical Physics
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