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A relay Mackey—Glass model with two delays

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Abstract

We study the dynamics of the generalized Mackey—Glass equation with two delays using the large-parameter method. After a special exponential replacement of variables, a singularly perturbed equation is obtained, for which we construct a meaningful limit object, a differential—difference relay equation with two delays. We prove that the relay equation has a simple periodic solution with one interval on the period where the solution is positive. To illustrate the obtained result, we numerically analyze the original singularly perturbed equation, for which we find a solution near the periodic solution of the limit relay equation.

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References

  1. M. C. Mackey and L. Glass, “Oscillation and chaos in physiological control systems,” Science197, 287–289 (1977).

    Article  ADS  Google Scholar 

  2. L. Glass and M. Mackey, From Clocks to Chaos: The Rhythms of Life, Princeton Univ. Press, Princeton (1988).

    MATH  Google Scholar 

  3. S. D. Glyzin, A. Yu. Kolesov, and N. Kh. Rozov, “An approach to modeling artificial gene networks,” Theor. Math. Phys.194, 471–490 (2018).

    Article  MathSciNet  Google Scholar 

  4. S. D. Glyzin, A. Yu. Kolesov, and N. Kh. Rozov, “Quasi-stable structures in circular gene networks,” Comput. Math. Math. Phys.58, 659–679 (2018).

    Article  MathSciNet  Google Scholar 

  5. L. Junges and J. A. C. Gallas, “Intricate routes to chaos in the Mackey—Glass delayed feedback system,” Phys. Lett. A376, 2109–2116 (2012).

    Article  ADS  Google Scholar 

  6. L. Berezansky and E. Braverman, “Mackey—Glass equation with variable coefficients,” Comput. Math. Appl.51, 1–16 (2006).

    Article  MathSciNet  Google Scholar 

  7. H. Su, X. Ding, and W. Li, “Numerical bifurcation control of Mackey—Glass system,” Appl. Math. Model.35, 3460–3472 (2011).

    Article  MathSciNet  Google Scholar 

  8. E. Liz, E. Trofimchuk, and S. Trofimchuk, “Mackey—Glass type delay differential equations near the boundary of absolute stability,” J. Math. Anal. Appl.275, 747–760 (2002).

    Article  MathSciNet  Google Scholar 

  9. X.-M. Wu, J.-W. Li, and H.-Q. Zhou, “A necessary and sufficient condition for the existence of positive periodic solutions of a model of hematopoiesis,” Comput. Math. Appl.54, 840–849 (2007).

    Article  MathSciNet  Google Scholar 

  10. E. P. Kubyshkin and A. R. Moryakova, “Bifurcation of periodic solutions of the Mackey—Glass equation,” Model. Anal. Inform. Sist.23, 784–803 (2016).

    Article  MathSciNet  Google Scholar 

  11. P. Grassberger and I. Procaccia, “Measuring the strangeness of strange attractors,” Phys. D9, 189–208 (1983).

    Article  MathSciNet  Google Scholar 

  12. A. Namajūnas, K. Pyragas, and A. Tamaševičius, “Stabilization of an unstable steady state in a Mackey—Glass system,” Phys. Lett. A, 204, 255–262 (1995)

    Article  ADS  Google Scholar 

  13. A. Namajūnas, K. Pyragas, and A. Tamaševičius, “An electronic analog of the Mackey—Glass system,” Phys. Lett. A, 201, 42–46 (1995).

    Article  ADS  Google Scholar 

  14. M. Tateno and A. Uchida, “Nonlinear dynamics and chaos synchronization in Mackey—Glass electronic circuits with multiple time-delayed feedback,” Nonlin. Theory Appl., IEICE3, 155–164 (2012).

    Article  ADS  Google Scholar 

  15. P. Amil, C. Cabeza, and A. C. Marti, “Exact discrete-time implementation of the Mackey—Glass delayed model,” IEEE Trans. on Circuits and Systems II: Express Briefs62, 681–685 (2015).

    Article  Google Scholar 

  16. P. Amil, C. Cabeza, C. Masoller, and A. Marti, “Organization and identification of solutions in the time-delayed Mackey—Glass model,” Chaos25, 043112 (2015); arXiv:1412.6360 (2014).

    Article  ADS  MathSciNet  Google Scholar 

  17. E. M. Shahverdiev, R. A. Nuriev, R. H. Hashimov, and K. A. Shore, “Chaos synchronization between the Mackey—Glass systems with multiple time delays,” Chaos, Solitons Fractals29, 854–861 (2006).

    Article  ADS  MathSciNet  Google Scholar 

  18. S. Sano, A. Uchida, S. Yoshimori, and R. Roy, “Dual synchronization of chaos in Mackey—Glass electronic circuits with time-delayed feedback,” Phys. Rev. E75, 016207 (2007).

    Article  ADS  Google Scholar 

  19. A. Wan and J. Wei, “Bifurcation analysis of Mackey—Glass electronic circuits model with delayed feedback,” Nonlinear Dynam.57, 85–96 (2009).

    Article  MathSciNet  Google Scholar 

  20. E. F. Mishchenko and N. Kh. Rozov, Differential Equations with Small Parameters and Relaxation Oscillations [in Russian], Nauka, Moscow (1975); English transl. (Math. Conc. Meth. Sci. Engin., Vol. 13), Plenum, New York (1980).

    MATH  Google Scholar 

  21. S. A. Kashchenko, “Stationary regimes of the equation describing fluctuations of an insect population,” Sov. Phys. Dokl.28, 935–936 (1983).

    ADS  MATH  Google Scholar 

  22. S. A. Kashchenko, “Steady states of a delay differential equation of an insect population’s dynamics,” Autom. Control Comput. Sci.48, 445–457 (2015).

    Article  Google Scholar 

  23. A. Yu. Kolesov, E. F. Mishchenko, and N. Kh. Rozov, “Relay with delay and its C1-approximation,” Proc. Steklov Inst. Math.216, 119–146 (1997).

    MATH  Google Scholar 

  24. S. D. Glyzin, A. Yu. Kolesov, and N. Kh. Rozov, “The theory of nonclassical relaxation oscillations in singularly perturbed delay systems,” Sb. Math.205, 781–842 (2014).

    Article  MathSciNet  Google Scholar 

  25. A. Yu. Kolesov and Yu. S. Kolesov, “Relaxational oscillations in mathematical models of ecology,” Proc. Steklov Inst. Math.199, 1–126 (1995).

    MATH  Google Scholar 

  26. A. Yu. Kolesov, E. F. Mishchenko, and N. Kh. Rozov, “A modification of Hutchinson’s equation,” Comput. Math. Math. Phys.50, 1990–2002 (2010).

    Article  MathSciNet  Google Scholar 

  27. A. A. Kashchenko, “Multistability in a system of two coupled oscillators with delayed feedback,” J. Differ. Equ.266, 562–579 (2019).

    Article  ADS  MathSciNet  Google Scholar 

  28. S. D. Glyzin, A. Yu. Kolesov, and N. Kh. Rozov, “Self-excited relaxation oscillations in networks of impulse neurons,” Russian Math. Surveys70, 383–452 (2015).

    Article  ADS  MathSciNet  Google Scholar 

  29. J. Schauder, “Der Fixpunktsatz in Funktionalraümen,” Studia Math.2, 171–180 (1930).

    Article  Google Scholar 

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Acknowledgments

The author thanks Sergey Dmitrievich Glyzin for the fruitful discussions about this work.

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Authors and Affiliations

  1. Demidov Yaroslavl State University, Yaroslavl, Russia

    M. M. Preobrazhenskaya

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  1. M. M. Preobrazhenskaya
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Correspondence to M. M. Preobrazhenskaya.

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The author declares no conflicts of interest.

Additional information

This research was supported by the President of the Russian Federation (Grant No. MK-1190.2020.1).

Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 203, No. 1, pp. 106–118, April, 2020.

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Preobrazhenskaya, M.M. A relay Mackey—Glass model with two delays. Theor Math Phys 203, 524–534 (2020). https://doi.org/10.1134/S004057792004008X

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  • DOI: https://doi.org/10.1134/S004057792004008X

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