Relaxation Oscillations in a Logistic Equation with Nonconstant Delay


A logistic equation with state- and parameter-dependent delay is considered. The existence of a nonlocal relaxation periodic solution of this equation is proved for sufficiently large parameter values. The proof is carried out by using the large parameter method. For large parameter values, asymptotic estimates of the main characteristics of this solution are also constructed.

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  1. 1.

    M. C. Mackey, “Commodity price fluctuations: price dependent delays and nonlinearities as explanatory factors,” J. Econom. Theory 48, 497–509 (1989).

    MathSciNet  Article  Google Scholar 

  2. 2.

    R. S. Valentine, “Profitability, stability and operability of liquid rocket engines,” Voprosy Raketnoi Tekhniki, No. 1, 29–59 (1973).

    Google Scholar 

  3. 3.

    Yu. S. Kolesov and D. I. Shvitra, “Mathematical modelling of the combustion process in the chamber of a liquid rocket engine,” Litovsk. Mat. Sborn. 15 (4) (1975).

  4. 4.

    T. Insperger, A. W. Barton, and G. Stepan, “Criticality of Hopf bifurcation in state-dependent delay model of turning processes,” Int. J. Non-Linear Mech. 43 (2), 140–149 (2008).

    Article  Google Scholar 

  5. 5.

    M. G. Zager, P. M. Schlosser, and H. T. Tran, “A delayed nonlinear PBPK model for genistein dosimetry in rats,” Bull. Math. Biol. 69 (1), 93–117 (2007).

    MathSciNet  Article  Google Scholar 

  6. 6.

    Jack K. Hale, Theory of Functional Differential Equations (Berlin, 1977).

  7. 7.

    Q. Hu and J. Wu, “Global Hopf bifurcation for differential equations with state-dependent delay,” J. Differential Equations 248 (12), 2801–2840 (2010).

    MathSciNet  Article  Google Scholar 

  8. 8.

    M. Brokate and F. Colonius, “Linearizing equations with state-dependent delays,” Appl. Math. Optim. 21, 45–52 (1990).

    MathSciNet  Article  Google Scholar 

  9. 9.

    K. L. Cooke and W. Z. Huang, “On the problem of linearization for state-dependent delay differential equations,” Proc. Amer. Math. Soc. 124 (5), 1417–1426 (1996).

    MathSciNet  Article  Google Scholar 

  10. 10.

    F. Hartung and J. Turi, “On differentiability of solutions with respect to parameters in state-dependent delay equations,” J. Differential Equations 135 (2), 192–237 (1997).

    MathSciNet  Article  Google Scholar 

  11. 11.

    I. S. Kashchenko and S. A. Kashchenko, “Local dynamics of an equation with a large state-dependent delay,” Dokl. Akad. Nauk 464 (5), 521–524 (2015) [Dokl. Math. 92 (2), 581–584 (2015)].

    MATH  Google Scholar 

  12. 12.

    Y. Kuang and H. L. Smith, “Slowly oscillating periodic solutions of autonomous state-dependent delay equations,” Nonlinear Anal. 19 (9), 855–872 (1992).

    MathSciNet  Article  Google Scholar 

  13. 13.

    M. C. Mackey and J. Belair, “Consumer memory and price fluctuations in commodity markets: an integrod-ifferential model,” J. Dynam. Differential Equations 1 (3), 299–325 (1989).

    MathSciNet  Article  Google Scholar 

  14. 14.

    V. O. Golubenets, “Local bifurcations analysis of a state-dependent delay differential equation,” Model. Anal. Inform. Sist. 22 (5), 711–722 (2015).

    MathSciNet  Article  Google Scholar 

  15. 15.

    J. Mallet-Paret, R. D. Nussbaum, and P. Paraskevopoulos, “Periodic solutions for functional differential equations with multiple state-dependent time lags,” Topol. Methods Nonlinear Anal. 3, 101–162 (1994).

    MathSciNet  Article  Google Scholar 

  16. 16.

    S. A. Kashchenko, “Asymptotics of solutions of the generalized Hutchinson’s equation,” Model. Anal. Inform. Sist. 19 (3), 32–61 (2012).

    Article  Google Scholar 

  17. 17.

    S. A. Kashchenko, “Asymptotics of the periodic solution of the generalized Hutchinson’s equation,” in studies in Stability and Oscillation Theory (YarGU, Yaroslavl, 1981), pp. 64–85 [in Russian].

    Google Scholar 

  18. 18.

    R. Edwards, Functional Analysis: Theory and Applications (New York, 1965; Mir, Moscow, 1969).

    Google Scholar 

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This work was supported by the Russian Foundation for Basic Research under grant 19-31-90082.

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Correspondence to V. O. Golubenets.

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Russian Text © The Author(s), 2020, published in Matematicheskie Zametki, 2020, Vol. 107, No. 6, pp. 833–847.

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Golubenets, V.O. Relaxation Oscillations in a Logistic Equation with Nonconstant Delay. Math Notes 107, 920–932 (2020).

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  • delay equation
  • relaxation solution
  • large parameter method