Abstract
The logistic equation supplemented with a summand characterizing delay is considered. The local and nonlocal dynamics of this equation are studied. For equations with delay, we use the standard Andronov—Hopf bifurcation methods and the asymptotic method developed by the author and based on the construction of special evolution equations defining the local dynamics of the equations containing delay. In addition, we study the existence and methods of constructing the asymptotics of nonlocal relaxation cycles. A comparison of the results obtained with those for the Hutchinson equation and some of its generalizations is given.
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References
S. Kakutani and L. Markus, “On the non-linear difference-differential equation y′(t) = (A-By(t-τ))y(t),” in Contributions to the Theory of Non-Linear Oscillations, Ann. Math. Stud., Vol. IV (PrincetonUniv. Press, Princeton, NJ, 1958), Vol. 41, pp. 1–18.
G. S. Jones, “The existence of periodic solutions of f′(x) = -αf(x - 1){1+ f(x)},” J. Math. Anal. Appl. 5 (3), 435–450 (1962).
S. A. Kashchenko, “Asymptotics of the solutions of the generalized Hutchinson equation,” Automat. Control Comput. Sci. 47 (7), 470–494 (2013).
S. A. Kashchenko, “On the periodic solutions of the equation x′(t) = -lx(t - 1)[1 + x(t)],” in Studies in Stability and Oscillation Theory (YarGU, Yaroslavl, 1978), pp. 110–117 [in Russian].
S. A. Kashchenko, “Asymptotics of the periodic solution of the generalized Hutchinson equation,” in Studies in Stability and Oscillation Theory (YarGU, Yaroslavl, 1981), pp. 22 [in Russian].
S. A. Kashchenko, “Application of the normalization method to the study of the dynamics of a differential-difference equation with a small factor multiplying the derivative,” Differ. Uravn. 25 (8), 1448–1451 (1989).
S. A. Kaschenko, “Normalisation in the systems with small diffusion,” Int. J. Bifurcation Chaos 6 (7), 1093–1109 (1996).
S. A. Kashchenko, “The Ginzburg–Landau equation as a normal form for a second-order difference-differential equation with a large delay,” Zh. Vychisl.Mat. iMat. Fiz. 38 (3), 457–465 (1998) [Comput. Math. and Math. Phys. 38 (3), 443–451 (1998)].
I. S. Kashchenko, “Asymptotic analysis of the behavior of solutions to equations with large delay,” Dokl.Ross. Akad. Nauk 421 (5), 586–589 (2008) [Dokl.Math. 78 (1), 570–573 (2008)].
I. S.Kashchenko and S. A. Kashchenko, “Dynamics of an equation with a large spatially distributed control parameter,” Dokl. Ross. Akad. Nauk 438 (1), 30–34 (2011) [Dokl.Math. 83 (3), 408–412 (2011)].
S. A. Kashchenko, “Investigation, by large parameter methods, of a systemof nonlinear differential-difference equations modeling a predator-prey problem,” Dokl. Ross. Akad. Nauk 266, 792–795 (1982) [Soviet Math. Dokl. 26, 420–423 (1982)].
J. Hale, Theory of Functional Differential Equations (Springer-Verlag, New York, 1977; Mir, Moscow, 1984).
Ph. Hartman, Ordinary Differential Equations (John Wiley, New York–London–Sydney, 1964; Mir, Moscow, 1970).
A. D. Bryuno, A Local Method of Nonlinear Analysis of Differential Equations (Nauka, Moscow, 1979) [in Russian].
N. N. Bautin and E. A. Leontovich, Methods of the Qualitative Study of Dynamical Systems in the Plane, inMathematical Reference Library (Nauka, Moscow, 1990), Vol. 11 [in Russian].
R. Edwards, Functional Analysis: Theory and Applications (New York, 1965; Mir, Moscow, 1969).
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Original Russian Text © S. A. Kashchenko, 2015, published in Matematicheskie Zametki, 2015, Vol. 98, No. 1, pp. 85–100.
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Kashchenko, S.A. Dynamics of the logistic equation with delay. Math Notes 98, 98–110 (2015). https://doi.org/10.1134/S0001434615070093
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DOI: https://doi.org/10.1134/S0001434615070093