Bifurcations in a delay logistic equation under small perturbations

S. A. Kashchenko

Izvestiâ vysših učebnyh zavedenij. Matematika, no. 10 (2020), pp. 47-64.

Résumé

The article considers the dynamic properties of a logistic equation with delay. The first section studies the local behavior of the original equation solutions with the help of bifurcational methods. The main attention is paid to the question of the influence of small perturbations with large delay on the dynamic properties of the solutions. Special nonlinear equations of the parabolic type are constructed. Their local dynamics describes the behavior of the solutions from the small neighborhood of the balance state for the original equation with delay. The second section studies the important for applications question of a parametric resonance at two-frequency disturbance with the help of asymptotic methods.

Détail

Keywords: dynamics, stability, asymptotics, parametric resonance.
Mots-clés : bifurcation

Bibliographie

[1] Wright E. M., “A non-linear difference-differential equation”, J. für die reine und angewandte Math., 194 (1955), 66–87 | MR | Zbl

[2] Kakutani S., Markus L., “On the non-linear difference-differential equation $y'(t)=[A - By(t-\tau )]y(t)$”, Contributions to the Theory of Nonlinear Oscillations, Ann. Math. Stud., 4, Princeton Univ. Press, Princeton, 1958, 1–18 | MR

[3] Kaschenko S. A., “K voprosu ob otsenke v prostranstve parametrov oblasti globalnoi ustoichivosti uravneniya Khatchinsona”, Nelineinye kolebaniya v zadachakh ekologii, YarGU, Yaroslavl, 1985, 55–62

[4] Jones G. S., “The existence of periodic solutions of $f^{\prime }(x) = -\alpha f(x - 1) [1 + f(x)]$”, J. Contemporary Math. Anal., 5 (1962), 435–450 | DOI | MR | Zbl

[5] Kaschenko S. A., “Asimptotika periodicheskogo resheniya obobschennogo uravneniya Khatchinsona”, Issledovaniya po ustoichivosti i teorii kolebanii, YarGU, Yaroslavl, 1981, 64–85

[6] Kashchenko S. A., “Asymptotics of the Solutions of the Generalized Hutchinson Equation”, Automatic Control and Computer Sci., 47:7 (2013), 470–494 | DOI

[7] Kuang Y., Delay Differential Equations : With Applications in Population Dynamics, Academic Press, Boston, 1993 | MR | Zbl

[8] Gourley S. A., Sou J. W.-H., Wu J.H., “Nonlocality of Reaction-Diffusion Equations Induced by Delay: Biological Modeling and Nonlinear Dynamics”, J. Math. Sci., 124:4 (2004), 5119–5153 | DOI | MR

[9] Hale J. K., Theory of functional differential equations, Springer Verlag, N.Y., 1977 | MR | Zbl

[10] Wu J., Theory and applications of partial functional differential equations, Springer Verlag, N.Y., 1996 | MR | Zbl

[11] Cushing J. M., Integrodifferential Equations and Delay Models in Population Dynamics, Springer, Berlin–Heidelberg, 1977 | MR | Zbl

[12] May R. M., Stability and complexity in model ecosystems, Princeton Univ. Press, Princeton, 2001 | Zbl

[13] Murray J. D., Mathematical Biology, v. II, Spatial Models and Biomedical Applications, Springer Verlag, N.Y., 2003 | MR | Zbl

[14] Kolesov A. Yu., Kolesov Yu. S., Relaksatsionnye kolebaniya v matematicheskikh modelyakh ekologii, Nauka, M., 1993

[15] Oster G., Guckenheimer J., “Bifurcation Phenomena in Population Models”, The Hopf Bifurcation and Its Appl., Springer Verlag, N.Y., 1976, 327–353 | DOI | MR

[16] Erneux T., Applied delay differential equations, Springer Verlag, N.Y., 2009 | MR | Zbl

[17] Kaschenko S. A., “Bifurkatsii v okrestnosti tsikla pri malykh vozmuscheniyakh s bolshim zapazdyvaniem”, Zhurn. vychisl. matem. i matem. fiz., 40:5 (2000), 693–702 | MR

[18] Kaschenko S. A., “Bifurcational Features in Systems of Nonlinear Parabolic Equations with Weak Diffusion”, International J. Bifurcation and Chaos in Appl. Sci. and Engineering, 15:11 (2005), 3595–3606 | DOI | MR | Zbl

[19] Kaschenko S. A., “Primenenie metoda normalizatsii k izucheniyu dinamiki differentsialno-raznostnykh uravnenii s malym mnozhitelem pri proizvodnoi”, Differents. uravneniya, 25:8 (1989), 1448–1451 | MR

[20] Kaschenko I. S., “Lokalnaya dinamika uravnenii s bolshim zapazdyvaniem”, Zhurn. vychisl. matem. i matem. fiz., 48:12 (2008), 2141–2150 | MR

[21] Kaschenko S. A., “Uravnenie Ginzburga–Landau — normalnaya forma dlya differentsialno-raznostnogo uravneniya vtorogo poryadka s bolshim zapazdyvaniem”, Zhurn. vychisl. matem. i matem. fiz., 38:3 (1998), 457–465 | MR

[22] Kashchenko A. A., “Analysis of Running Waves Stability in the Ginzburg–Landau Equation with Small Diffusion”, Automatic Control and Computer Sci., 49:7 (2015), 514–517 | DOI

[23] Kashchenko I. S., Kashchenko S. A., “Infinite Process of Forward and Backward Bifurcations in the Logistic Equation with Two Delays”, Nonlinear Phenomena in Complex Syst., 22:4 (2019), 407–412 | DOI | Zbl

[24] Kaschenko S. A., Kolesov Yu. S., “Raskachivanie «kachelei» pri pomoschi dvukhchastotnoi sily”, Issledov. po ustoichivosti i teorii kolebanii, YarGU, Yaroslavl, 1978, 19–25

[25] Kaschenko S. A., Kolesov Yu. S., “Parametricheskii rezonans v sistemakh s zapazdyvaniem pri dvukhchastotnom vozmuschenii”, Sib. matem. zhurn., 21:2 (1980), 113–118 | MR

[26] Bykova N. D., Glyzin S. D., Kashchenko S. A., “Parametric Resonance in the Logistic Equation with Delay under a Two-Frequency Perturbation”, Nonlinear Phenomena in Complex Syst., 19:1 (2016), 80–87 | MR | Zbl

[27] Bautin N. N., Leontovich E. A., Metody i priemy kachestvennogo issledovaniya dinamicheskikh sistem na ploskosti, Nauka, M., 1990

[28] Bogolyubov N. N., Mitropolskii Yu. A., Asimptoticheskie metody v teorii nelineinykh kolebanii, Gostekhizdat, M., 1955 | MR

[29] Vasileva A. B., Butuzov V. F., Asimptoticheskie razlozheniya reshenii singulyarno vozmuschennykh uravnenii, Nauka, M., 1973