Finite-dimensional maps describing the dynamics of a logistic equation with delay
- Authors: Glyzin S.D.1, Kashchenko S.A.1,2
-
Affiliations:
- P.G. Demidov Yaroslavl State University
- National Research Nuclear University MEPhI (Moscow Engineering Physics Institute)
- Issue: Vol 487, No 6 (2019)
- Pages: 611-616
- Section: Mathematics
- URL: https://journals.eco-vector.com/0869-5652/article/view/15950
- DOI: https://doi.org/10.31857/S0869-56524876611-616
- ID: 15950
Cite item
Full Text
Abstract
This article discusses a family of maps that are used in the numerical simulation of a logistic equation with delay. This equation and presented maps are widely used in problems of mathematical ecology as models of the dynamics of populations. The paper compares the dynamic properties of the trajectories of these mappings and the original equation with delay. It is shown that the behavior of the solutions of maps can be quite complicated, while the logistic equation with delay has only a stable equilibrium state or cycle.
Keywords
About the authors
S. D. Glyzin
P.G. Demidov Yaroslavl State University
Author for correspondence.
Email: glyzin.s@gmail.com
Russian Federation, 14, Sovetskaya street, Yaroslavl, 150003
S. A. Kashchenko
P.G. Demidov Yaroslavl State University; National Research Nuclear University MEPhI (Moscow Engineering Physics Institute)
Email: kasch@uniyar.ac.ru
Russian Federation, 14, Sovetskaya street, Yaroslavl, 150003; 31, Kashirskoe shosse, Moscow, 115409
References
- Hale J. K. Theory of Functional Differential Equations. N.Y.: Springer Verlag, 1977. 626 p.
- Kuang Y. Delay Differential Equations with Applications in Population Dynamics. Boston: Academic Press, 1993. 398 p.
- Шарковский А. Н. Сосуществование циклов непрерывного отображения прямой в себя // Украин. матем. журн. 1964. Т. 16. № 1. С. 61-71.
- Feigenbaum M. J. Quantitative Universality for a Class of Nonlinear Transformations // J. Stat. Phys. 1978. V. 19. № 1. P. 25-52.
- Marsden J. E., McCracken M. The Hopf Bifurcation and Its Applications // Appl. Math. Sci. 19. Springer-Verlag, 1976.
- Шноль Э. Э. Об устойчивости неподвижных точек двумерных отображений // Дифференц. уравнения. 1994. Т. 30. № 7. С. 1156-1167.
- Sacker R. J. On Invariant Surfaces and Bifurcation of Periodic Solutions of Ordinary Differential Equations. 1964. Report IMM-NYU333, New York University.
- Kashchenko I. S., Kashchenko S. A. Normal and Quasinormal Forms for Systems of Difference and Diffe-rential-Difference Equations // Communications in Nonlinear Science and Numerical Simulation. 2016. Sept. V. 38. P. 243-256.
- Henon M. A Two-Dimensional Mapping with a Strange Attractor // Communications in Mathematical Phy-sics. 1976. V. 50 (1). P. 69-77.
- Ruette S. Chaos on the Interval / University Lecture Series. AMS, 2017. V. 67.
- Глызин Д. С., Глызин С. Д., Колесов А. Ю., Розов Н. Х. Метод динамической перенормировки для нахождения максимального ляпуновского показателя хаотического аттрактора // Дифференц. уравнения. 2005. Т. 41. № 2. С. 268-273.
Supplementary files
![](/img/style/loading.gif)