Bifurcation of Periodic Solutions of the Mackey– Glass Equation
We study the bifurcation of the equilibrium states of periodic solutions for the Mackey– Glass equation. This equation is considered as a mathematical model of changes in the density of white blood cells. The equation written in dimensionless variables contains a small parameter at the derivative, w...
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| Format: | Article |
| Language: | English |
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Yaroslavl State University
2016-12-01
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| Series: | Моделирование и анализ информационных систем |
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| Online Access: | https://www.mais-journal.ru/jour/article/view/415 |
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| author | E. P. Kubyshkin A. R. Moryakova |
| author_facet | E. P. Kubyshkin A. R. Moryakova |
| author_sort | E. P. Kubyshkin |
| collection | DOAJ |
| description | We study the bifurcation of the equilibrium states of periodic solutions for the Mackey– Glass equation. This equation is considered as a mathematical model of changes in the density of white blood cells. The equation written in dimensionless variables contains a small parameter at the derivative, which makes it singular. We applied the method of uniform normalization, which allows us to reduce the study of the solutions behavior in the neighborhood of the equilibrium state to the analysis of the countable system of ordinary differential equations. We poot out the equations in ”fast” and ”slow” variables from this system. Equilibrium states of the ”slow” variables equations determine the periodic solutions. The analysis of equilibrium states allows us to study the bifurcation of periodic solutions depending on the parameters of the equation and their stability. The possibility of simultaneous bifurcation of a large number of stable periodic solutions is shown. This situation is called the multistability phenomenon. |
| format | Article |
| id | doaj-art-ab45d84d09d9410fbcb50f3b3b5abbd3 |
| institution | Matheson Library |
| issn | 1818-1015 2313-5417 |
| language | English |
| publishDate | 2016-12-01 |
| publisher | Yaroslavl State University |
| record_format | Article |
| series | Моделирование и анализ информационных систем |
| spelling | doaj-art-ab45d84d09d9410fbcb50f3b3b5abbd32025-08-04T14:06:42ZengYaroslavl State UniversityМоделирование и анализ информационных систем1818-10152313-54172016-12-0123678480310.18255/1818-1015-2016-6-784-803344Bifurcation of Periodic Solutions of the Mackey– Glass EquationE. P. Kubyshkin0A. R. Moryakova1P.G. Demidov Yaroslavl State UniversityP.G. Demidov Yaroslavl State UniversityWe study the bifurcation of the equilibrium states of periodic solutions for the Mackey– Glass equation. This equation is considered as a mathematical model of changes in the density of white blood cells. The equation written in dimensionless variables contains a small parameter at the derivative, which makes it singular. We applied the method of uniform normalization, which allows us to reduce the study of the solutions behavior in the neighborhood of the equilibrium state to the analysis of the countable system of ordinary differential equations. We poot out the equations in ”fast” and ”slow” variables from this system. Equilibrium states of the ”slow” variables equations determine the periodic solutions. The analysis of equilibrium states allows us to study the bifurcation of periodic solutions depending on the parameters of the equation and their stability. The possibility of simultaneous bifurcation of a large number of stable periodic solutions is shown. This situation is called the multistability phenomenon.https://www.mais-journal.ru/jour/article/view/415the mackey–glass equationperiodic solutionsmultistability bifurcation |
| spellingShingle | E. P. Kubyshkin A. R. Moryakova Bifurcation of Periodic Solutions of the Mackey– Glass Equation Моделирование и анализ информационных систем the mackey–glass equation periodic solutions multistability bifurcation |
| title | Bifurcation of Periodic Solutions of the Mackey– Glass Equation |
| title_full | Bifurcation of Periodic Solutions of the Mackey– Glass Equation |
| title_fullStr | Bifurcation of Periodic Solutions of the Mackey– Glass Equation |
| title_full_unstemmed | Bifurcation of Periodic Solutions of the Mackey– Glass Equation |
| title_short | Bifurcation of Periodic Solutions of the Mackey– Glass Equation |
| title_sort | bifurcation of periodic solutions of the mackey glass equation |
| topic | the mackey–glass equation periodic solutions multistability bifurcation |
| url | https://www.mais-journal.ru/jour/article/view/415 |
| work_keys_str_mv | AT epkubyshkin bifurcationofperiodicsolutionsofthemackeyglassequation AT armoryakova bifurcationofperiodicsolutionsofthemackeyglassequation |