Stability of equilibrium and periodic solutions of a delay equation modeling leukemia
Anca-Veronica Ion1, Raluca-Mihaela Georgescu2
| Аннотация | We consider a delay differential equation that occurs in the study of chronic myelogenous leukemia. The equation was investigated numerically in \cite{IonPM-M}, \cite{IonPM-B-M}, where also some conclusions regarding the stability of equilibria are given. In \cite{IonI09} the first author studied in detail the stability of the two equilibrium points and obtained results that do not agree to those of \cite{IonPM-M}, \cite{IonPM-B-M}. In the present work, we shortly remind the results of \cite{IonI09} and then concentrate on the study of stability of periodic solutions emerged by Hopf bifurcation from the non-trivial equilibrium point. We give the algorithm for the construction of a center manifold at a typical point (in the parameter space) of Hopf bifurcation, and of a unstable manifold in the vicinity of such a point (where such a manifold exists). Then we find the normal form of the equation restricted to the center manifold, by computing the first Lyapunov coefficient. The normal form allows us to establish the stability properties of the periodic solutions occurred by Hopf bifurcation. \par \textbf{Acknowledgement.} Work supported by Grant 11/05.06.2009 within the framework of the Russian Foundation for Basic Research - Romanian Academy collaboration. \par \textbf{AMS MSC 2000:} 65L03, 37C75, 37G05, 37G15. |
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| Ключевые слова | delay differential equations; stability; Hopf bifurcation; normal forms. |
1"Gh. Mihoc-C. Iacob" Institute of Mathematical Statistics and Applied Mathematics of the Romanian Academy, Bucharest, Romania.
2University of Pitec sti, Romania.
Цитирование: Anca-Veronica Ion, Raluca-Mihaela Georgescu Stability of equilibrium and periodic solutions of a delay equation modeling leukemia // Журнал Средневолжского математического общества. 2009. Т. 11, № 2. С. 152–163.
