Bifurcation of Periodic Solutions of the Mackey– Glass Equation

E P Kubyshkin,A R Moryakova

doi:10.18255/1818-1015-2016-6-784-803

Abstract

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.

Highlights

We study the bifurcation of the equilibrium states of periodic solutions for the Mackey– Glass equation
В качестве начальных значений при интегрировании уравнения (4) выбирались функции, полученные согласно (94)–(95)

Summary

Introduction

Ниже изучаются периодические решения уравнения (2), бифурцирующие из состояния равновесия (3) при изменении параметров ε1 и β, исследуется их устойчивость. Для определения корней уравнения (7) достаточно рассмотреть последовательность уравнений λ + ln (1 + ε1λ) = ln (1 + ε2) + iπk, k = 1, 3, 5, .

Results

Conclusion