Bifurcation Features of Periodic Solutions of the Mackey–Glass Equation

Abstract

Bifurcations of periodic solutions of the well-known Mackey–Glass equation from its unique equilibrium state under varying equation parameters are considered. The equation is used as a mathematical model of variations in the density of white blood cells (neutrophils). Written in dimensionless variables, the equation contains a small parameter multiplying the derivative, which makes this equation singular. It is shown that the behavior of solutions to the equation with initial data from a fixed neighborhood of the equilibrium state in the equation phase space is described by a countable system of nonlinear ordinary differential equations. This system has a minimal structure and is called the normal form of the equation in the neighborhood of the equilibrium state. One fast variable and a countable number of slow variables can be extracted from this system of equations. As a result, the averaging method can be applied to the obtained system. It is shown that the equilibrium states of the averaged system of equations in slow variables are associated with periodic solutions of the same stability type in the original equation. The possibility of simultaneous bifurcation of a large number of periodic solutions (multistability bifurcation) is shown. It is also shown that, with a further increase in the bifurcation parameter, each of the periodic solutions exhibits the transition to a chaotic attractor through a series of period-doubling bifurcations. Thus, the behavior of the solutions of the Mackey–Glass equation is characterized by chaotic multistability.