Hopf points of codimension two in a delay differential equation modeling leukemia

TL;DR

This paper investigates degenerate Hopf bifurcation points in a delay differential equation model of leukemia, using center manifold approximation and Lyapunov coefficients to classify bifurcations in biologically relevant parameter regions.

Contribution

It identifies and characterizes degenerate Hopf points in a leukemia model, advancing understanding of complex bifurcation phenomena in delay differential equations.

Findings

Locates points with zero first Lyapunov coefficient in parameter space

Computes second Lyapunov coefficient to classify bifurcation type

Highlights regions of biological significance with degenerate bifurcations

Abstract

This paper continues the work contained in two previous papers, devoted to the study of the dynamical system generated by a delay differential equation that models leukemia. Here our aim is to identify degenerate Hopf bifurcation points. By using an approximation of the center manifold, we compute the first Lyapunov coefficient for Hopf bifurcation points. We find by direct computation, in some zones of the parameter space (of biological significance), points where the first Lyapunov coefficient equals zero. For these we compute the second Lyapunov coefficient, that determines the type of the degenerate Hopf bifurcation.