Bifurcation of Periodic Delay Differential Equations at Points of 1:4 Resonance

TL;DR

This paper analyzes the bifurcation behavior of a scalar delay differential equation at 1:4 resonance points, deriving conditions for stability and explicitly calculating normal form coefficients.

Contribution

It provides a detailed analysis of bifurcations at 1:4 resonance in delay differential equations, including explicit normal form coefficients and stability conditions.

Findings

Resonant bifurcation occurs at critical parameter values.

Explicit formulas for third order normal form coefficients are derived.

The 1:4 resonance does not affect certain delay equations of specific form.

Abstract

The time-periodic scalar delay differential equation $\dot x(t)=\gamma f(t,x(t-1))$ is considered, which leads to a resonant bifurcation of the equilibrium at critical values of the parameter. Using Floquet theory, spectral projection and center manifold reduction, we give conditions for the stability properties of the bifurcating invariant curves and four-periodic orbits. The coefficients of the third order normal form are derived explicitly. We show that the 1:4 resonance has no effect on equations of the form $\dot z(t)=-\gamma r(t)g(x(t-1))$.