Calculation of explicit expressions for the Hopf bifurcation limit cycles in delay-differential equations

TL;DR

This paper develops a systematic method to derive explicit power series approximations for limit cycle solutions at Hopf bifurcations in delay differential equations, enhancing analytical understanding of complex dynamical behaviors.

Contribution

It extends previous methodologies by providing a detailed iterative algorithm for approximating limit cycles in nonlinear delay differential equations with guaranteed error bounds.

Findings

Applied to a traffic flow model, explaining density waves.

Analyzed an epidemic model, capturing disease propagation dynamics.

Demonstrated high accuracy of the power series approximations.

Abstract

This paper introduces a methodology to derive explicit power series approximations for the limit cycle periodic solutions of the Hopf bifurcation in autonomous discrete delay differential equations (DDE). The procedure extends the methodology introduced by Casal and Freedman in 1980 by providing a detailed algorithm that iteratively performs systematic calculations up to any desired order of approximation, ensuring a specific error tolerance for any nonlinear DDE presenting a Hopf bifurcation. The methodology is applied to two relevant delay-differential models to illustrate its features: a recently introduced car-following mobility model, whose oscillations are a plausible explanation for the density waves and congestion in road traffic, and a SIR epidemic model for propagation of diseases with temporary immunity.