Pseudospectral approximation of Hopf bifurcation for delay differential equations

TL;DR

This paper presents a pseudospectral method to approximate delay differential equations by ODEs, enabling efficient bifurcation analysis, especially of Hopf bifurcations, with proven convergence as the approximation dimension increases.

Contribution

It introduces a pseudospectral approximation technique for DDEs that preserves structural properties and links bifurcation coefficients of DDEs and ODEs, with convergence proof.

Findings

The method efficiently analyzes DDE bifurcations.

Structural similarity between DDE and ODE generators is demonstrated.

Convergence of bifurcation coefficients is proven as approximation dimension increases.

Abstract

Pseudospectral approximation reduces DDE (delay differential equations) to ODE (ordinary differential equations). Next one can use ODE tools to perform a numerical bifurcation analysis. By way of an example we show that this yields an efficient and reliable method to qualitatively as well as quantitatively analyse certain DDE. To substantiate the method, we next show that the structure of the approximating ODE is reminiscent of the structure of the generator of translation along solutions of the DDE. Concentrating on the Hopf bifurcation, we then exploit this similarity to reveal the connection between DDE and ODE bifurcation coefficients and to prove the convergence of the latter to the former when the dimension approaches infinity.