Bifurcation Analysis of Systems with Delays: Methods and Their Use in Applications

TL;DR

This paper reviews bifurcation analysis methods for delay differential equations, highlighting the capabilities of DDE-BIFTOOL and demonstrating their application through case studies on climate models and state-dependent delays.

Contribution

It introduces advanced bifurcation analysis techniques for delay systems and showcases their practical use in complex real-world models.

Findings

DDE-BIFTOOL enables numerical continuation of steady states and periodic orbits.

The tool detects bifurcations of codimension one and two, aiding stability analysis.

Case studies illustrate insights into El Nino dynamics and state-dependent delays.

Abstract

This chapter presents a dynamical systems point of view of the study of systems with delays. The focus is on how advanced tools from bifurcation theory, as implemented for example in the package DDE-BIFTOOL, can be applied to the study of delay differential equations (DDEs) arising in applications, including those that feature state-dependent delays. We discuss the present capabilities of the most recent release of DDE-BIFTOOL. They include the numerical continuation of steady states, periodic orbits and their bifurcations of codimension one, as well as the detection of certain bifurcations of codimension two and the calculation of their normal forms. Two longer case studies, of a conceptual DDE model for the El Nino phenomenon and of a prototypical scalar DDE with two state-dependent feedback terms, demonstrate what kind of insights can be obtained in this way.