Normal form computations for Delay Differential Equations in DDE-BIFTOOL

Abstract

Delay Differential Equations (DDEs) appear in many applications, including neuroscience, ecology, and engineering. The analysis of one- and two-parameter families of bifurcations is based on computing normal forms of ODEs without delays describing the dynamics on center manifolds. We give an overview of so-called sun-star calculus of dual semigroups necessary to derive symbolic formulas for the critical normal form coefficients for the Hopf, Generalized Hopf, Zero-Hopf and Double Hopf bifurcations. We then discuss their implementation in the Matlab package DDE-BIFTOOL. Additionally, detection of these bifurcations was implemented. We demonstrate the new features by detecting bifurcations and computing their normal form coefficients in several DDE models.