Explicit formulas for computing the normal form of Bogdanov–Takens bifurcation in delay differential equations

This paper considers the computation of normal form associated with codimension-two Bogdanov–Takens (BT) bifurcation in delay differential equations. The main attention is focused on dynamical systems described by delay differential equations having a double-zero eigenvalue with geometric multiplicity one, which is usually called non-semisimple double-zero eigenvalue. Explicit formulas for computing the normal form of such systems with two unfolding parameters are obtained by applying center manifold reduction and the method of normal forms. In particular, the normal form associated with the flow on a center manifold up to third-order terms are derived. As an application, an Oregonator oscillator having such a BT singularity with delay is analyzed using our explicit formulas.