EXPLICIT FORMULAS FOR CALCULATING THE NORMAL FORM OF ZERO-HOPF BIFURCATION IN A CLASS OF NEUTRAL DIFFERENTIAL EQUATIONS

This paper explores the computation of the normal form linked to a codimension-two zero-Hopf bifurcation occurring in a neutral functional differential equation containing multiple delays. The primary focus lies on dynamical systems governed by these equations, possessing a simple zero and the pair of purely imaginary eigenvalues, commonly referred to as a zero-Hopf eigenvalue. We derive explicit formulas for computing the normal form of such systems, considering two unfolding parameters, through the application of center manifold reduction and the method of normal forms. Specifically, we obtain the normal form associated with the dynamics on a center manifold, extending up to third-order terms. As an illustration, we employ our explicit formulas to analyze a class of neutral neural networks that exhibit this type of zero-Hopf singularity.