Normal forms of Hopf–Bogdanov–Takens bifurcation for retarded differential equations

Abstract

This article explores the process of computing normal forms related to a codimension-three Hopf–Bogdanov–Takens (H–B–T) bifurcation in the framework of retarded functional differential equations. The focus is on the behavior of dynamic systems defined by such equations, which exhibit a pair of purely imaginary roots along with a double zero root, referred to as the H–B–T eigenvalue. By employing center manifold reduction alongside the normal form technique, explicit formulas are derived to facilitate the computation of the normal forms for these systems, integrating three parameters for unfolding. To demonstrate the relevance of our results, we apply the analysis to a particular type of bidirectional associative memory network composed of three neurons, where we explore and illustrate the system’s dynamic behavior through an illustrative example and associated numerical simulations.