Toroidal normal forms for bifurcations in retarded functional differential equations II: Saddle-node/multiple Hopf interaction

TL;DR

This paper develops a framework for analyzing the realizability of bifurcation normal forms in retarded functional differential equations, focusing on saddle-node and multiple Hopf interactions with non-semisimple symmetry groups.

Contribution

It introduces a novel approach to the realizability problem for non-semisimple cases involving nilpotency and Jordan blocks, extending previous semisimple analyses.

Findings

Framework for non-semisimple realizability problem established

Solution provided for saddle-node/multiple Hopf bifurcation interactions

Potential applicability to general repeated eigenvalue cases with Jordan blocks

Abstract

In this paper, we study the realizability problem for retarded functional differential equations near an equilibrium point undergoing a nonlinear mode interaction between a saddle-node bifurcation and a non-resonant multiple Hopf bifurcation. In contrast to the case of transcritical/multiple Hopf interaction which was studied in an earlier paper, the analysis here is complicated by the presence of a nilpotency which introduces a non-compact component in the symmetry group of the normal form. We present a framework to analyse the realizability problem in this non-semisimple case which exploits to a large extent our previous results for the realizability problem in the semisimple case. Apart from providing a solution to the problem of interest in this paper, it is believed that the approach used here could potentially be adapted to the study of the realizability problem for toroidal normal forms in the general case of repeated eigenvalues with Jordan blocks.