Hopf Bifurcation Analysis of Distributed Delay Equations with Applications to Neural Networks

Abstract

In this paper, we study how to capture smooth oscillations arising from delay-differential equations with distributed delays. For this purpose, we introduce a modified version of the frequency-domain method based on the Graphical Hopf Bifurcation Theorem. Our approach takes advantage of a simple interpretation of the distributed delay effect by means of some Laplace-transformed properties. Our theoretical results are illustrated through an example of two coupled neurons with distributed delay in their communication channel. For this system, we compute several bifurcation diagrams and approximations of the amplitudes of periodic solutions. In addition, we establish analytical conditions for the appearance of a double zero bifurcation and investigate the unfolding by the proposed methodology.