Asymptotic Analysis of Bifurcations in Feedforward Networks

Abstract

A feedforward network is a unidirectionally coupled chain of dynamical systems in which the first cell is coupled to itself, and each successive cell is coupled to the next one. Feedforward networks have gained considerable interest because of their potential to enhance signal amplification and to manipulate the frequency of oscillations. Indeed, it has been shown that the growth rate of the bifurcation undergone by the final cell is much larger than the expected square root growth rate associated with the standard Hopf bifurcation. In this paper, we present a new approach to studying this growth rate phenomenon. We employ a two-time-scale analysis and asymptotic approximations to detect behavior associated with the growth rate phenomenon that has not been previously observed. In particular, we show that the Hopf bifurcation is not the only bifurcation capable of exhibiting this large growth rate behavior. Using asymptotic methods we show that it is not a special property of the Hopf bifurcation that allows for this accelerated growth rate; it is a combination of the unidirectional coupling and the higher-degree nonlinearities that cause this effect. Furthermore, we show that this large growth rate need not persist away from the bifurcation. In fact, the growth rate is asymptotic to the standard square root growth rate as the bifurcation parameter increases.