Much work has already been done on studying feedforward networks of dynamical systems that undergo a Hopf bifurcation (see for example [10],[14],[15], [16]). In particular, it has been noted and proven several times 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 thesis, we present a new approach to studying this growth rate phenomenon. By choosing a particular normal form for the bifurcation, we may use tools such as the two timescale method and asymptotic approximations to detect behavior associated to this 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 this phenomenon 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 our model (and the model used by several authors), the growth rate is asymptotic to the standard square root growth rate as the bifurcation parameter increases.
