Global Hopf Bifurcation for Volterra Integral Equations

This paper investigates global bifurcation of time periodic orbits for autonomous systems of integral equations of convolution type depending on a real parameter $\lambda $. An easy criterion for global bifurcation is derived: if—for simplicity—there exists only one stationary, nondegenerate solution for all $\lambda $, then it is sufficient that the linear unstable dimensions for $\lambda $ near $ - \infty $ resp. $ + \infty $ differ from each other. The theorem requires the integral kernels to be integrable with some exponential weight. The proof then relies on approximation by ordinary differential equations. Applications are given to oscillations in a model for epidemics, and to a model for circular neural nets.