Biological Network Dynamics: Poincaré–Lindstedt Series and the Effect of Delays on Local Hopf Bifurcations

We study local Hopf bifurcations in an activator–inhibitor system without diffusion which can be modeled as a delay differential equation with two discrete delays. The main result of this paper is the local existence of the Poincaré–Lindstedt series to all orders for the bifurcating periodic solutions. The model has a nonpolynomial type of nonlinearity, and yet this allows us to exploit the use of Fourier–Taylor series to develop order-by-order calculations that lead to linear recurrence equations for the coefficients of the Poincaré–Lindstedt series. As applications, we implement the computation of the coefficients of these series for any finite order, and use a pseudo-arclength continuation to compute local branches of small-amplitude periodic solutions.