Hopf bifurcation in numerical approximation of the sunflower equation (Q854418)

Initial value problems of the sunflower equation are considered, which represents a scalar delay differential equation of second order. The analysis is based on an equivalent system of two ordinary differential equations (ODEs) of first order with delay. Hopf bifurcation arises for a specific class of parameters. The authors consider the explicit Euler method to obtain a numerical approximation of an initial value problem. The resulting difference equations are analysed in detail. Assuming a Hopf bifurcation at some parameter in the system of ODEs, the authors prove that the corresponding system of difference equations exhibits a Hopf bifurcation for sufficiently small step sizes, too. The difference between both bifurcation points decreases linearly in dependence on the step size applied in the Euler method. Moreover, a proof is given that the direction and stability properties of the two Hopf bifurcations coincide for sufficiently small step sizes.