How do numerical methods perform for delay differential equations undergoing a Hopf bifurcation? (Q1841961)

This paper considers the behaviour of \(\theta\)-methods applied with a constant stepsize of integer fractions to delay differential equations undergoing a Hopf bifurcation. Three different approaches are adopted (a boundary locus approach, a direct bifurcation analysis of the difference scheme using a fixed-point iteration and a projection approach) which lead to the conclusion that \(\theta\)-methods retain Hopf bifurcations and preserve their type for sufficiently small values of the stepsize.