Preservation of Hopf bifurcation for neutral delay-differential equations by \(\theta\)-methods
The authors discuss the behaviour of the classical \(\theta\)-method when applied to a certain neutral delay differential equation that exhibits a Hopf bifurcation. In particular, they demonstrate that the \(\theta\)-method preserves the bifurcation and the asymptotic stability of the exact solution if the step size \(h\) satisfies \(h = 1/m\) and \(m \in \mathbb N\) is sufficiently large. A part of the paper is devoted to showing how large \(m\) must be chosen to this end for any given value of the parameter \(\theta\). In particular, it turns out that any \(m \geq 1\) can be used in the practically very important case \(\theta = 1/2\).
scientific article; zbMATH DE number 6155455