Numerical Hopf bifurcation for a class of delay differential equations
Parameter-dependent delay differential equations (DDEq) \[ x'(t)=f(x(t),x(t-\tau),\lambda),\quad \lambda\in \mathbb{R},\quad t\geq 0,\quad x(t)=\varphi(t),\quad -\tau\leq t\leq 0,\quad \tau>0, \] (\(\varphi\) is the continuous initial function) containing a Hopf bifurcation are considered. In previous work [\textit{N. J. Ford, V. Wulf}, The use of boundary locus plots in the identification of bifurcating points in the numerical approximation of delay differential equations, MCCM Tech. Report 322, Manchester Univ., ISSN 1360-1725 (1998)] the authors have shown that strictly stable linear multistep methods applied to DDEqs with Hopf bifurcation yield difference equations which contain a Hopf bifurcation of the same type. The results of the present article extend the authors' previous analysis relating to the discretization of the delay logistic equation to a wider class of problems.
scientific article; zbMATH DE number 1423347