Numerical Hopf bifurcation of linear multistep methods for a class of delay differential equations

摘要

In this paper, we consider the discretization of parameter-dependent delay differential equation of the formy′(t)=f(y(t),y(t-1),τ),τ⩾0,y∈Rd.It is shown that if the delay differential equation undergoes a Hopf bifurcation at τ=τ∗, then the discrete scheme undergoes a Hopf bifurcation at τ(h)=τ∗+O(hp) for sufficiently small step size h, where p⩾1 is the order of the strictly stable linear multistep method. The direction of numerical Hopf bifurcation and stability of bifurcating invariant curve are the same as that of the corresponding delay differential equation.