Preservation of Hopf bifurcation for neutral delay-differential equations by θ-methods

This paper considers the preservation of Hopf bifurcation of some neural delay-differential equations by θ-method. By analyzing the dynamics of the numerical discrete system derived by θ-method, we show that θ-method could inherit the Hopf bifurcation and the asymptotical stability for sufficiently small stepsize h=1/m, where m is a positive integer. In particular, for θ=1/2 the result holds for any stepsize h=1/m. Furthermore, the stability of the closed invariant curve is established. Finally, some numerical examples are illustrated to support the analytic results.