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 @q-method. By analyzing the dynamics of the numerical discrete system derived by @q-method, we show that @q-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 @q=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.

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