Hopf Bifurcations for Neutral Functional Differential Equations with Infinite Delays

In the theory of linear autonomous neutral functional di¤erential equations with infinite delay, the spectrum distribution of the infinitesimal generator of its solution operators is studied under a certain phase space. Thereafter, we prove the representation theorem of the solution operators, which is later employed to obtain exponential dichotomy properties in terms of semigroup theory. Formal adjoint theory for linear autonomous NFDEs with infinite delay is established including such topics as formal adjoint equations, the relationship between the formal adjoint and true adjoint, and decomposing the phase space with formal adjoint equation. Finally, the algorithm for calculating the Hopf bifurcation properties for nonlinear NFDEs with infinite delay is presented based on the theory of linear equations.