Local dynamics of an equation with a large state-dependent delay (Q906099)

In this paper, the authors deal with two differential equations with a large density dependent delay. The first equation is a first order scalar DDE and the second is a second order DDE with a damping term. They analyze the behaviors of solutions to these two equations in a neighborhood of an equilibrium by explore the stability of the equilibrium, which is related to the global dynamics of a nonlinear parabolic equation with periodic or anti-periodic boundary solutions which is known as a quasi-normal form. Asymptotic formulas are presented that relate the solutions of the quasi-normal form and the original equation.