On Poisson's state-dependent delay (Q1943279)

The study of state-dependent delay differential equations has been a very active research area in the theory of functional differential equations for the last decade. The author of this paper is one of the leading researchers in this area and has developed a dynamical systems framework for such equations. In this paper, the author investigates the equation \[ [x(t+(xx')(t))]^2 - ([x'(t)]^2+1)[x(t)]^2 = -A, \] which was first presented by Poisson about two centuries ago and is not covered by the recent results on initial value problems for differential equations with state-dependent delay. It is shown that the equation generates a semiflow of differentiable solution operators, on a manifold of differentiable functions and away from a singular set. It is also shown that initial data in the singular set produce multiple solutions.