Solution manifolds which are almost graphs

Abstract

Differential equations with state-dependent delays which generalize the scalar example (0) x ′ ( t ) = g ( x ( t ) , x ( t − d ( x t ) ) ) where g : R 2 → R and d : C ( [ − r , 0 ] , R ) → [ 0 , r ] are continuously differentiable, and with x t : [ − r , 0 ] → R given by x t ( s ) = x ( t + s ) , define semiflows of differentiable solution operators on an associated submanifold of the state space C 1 = C 1 ( [ − r , 0 ] , R n ) . When written in the general form x ′ ( t ) = f ( x t ) with a map f : C 1 ⊃ U → R n then the associated manifold is X f = { ϕ ∈ U : ϕ ′ ( 0 ) = f ( ϕ ) } . We obtain results on the nature of X f . • If all delays in the system are bounded away from zero then a projection C 1 → C 1 onto the subspace H = { ϕ ∈ C 1 : ϕ ′ ( 0 ) = 0 } defines a diffeomorphism of X f onto an open subset of H . In other words, X f is a graph over H . • There exist g and d with d ( ϕ ) > 0 everywhere and inf ⁡ d = 0 so that the manifold X f associated with Eq. (0) does not admit any graph representation. • If all delays in the system are strictly positive (but not necessarily bounded away from zero) then X f has an “almost graph representation” which implies that it is diffeomorphic to an open subset of H .