Merging homoclinic solutions due to state-dependent delay

For π2<α<5π2 the linear differential equationx′(t)=−αx(t−1) with constant time lag is unstable and hyperbolic. For α   sufficiently close to 5π2 we construct a state-dependent delay dU(ϕ)∈(0,2)dU(ϕ)∈(0,2) on an open set in the space C1([−2,0])C1([−2,0]), with dU(ϕ)=1dU(ϕ)=1 for ϕ close to 0, so that the nonlinear equationx′(t)=−αx(t−dU(xt)) has a one-parameter-family of entire solutions c[r]:R→Rc[r]:R→R, |r|0ty>0, c[r](t)=c[0](t)c[r](t)=c[0](t) for |r|