Blowing-Up Asymptotically Autonomous Planar Vector Fields: Infinite Delay Equations and Invariant Manifolds

We show the existence of nonautonomous invariant manifolds for planar, asymptotically autonomous differential equations, that have equilibrium solutions with zero Lyapunov spectrum. These invariant manifolds correspond to the stable and unstable manifold of a desingularized equation, that we obtain by using the blow-up method. More precisely, the blow-up method is extended to the nonautonomous setting and transforms the original finite-dimensional ordinary differential equation into an infinite-dimensional delay equation with infinite delay, but hyperbolic structure. In the technical construction of the invariant manifolds for the delay equation, we have to carefully study the effect of the time reparametrization used for desingularization in the blown-up space to guarantee sufficient regularity. This allows us to employ a Lyapunov–Perron argument to obtain existence of an invariant manifold. We combine the last step with an implicit function argument to study differentiability of the manifold. Finally, we reverse the blow-up transformation of space and time variables obtaining invariant manifolds of the initially considered equation.