Parameterization of Unstable Manifolds for DDEs: Formal Series Solutions and Validated Error Bounds

This paper studies the local unstable manifold attached to an equilibrium solution of a system of delay differential equations (DDEs). Two main results are developed. The first is a general method for computing the formal Taylor series coefficients of a function parameterizing the unstable manifold. We derive linear systems of equations whose solutions are the Taylor coefficients, describe explicit formulas for assembling the linear equations for DDEs with polynomial nonlinearities. We also discuss a scheme for transforming non-polynomial DDEs into polynomial ones by appending auxiliary equations. The second main result is an a-posteriori theorem which—when combined with deliberate control of rounding errors—leads to mathematically rigorous computer assisted convergence results and error bounds for the truncated series. Our approach is based on the parameterization method for invariant manifolds and requires some mild non-resonance conditions between the unstable eigenvalues.