Parameterization method for unstable manifolds of delay differential equations

This work is concerned with efficient numerical methods for computing high order Taylor and Fourier-Taylor approximations of unstable manifolds attached to equilibrium and periodic solutions of delay differential equations. In our approach we first reformulate the delay differential equation as an ordinary differential equation on an appropriate Banach space. Then we extend the Parameterization Method for ordinary differential equations so that we can define operator equations whose solutions are charts or covering maps for the desired invariant manifolds of the delay system. Finally we develop formal series solutions of the operator equations. Order-by-order calculations lead to linear recurrence equations for the coefficients of the formal series solutions. These recurrence equations are solved numerically to any desired degree. The method lends itself to a-posteriori error analysis, and recovers the dynamics on the manifold in addition to the embedding. Moreover, the manifold is not required to be a graph, hence the method is able to follow folds in the embedding. In order to demonstrate the utility of our approach we numerically implement the method for some 1, 2, 3 and 4 dimensional unstable manifolds in problems with constant, and (briefly) state dependent delays.