** Authors: ** Jan Bouwe van den Berg

Jason D. Mireles-James

Jean-Philippe Lessard

Konstantin Mischaikow

** Abstract:** In this paper we propose a rigorous numerical
technique for the computation of symmetric connecting orbits for
ordinary dierential equations.
The idea is to solve a projected boundary value problem (BVP)
in a function space via a fied point argument. The formulation
of the projected BVP involves a high-order parameterization
of the invariant manifolds at the steady states. Using this
parameterization, one can obtain explicit exponential
asymptotic bounds for the coefficients of the expansion of the
manifolds. Combining these bounds with piecewise linear
approximations, one can construct a contraction in a function
space whose unique fixed point corresponds to the wanted
connecting orbit. We have implemented the method to demonstrate
its effectiveness, and we have used it to prove the existence
of a family of even homoclinic orbits for the Gray-Scott
system.