Error Estimates of a Computational Method for Generalised Connecting Orbits

摘要

We provide error estimates for an approximation method to compute simultaneously solutions of twodynamical systems each with given asymptotic behaviour and both coupled only by conditions on initial values. Themethod applies to compute connecting orbits — point–to–point, point–to–periodic and periodic–to–periodic — as in theliterature and in numerical applications. Since our set–up is more general, we call solutions of our systems generalisedconnecting orbits and provide further applications like Skiba points in economic models or solutions with a discontinuity.By specifying the asymptotic rates our method also applies to the computation of solutions converging in a strongly stablemanifold. The numerical analysis shows that the error decays exponentially with the length of the approximation intervalseven in the strongly stable case and for periodic solutions. For orbits connecting hyperbolic equilibria this is in agreementwith known results in the literature. In our method we select appropriate asymptotic boundary conditions which dependtypically on parameters. In order to solve these types of boundary value problems we set up an iterative procedure whichis called boundary corrector method.