ABSTRACT

We provide error estimates for an approximation method to compute simultaneously solutions of two dynamical systems each with given asymptotic behaviour and both coupled only by conditions on initial values. The method applies to compute connecting orbits — point–to–point, point–to–periodic and periodic–to–periodic — as in the literature and in numerical applications. Since our set–up is more general, we call solutions of our systems generalised connecting 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 stable manifold. The numerical analysis shows that the error decays exponentially with the length of the approximation intervals even in the strongly stable case and for periodic solutions. For orbits connecting hyperbolic equilibria this is in agreement with known results in the literature. In our method we select appropriate asymptotic boundary conditions which depend typically on parameters. In order to solve these types of boundary value problems we set up an iterative procedure which is called boundary corrector method.

Keywords:

Numerical method, point–to–periodic, periodic–to–periodic, asymptotic rate, asymptotic boundary condition, error estimates.