# Preprint of the project: DFG Priority Research Program: DANSE

## Connecting orbits in highdimensional dynamical systems

472/01 Thorsten Pampel (Göke).
Approximation of generalized connecting orbits with asymptotic rate

We set up the concept of connecting orbits of a generalized form which allows for discontinuities in the system or the solution at time t=0. Moreover, it is possible to select solutions which converge in a strong stable manifold by specifying the asymptotic rates. We embed connecting orbits as defined in the literature, and provide further applications which have the structure of such generalized connecting orbits, e. g. the computation of so called Skiba points'' in optimization problems. We develop a numerical method for computing generalized connecting orbits and derive error estimates. In particular, we show that the error decays exponentially with the length of the approximation interval, even in the strongly stable case and for periodic solutions. This is in agreement with known results for orbits connecting hyperbolic equilibria. For our method, we select appropriate asymptotic boundary conditions, which depend typically on parameters. In order to solve these type of boundary value problems we set up an efficient iterative procedure, called boundary corrector method. As an example we detect point to periodic connecting orbits in the Lorenz system.