## 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.