Preprint des Projektes: DFG Schwerpunkt Programm: Ergodentheorie, Analysis und effiziente Simulation dynamischer Systeme

Verbindungsorbits in hochdimensionalen dynamischen Systemen

25/00 Thorsten Pampel (Göke).
Numerical approximation of connecting orbits with asymptotic rate

In this paper we set up and analyze a numerical method for so called connecting orbits with asymptotic rate in parameterized dynamical systems. A connecting orbit with asymptotic rate has its initial value in a given submanifold of the phase space (or its cross product with parameter space) and it converges with an exponential rate to a given orbit, e. g. a steady state or a periodic orbit. It is well known that orbits with asymptotic rate can be used to foliate stable or strong stable manifolds of invariant sets. We show that the problem of determining a connecting orbit with asymptotic rate is well-posed if a certain transversality condition is made and a specific relation between the number of stable dimensions and the number of parameters holds. For the proof we employ the implicit function theorem in spaces of exponentially decaying functions. Using asymptotic boundary conditions we truncate the original problem to a finite interval and show that the error decays exponentially. Typically the asymptotic boundary conditions by themselves are the result of a boundary value problem, e. g. if the limiting orbit is periodic. Thus it is expensive to calculate them in a parameter dependent way during the approximation procedure. To avoid this we develop a boundary corrector method which turns out to be nearly optimal after very few steps.