Collection of PhD theses of (former) members of the research group

PhD thesis

tp01 Thorsten Pampel (Göke).
Numerical approximation of generalized connecting orbits

We set up the concept of connecting orbits with asymptotic rate and generalized connecting orbits for continuous, parameterdependend dynamical systems and analyse an approximation method. In particular, we get error estimates for the method, known in the literature only in special cases. 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.
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.
We set up the concept of generalized connecting orbits 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 and set up a version of the boundary corrector method. 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. As an example we detect point to periodic connecting orbits in the Lorenz system.
We apply our method to the models of optimal investment and of resource management. In both cases the time horizon is infinite and the optimal control variables are continuous. we detect periodic solutions and Skiba points, where optimal solutions have different stationary or periodic asymptotic behavior.

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