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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