Continuation of Sets of Constrained Orbit Segments

Sets of constrained orbit segments of time continuous flows are collections of trajectories that represent a whole or parts of an invariant set. A non-trivial but simple example is a homoclinic orbit. A typical representation of this set consists of an equilibrium point of the flow and a trajectory that starts close and returns close to this fixed point within finite time. More complicated examples are hybrid periodic orbits of piecewise smooth systems or quasi-periodic invariant tori. Even though it is possible to define generalised two-point boundary value problems for computing sets of constrained orbit segments, this is very disadvantageous in practice. In this talk we will present an algorithm that allows the efficient continuation of sets of constrained orbit segments together with the solution of the full variational problem.