We propose to compute approximations to invariant sets in dynamical systems by
minimizing an appropriate distance between a suitably selected finite set of
points and its image under the dynamics. We demonstrate, through computational
experiments, that this approach can successfully converge to approximations of
(maximal) invariant sets of arbitrary topology, dimension, and stability, such
as, e.g., saddle type invariant sets with complicated dynamics. We further
propose to extend this approach by adding a Lennard-Jones type potential term
to the objective function, which yields more evenly distributed approximating
finite point sets, and illustrate the procedure through corresponding
numerical experiments. In the phase space of any nonlinear dynamical system,
the “skeleton” of the global dynamical behavior consists of the invariant sets
of the system, e.g., fixed points, periodic orbits, general recurrent sets,
and the connecting orbits/invariant manifolds between them. Computational
methods for approximating invariant sets have been, and will continue to be, a
major part of the “toolkit” of every dynamical systems researcher, whether on
the mathematical or on the modeling side. In this contribution we devise and
implement a new variational approach for this task, which is able to compute
invariant sets of arbitrary dimension, topology, and stability type. In
addition—and in contrast to classical techniques—our method provides an
approximate parametrization of the invariant set, which can be (smoothly)
followed in parameter space. I. INTRODUCTIO
