In dynamical systems given by an ODE, one is interested in the basin
of attraction of invariant sets, such as equilibria or periodic
orbits. The basin of attraction consists of solutions which converge
towards the invariant set. To determine the basin of attraction, one
can use a solution of a certain linear PDE which can be approximated
by meshless collocation.
The basin of attraction of an equilibrium can be determined through
sublevel sets of a Lyapunov function, i.e. a scalar-valued function
which is decreasing along solutions of the dynamical system. One
method to construct such a Lyapunov function is to solve a certain
linear PDE approximately using Meshless Collocation. Error estimates
ensure that the approximation is a Lyapunov function.
The basin of attraction of a periodic orbit can be analysed by Borg’s
criterion measuring the time evolution of the distance between
adjacent trajectories with respect to a certain Riemannian metric.
The sufficiency and necessity of this criterion will be discussed,
and methods how to compute a suitable Riemannian metric using
Meshless Collocation will be presented in this talk.