Determination of areas and basins of attraction in planar dynamical systems using meshless collocation

This work is focused on the approximation of sets of attractive solutions of planar dynamical systems. Existing work has shown that for many dynamical systems a Riemannian contraction metric can be used to determine sets of solutions with certain attraction properties. For autonomous dynamical systems in R² it has been shown that the Riemannian contraction metric can be reduced to a scalar weight function W. In this work we show that a similar result holds true for finite-time dynamical systems with one spatial dimension. We show how meshless collocation can be used to construct an approximation of W. The approximated weight function can then be used to determine subsets of the area of exponential attraction. This is the first time a method has been introduced to approximate finite-time areas of exponential attraction. We also give a convergence proof for the method. For autonomous dynamical systems in R² there already exists a method that uses W to determine a subset of the basin of attraction of an exponentially stable periodic orbit, O. However that method relies on properties of O being known. We show that the existing equation for W can be manipulated so that no knowledge of the periodic orbit is required to approximate W. We present a method that utilises meshless collocation to approximate W and show that the method is convergent. The approximant of W is then used to determine subsets of the basin of attraction of O.