On invariant manifolds of saddle points for 3D multistable models

Abstract

In dynamical systems a particular solution is completely determined by the parameters considered and the initial conditions. Indeed, when the model shows a multistability, starting from different initial state, the trajectories can evolve towards different attractors. The invariant manifolds of the saddle points separate the vector field into the basins of attraction of different stable equilibria. The aim of this work is the reconstruction of these separation surfaces in order to know in advance the geometry of the basins. In this paper three-dimensional models with three or more stable fixed points is investigated. To this purpose a procedure for the detection of the scattered data lying on the manifolds is proposed. Then a Moving Least Squares meshfree method is involved to approximate the surfaces. Numerical results are presented in order to assess the method.