Approximation of the basins of attraction in dynamical systems

In a mathematical model involving a set of ordinary differential equations, a particular solution of the system is completely determined by the initial condition. When the omega limit set reduces to a point, the solution settles at steady state. The possible steady states of the system are completely determined by its parameters. However, it is possible that more steady states can originate from different initial conditions but with the same parameter set (multi-stability). In that case the outcome of such model depends on the initial condition. Therefore, it is important to assess the domain of attraction for each possible attractor. We have already analyzed the problem of the reconstruction of the basins of attraction for competition models presenting bistability in [1, 2, 3]. Here we focus instead on systems of two and three ordinary differential equations presenting bior tri-stability, aiming again at assessing the basins of attraction of these different stable equilibria. In particular our algorithm is general and robust enough so as to accommodate the three equilibria case as well as the particular case in which two of these equilibria coalesce. For this purpose we have implemented several Matlab functions for the approximation of the points lying on the curves or on the surfaces determining the basins of attraction, obtained by a bisection algorithm, and for the graphical representation of such curves and surfaces. We approximate the latter with the implicit Partition of Unity method, using as local approximants the compactly supported Wendland’s functions [4].