Structure of basin boundaries of attractors of ODE’s on S2

It has been shown that, for a dynamical system generated via autonomous ordinary differential equations, many counterpart results on the complexity of basins of attraction hold as in many well known cases of dynamical systems described by maps studied in the literature. Even though the basins of attraction have no fractal boundaries in a strict sense, they can be of fine structure and intertwine each other, thus making predictability practically impossible due to the small uncertainties in initial conditions. We prove here that in the presence of more than two attractors and saddle point(s), this intertwined structure of basins of attraction can occur.