Unbounded basins of attraction of limit cycles

Consider a dynamical system given by a system of autonomous ordinary differential equations. In this paper we provide a sufficient local condition for an unbounded subset of the phase space to belong to the basin of attraction of a limit cycle. This condition also guarantees the existence and uniqueness of such a limit cycle, if that subset is compact. If the subset is unbounded, the positive orbits of all points of this set either are unbounded or tend to a unique limit cycle.