  We provide explicit criteria for blow-up solutions of autonomous ordinary
differential equations. Ideas are based on the quasi-homogeneous
desingularization (blowing-up) of singularities and compactifications of phase
spaces, which suitably desingularize singularities at infinity. We derive
several type of compactifications and show that dynamics at infinity is
qualitatively independent of the choice of such compactifications. We also show
that hyperbolic invariant sets, such as equilibria and periodic orbits, at
infinity induce blow-up solutions with specific blow-up rates. In particular,
blow-up solutions can be described as trajectories on stable manifolds of
equilibria at infinity for associated vector fields. Finally, we demonstrate
blow-up solutions of several differential equations.
