On blow-up solutions of differential equations with Poincar\'e-type compactifications

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.