  Geometric treatments of blow-up solutions for autonomous ordinary
differential equations and their blow-up rates are concerned. Our approach
focuses on the type of invariant sets at infinity via compactifications of
phase spaces, and dynamics on their center-stable manifolds. In particular, we
show that dynamics on center-stable manifolds of invariant sets at infinity
with appropriate time-scale desingularizations as well as blowing-up of
singularities characterize dynamics of blow-up solutions as well as their
rigorous blow-up rates.
