  We describe blow-up behavior for ODEs by means of dynamics at infinity with
complex asymptotic behavior in autonomous systems, as well as in nonautonomous
systems. Based on preceding studies, a variant of closed embeddings of phase
spaces and the time-scale transformation determined by the structure of vector
fields at infinity reduce our description of blow-ups to unravel the shadowing
property of (pre)compact trajectories on the horizon, the geometric object
expressing the infinity, with the specific convergence rates. Geometrically,
this description is organized by asymptotic phase of invariant sets on the
horizon. Blow-up solutions in nonautonomous systems can be described in a
similar way. As a corollary, normally, or partially hyperbolic invariant
manifolds on the horizon possessing asymptotic phase are shown to induce
blow-ups.
