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In this paper, blow-up solutions of autonomous ordinary differentialequations (ODEs) which are unstable under perturbations of initial points,referred to as saddle-type blow-up solutions, are studied. Combining dynamicalsystems machinery (e.g., compactifications, time-scale desingularizations ofvector fields) with tools from computer-assisted proofs (e.g., rigorousintegrators, the parameterization method for invariant manifolds), theseblow-up solutions are obtained as trajectories on local stable manifolds ofhyperbolic saddle equilibria at infinity. With the help of computer-assistedproofs, global trajectories on stable manifolds, inducing blow-up solutions,provide a global picture organized by global-in-time solutions and blow-upsolutions simultaneously. Using the proposed methodology, intrinsic features ofsaddle-type blow-ups are observed: locally smooth dependence of blow-up timeson initial points, level set distribution of blow-up times, and decompositionof the phase space playing a role as separatrixes among solutions, where themagnitude of initial points near those blow-ups does not matter for asymptoticbehavior. Finally, singular behavior of blow-up times on initial pointsbelonging to different family of blow-up solutions is addressed.

Saddle-Type Blow-Up Solutions with Computer-Assisted Proofs: Validation and Extraction of Global Nature