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

Abstract In this paper, blow-up solutions of autonomous ordinary differential equations (ODEs) which are unstable under perturbations initial points, referred to as saddle-type , studied. Combining dynamical systems machinery (e.g., compactifications, timescale desingularizations vector fields) with tools from computer-assisted proofs rigorous integrators, the parameterization method for invariant manifolds), these obtained trajectories on local stable manifolds hyperbolic saddle equilibria at infinity. With help proofs, global manifolds, inducing solutions, provide a picture organized by global-in-time and simultaneously. Using proposed methodology, intrinsic features blow-ups observed: locally smooth dependence times level set distribution decomposition phase space playing role separatrixes among where magnitude points near those does not matter asymptotic behavior. Finally, singular behavior belonging different family is addressed.