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 of initial conditions are studied. Combining dynamical systems machinery (e.g. phase space compactifications, time-scale desingularizations of vector fields) with tools from computer-assisted proofs (e.g. rigorous integrators, parameterization method for invariant manifolds), these unstable blow-up solutions are obtained as trajectories on stable manifolds of hyperbolic (saddle) equilibria at infinity. In this process, important features are obtained: smooth dependence of blow-up times on initial points near blow-up, level set distribution of blow-up times, singular behavior of blow-up times on unstable blow-up solutions, organization of the phase space via separatrices (stable manifolds). In particular, we show that unstable blow-up solutions themselves, and solutions defined globally in time connected by those blow-up solutions can separate initial conditions into two regions where solution trajectories are either globally bounded or blow-up, no matter how the large initial points are.

Singularities and heteroclinic connections in complex-valued evolutionary equations with a quadratic nonlinearity

“…Such an approach could be directly applied to prove blowup for other initial data, but only if the blowup set is robust and not isolated. In finite dimensional ODEs the recent work [LMT21] has demonstrated computer-assisted proofs of unstable blowup. However it is unclear how to extend this technique to the infinite dimensional case.…”