Numerical validation of blow-up solutions with quasi-homogeneous compactifications
It offers a rigorous numerical validation framework for blow-up phenomena, which is important for researchers studying singularities in differential equations.
The paper presents a numerical method to rigorously validate blow-up solutions in finite-dimensional vector fields with asymptotic quasi-homogeneity, providing rigorous bounds on blow-up times and solution profiles.
We provide a numerical validation method of blow-up solutions for finite dimensional vector fields admitting asymptotic quasi-homogeneity at infinity. Our methodology is based on quasi-homogeneous compactifications containing a new compactification, which shall be called a quasi-parabolic compactification. Divergent solutions including blow-up solutions then correspond to global trajectories of associated vector fields with appropriate time-variable transformation tending to equilibria on invariant manifolds representing infinity. We combine standard methodology of rigorous numerical integration of differential equations with Lyapunov function validations around equilibria corresponding to divergent directions, which yields rigorous upper and lower bounds of blow-up times as well as rigorous profile enclosures of blow-up solutions.