Rigorous numerics of finite-time singularities in dynamical systems - methodology and applications
For researchers studying dynamical systems with finite-time singularities, this work provides a validated numerical framework that extends rigorous numerics to a broader class of problems, though it is an incremental methodological contribution.
This paper develops a rigorous numerical computation procedure for finite-time singularities in dynamical systems, combining time-scale desingularization, Lyapunov function validation, and standard integration. The method is applied to finite-time extinction, traveling waves with compact support, and singular canards, yielding validated enclosures and composite wave solutions for degenerate parabolic equations.
This paper aims at providing rigorous numerical computation procedure for finite-time singularities in dynamical systems. Combination of time-scale desingularization as well as Lyapunov functions validation on stable manifolds of invariant sets for desingularized vector fields with standard integration procedure for ordinary differential equations give us validated trajectories of dynamical systems involving finite-time singularities. Our focus includes finite-time extinction, traveling wave solutions with half-line or compact support, and singular canards in fast-slow systems, including rigorous validations of enclosures of extinction, finite-passage times or size of supports for compactons. Such validated solutions lead to a plenty of composite wave solutions for degenerate parabolic equations, for example, with concrete information of profiles and evolutions. The present procedure also provides a universal aspect of finite-time singularities with rigorous numerics, combining with rigorous numerics of blow-up solutions in preceding works.