CRM CAMP in Nonlinear Analysis

Computer-Assisted Mathematical Proofs in Nonlinear Analysis

May 4, 2021 from 10:00 to 11:00 (Montreal/EST time) Zoom meeting

Rigorous numerics of blow-up solutions for autonomous ODEs

Seminar presented by Kaname Matsue (Kyushu University, Japan)

In this talk, I willl discuss about recent studies concerning rigorous numerics of blow-up solutions for autonomous ODEs in a systematic way under a mild assumption of vector fields. The fundamental tools used here are “compactifications” of phase spaces which map the infinity to the boundary of transformed phase spaces (the “horizon”), and “time-scale desingularizations determined by the original vector fields”. Blow-up solutions are then essentially transformed into solutions on stable manifolds of invariant sets on the horizon. In particular, rigorous enclosures of blow-up solutions and their blow-up times can be validated by means of standard machineries of dynamical systems such as ODE integrators, locally defined Lyapunov functions and parameterization of invariant manifolds. Dynamical system approach shown here reveals many quantitative and qualitative nature of blow-up behavior for various concrete dynamical systems. A series of works presented in the present talk (involving rigorous numerics) are based on joint works with Profs. Akitoshi Takayasu, Nobito Yamamoto and Jean-Philippe Lessard.