CRM CAMP in Nonlinear Analysis

Computer-Assisted Mathematical Proofs in Nonlinear Analysis

August 4, 2020 from 10:00 to 11:00 (Montreal/EST time) Zoom meeting

Rigorous numerical investigation of chaos and stability of periodic orbits in the Kuramoto-Sivashinsky PDE

Seminar presented by Daniel Wilczak (Jagiellonian University, Poland)

We give a computer-assisted proof of the existence of symbolic dynamics for a certain Poincaré map in the one-dimensional Kuramoto-Sivashinsky PDE. In particular, we show the existence of infinitely many (countably) periodic orbits (POs) of arbitrary large principal periods. We provide also a study of the stability type of some POs and show the existence of a countable infinity of geometrically different homoclinic orbits to a periodic solution. The proof utilizes pure topological results (variant of the method of covering relations on compact absolute neighbourhood retracts) with rigorous integration of the PDE and the associated variational equation. This talk is based on the recent results [1,2].

[1] D. Wilczak and P. Zgliczyński. A geometric method for infinite-dimensional chaos: symbolic dynamics for the Kuramoto-Sivashinsky PDE on the line, Journal of Differential Equations, Vol. 269 No. 10 (2020), 8509-8548.
[2] D. Wilczak and P. Zgliczyński. A rigorous C1-algorithm for integration of dissipative PDEs based on automatic differentiation and the Taylor method, in preparation.