A Posteriori Verification of Invariant Objects of Evolution Equations: Periodic Orbits in the Kuramoto--Sivashinsky PDE

Abstract

In this paper, a method for computing periodic orbits of the Kuramoto--Sivashinsky PDE via rigorous numerics is presented. This is an application and an implementation of the theoretical method introduced in [J.-L. Figueras, M. Gameiro, J.-P. Lessard, and R. de la Llave, “A framework for the numerical computation and a posteriori verification of invariant objects of evolution equations,” SIAM J. Appl. Dyn. Syst., to appear]. Using a Newton--Kantorovich-type argument (the radii polynomial approach), existence of solutions is obtained in a weighted $\ell^\infty$ Banach space of Fourier coefficients. Once a proof of a periodic orbit is done, an associated eigenvalue problem is solved and Floquet exponents are rigorously computed, yielding proofs that some periodic orbits are unstable. Finally, a predictor-corrector continuation method is introduced to rigorously compute global smooth branches of periodic orbits. An alternative approach and independent implementation of [J.-L. Figueras, M. Gameiro, J.-P. Less...