Rigorous validation of a Hopf bifurcation in the Kuramoto-Sivashinsky PDE

2020/09/28 by Berg, Jan Bouwe van den, Queirolo, Elena
#37L65 (Secondary) #37M20 (Primary) 35B32 #Dynamical Systems (math.DS) #FOS: Mathematics #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.2009.13597

Abstract

We use computer-assisted proof techniques to prove that a branch of non-trivial equilibrium solutions in the Kuramoto-Sivashinsky partial differential equation undergoes a Hopf bifurcation. Furthermore, we obtain an essentially constructive proof of the family of time-periodic solutions near the Hopf bifurcation. To this end, near the Hopf point we rewrite the time periodic problem for the Kuramoto-Sivashinsky equation in a desingularized formulation. We then apply a parametrized Newton-Kantorovich approach to validate a solution branch of time-periodic orbits. By construction, this solution branch includes the Hopf bifurcation point.

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