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

TL;DR

This paper rigorously proves the occurrence of a Hopf bifurcation in the Kuramoto-Sivashinsky PDE using computer-assisted methods, and constructs a family of time-periodic solutions near the bifurcation point.

Contribution

It provides a rigorous, computer-assisted proof of a Hopf bifurcation and constructs time-periodic solutions in the Kuramoto-Sivashinsky PDE, which was not previously established.

Findings

Validated a Hopf bifurcation in the Kuramoto-Sivashinsky PDE.

Constructed a family of time-periodic solutions near the bifurcation point.

Applied a parametrized Newton-Kantorovich method for solution validation.

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.