Plane Kolmogorov flows and Takens–Bogdanov bifurcation without parameters: The singly reversible case

Abstract

We consider the Kolmogorov problem of viscous incompressible planar fluid flow under external spatially periodic forcing. Looking for time-independent bounded solutions near the critical Reynolds number, we obtain a dynamical system on a 6-dimensional center manifold. The dynamics is generated by translations in the unbounded spatial direction. Reduction by first integrals yields a 3-dimensional reversible system with a line of equilibria. This line of equilibria is neither induced by symmetries, nor by first integrals. At isolated points, normal hyperbolicity of the line fails due to a transverse double eigenvalue zero. We investigate such a “Takens–Bogdanov bifurcation without parameters” by blow-up and averaging techniques. In particular we describe the complete set ℬ of all small bounded solutions. In the case of a double symmetry of the external force, which leads to a bi-reversible problem, the authors have proved in Asymptot. Anal. 60(3,4) (2008), 185–211, that ℬ consists of periodic profiles, homoclinic pulses and a heteroclinic front–back pair. In the present article we study the more complicated case where only one symmetry is present. Then ℬ consist entirely of trivial equilibria and multipulse heteroclinic pairs. The latter form a very complicated, albeit non-recurrent, set. Graphics of simplest case scenarios for ℬ are included.