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

Article type: Research Article

Authors: Afendikov, Andrei | Fiedler, Bernold | Liebscher, Stefan

Affiliations: Keldysh Institute of Applied Mathematics, Moscow, Russia. E-mail: [email protected] | Free University of Berlin, Institute of Mathematics I, Arnimallee 2-6, D-14195 Berlin, Germany. E-mails: {fiedler, liebsch}@math.fu-berlin.de

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 use the Kirchgässner reduction to obtain a spatial 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. In particular we describe the complete set ℬ of all small bounded solutions. In the classical Kolmogorov case, ℬ consists of periodic profiles, homoclinic pulses and a heteroclinic front–back pair. This is a consequence of the symmetry of the external force.

DOI: 10.3233/ASY-2008-0901

Journal: Asymptotic Analysis, vol. 60, no. 3-4, pp. 185-211, 2008