On the exchange of stability for laminar flows and subharmonic bifurcations.

We consider steady water waves in a two-dimensional channel. The water motion is rotational with constant vorticity. We consider an analytic branch of Stokes waves started from a subcritical laminar flow, where the period is considered as the bifurcation parameter. The first eigenvalue of the Frechet derivative on this branch is always negative. The main object of our study is the second eigenvalue of the Frechet derivative at this branch in a neighborhood of the laminar flow. This is a small eigenvalue, and the positive sign corresponds to the confirmation of the principle of exchange of stability and the negative sign to its violation. We consider the dependence of the sign on the depth of the laminar flow and the value of the constant vorticity. We discuss the connection of the sign of the second eigenvalue with subharmonic bifurcations. We also verify the property of formal stability by a description of the domain of the parameters of the problem, where this property holds. This is joint work with Oleg Motygin (Institute for Problems in Mechanical Engineering, Russian Academy of Sciences, St Petersburg).