Loss of stability and bifurcation of auto-oscillatory regimes close to Poiseuille flow (Q1183858)

Bifurcation of Poiseuille flow in a flat channel is used as an example to analyze the problem of determining variables that permit study of bifurcation of a main steady flow of a viscous incompressible liquid for parameters close to the values of the coordinates of a point on the curve of neutral stability at which the first Lyapunov exponent \(d_ 0\) vanishes and there is a changeover from subcritical to supercritical bifurcation. For Poiseuille flow, such a point \((R_ 2,\alpha_ 2)\), where \(R_ 2\) is the Reynolds number, and \(\alpha_ 2\) is the wave number, occurs on the lower branch of the neutral curve. In this paper, it is shown by the Lyapunov-Schmidt method that for \(\alpha<\alpha_ 2\) the stable time-periodic solution that bifurcates into the subcritical region loses stability in the case of slight supercriticality, and a fold singularity is formed in the amplitude surface.