On the exchange of stability for the subcritical laminar flow

We consider steady water waves in a two-dimensional channel bounded below by a flat, rigid bottom and above by a free surface. Surface tension is neglected, and the flow is rotational with constant vorticity $a$. We analyze an analytic branch of Stokes waves bifurcating from a subcritical laminar flow, with the wave period serving as the bifurcation parameter. Along this branch, the first eigenvalue of the Fr\'{e}chet derivative remains negative. Our main focus is the second eigenvalue; its sign plays a crucial role in the analysis of subharmonic bifurcations. This small eigenvalue determines the validity of the principle of exchange of stabilities: a positive sign confirms it, while a negative sign indicates its violation. Furthermore, a positive second eigenvalue corresponds to an increasing period along the bifurcation curve near the critical point, whereas a negative sign implies period decrease. We investigate how the sign of the second eigenvalue depends on the Bernoulli constant $R$ (equivalently, the laminar flow depth $d$) and the vorticity $a$. We show that for each $a$ there exists a critical depth $d_0(a)$ such that the second eigenvalue is positive for $d<d_0(a)$ and negative for $d>d_0(a)$. In the laminar flow, a stagnation point forms when the depth exceeds a threshold $d_s(a)$. We demonstrate that $d_0(a) < d_s(a)$ for $a > a_0 \approx -1.01803$, whereas $d_0(a) > d_s(a)$ for $a < a_0$. We also verify the property of formal stability by a description of the domain in $(a,d)$ variables, where this property holds. Numerical illustrations of these properties are presented in the paper.