Bifurcation analysis of Kolmogorov flows

We examine the bifurcation curves of solutions to the Kolmogorov problem and present the exact formula for the second derivatives of their components concerning Reynolds numbers at bifurcation points. Using this formula, we show the supercriticality of these curves in the case where the ratio of periodicities in two directions is close to one. In order to prove this, we construct an inverse matrix of infinite order, whose elements are given by sequences generated by continued fractions. For this purpose, we investigate some fundamental properties of these sequences such as quasi-monotonicity and exponential decay from general viewpoints