 Primary periodic bifurcation of the Kuramoto-Sivashinsky equation

Fix zero mean and period $P$ in the Kuramoto-Sivashinsky equation. At $P=2\pi$ the first Fourier mode becomes neutral and all higher modes are damped. For $k=2\pi /P$, expand $u=a\sin (kx)+b\sin (2kx)+\cdots$. The first equations are $\dot{a}={\lambda }_{1}a+kab/2+\cdots$, $\dot{b}={\lambda }_{2}b-k{a}^2/2+\cdots$, with ${\lambda }_j=(jk{)}^2-(jk{)}^4$. Slaving the second harmonic gives $\dot{a}={\lambda }_{1}a+k^2{a}^3/(4{\lambda }_2)+\cdots$. Since ${\lambda }_2=-12$ at onset, this is a supercritical bifurcation. The small nonzero branch is stable in the sense of orbital stability against disturbances of the same fixed period, modulo translation; stability against disturbances of larger period does not follow from this local argument.