Consider $u_t=u_{xx}+u(1-u^2), x\in\mathbb{R}$
with initial conditions $u_0(x)=u_0(x+\frac{2\pi}{k})$.
Now, in this paper what seems to happen is that $k$ is considered to be the bifurcation parameter and the equlibria are the periodic solutions, isn't it? (Figure 1.1)
I am wondering about the fact that the bifurcation parameter seems to be in the initial conditions and not in the equation itself.
Is that something special or how would one handle this (I have never seen something like this).