Classification of global attractors for ${\mathbb{S}}^1$-equivariant parabolic equations: a survey

We survey the global dynamics of semiflows generated by scalar semilinear parabolic equations which are $\mathbb{S}\mathbb{O}(2)$ equivariant under spatial shifts of $x\in {\mathbb{S}}^1=\mathbb{R}\mathrm{/}2\pi \mathbb{Z}$, i.e. $$u_t=u_{xx}+f(u,u_x),x\in {\mathbb{S}}^1\mathrm{.}$$ For dissipative $C^2$ nonlinearities $f$, the semiflow possesses a compact global attractor $\mathcal{A}={\mathcal{A}}^{\mathcal{P}}$ which we call Sturm attractor. The Sturm attractor ${\mathcal{A}}^{\mathcal{P}}$ decomposes as $${\mathcal{A}}^{\mathcal{P}}=\mathcal{E}\cup {\mathcal{F}}^{\mathcal{P}}\cup {\mathcal{R}}^{\mathcal{P}}\cup {\mathcal{H}}^{\mathcal{P}},$$ where ${\mathcal{H}}^{\mathcal{P}}$ denotes heteroclinic orbits between distinct elements of spatially homogeneous equilibria $\mathcal{E}$, rigidly rotating waves ${\mathcal{R}}^{\mathcal{P}}$ and, as their non-rotating counterparts, frozen waves ${\mathcal{F}}^{\mathcal{P}}$. We therefore represent ${\mathcal{A}}^{\mathcal{P}}$ by its connection graph ${\mathcal{C}}^{\mathcal{P}}$, with vertices in $\mathcal{E},{\mathcal{F}}^{\mathcal{P}},{\mathcal{R}}^{\mathcal{P}}$ and edges ${\mathcal{H}}^{\mathcal{P}}$. Under mild hyperbolicity assumptions, the directed graphs ${\mathcal{C}}^{\mathcal{P}}$ are finite and transitive. For illustration, we enumerate all 21 connection graphs ${\mathcal{C}}^{\mathcal{P}}$ with up to seven vertices. The result uses a lap signature of period maps associated to integrable versions of the steady state ODE of our PDE. As an example, we freeze and reconstruct the connection graph of the Vas tulip attractor, known from delay differential equations, in our PDE setting.