Classification of connection graphs of global attractors for $S^{1}$-equivariant parabolic equations

We consider the characterization of global attractors \mathcal{A}_{f} for semiflows generated by scalar one-dimensional semilinear parabolic equations of the form u_{t} = u_{xx}+ f(u,u_{x}) , defined on the circle x\in S^{1} , for a class of reversible nonlinearities. Given two reversible nonlinearities, f_{0} and f_{1} , with the same lap signature, we prove the existence of a reversible homotopy f_{\tau}, 0\le\tau\le 1 , which preserves all heteroclinic connections. Consequently, we obtain a classification of the connection graphs of global attractors in the class of reversible nonlinearities. We also describe bifurcation diagrams which reduce a global attractor \mathcal{A}_{1} to the trivial global attractor \mathcal{A}_{0}=\{0\} .