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

We consider the characterization of global attractors $A_f$ 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 $A_1$ to the trivial global attractor $A_0=\{0\}$.