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arXiv · 2403.15581

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

Carlos Rocha
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Abstract

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 $A_0=\{0\}$.

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Carlos Rocha. 2024-03-22. Classification of connection graphs of global attractors for $S^1$-equivariant parabolic equations. https://arxiv.org/abs/2403.15581

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