Classification of connection graphs of global attractors for S¹-equivariant parabolic equations

Carlos Rocha, Instituto Superior Técnico de Lisboa

Abstract:

We consider the characterization of global attractors A_f for semiflows generated by scalar one-dimensional semilinear parabolic equations of the form

uₜ = uₓₓ + f(u, uₓ),

defined on the circle x ∈ S¹, for a class of reversible nonlinearities.

Given two reversible nonlinearities, f₀ and f₁, with the same lap signature, we prove the existence of a reversible homotopy f_τ, for 0 ≤ τ ≤ 1, which preserves all heteroclinic connections. Consequently, we obtain a classification of the connection graphs of global attractors within the class of reversible nonlinearities.

We also describe bifurcation diagrams that reduce a global attractor A₁ to the trivial global attractor A₀ = {0}.