Encontros IST-IME 

09/09/2016, 09:00 — 09:40 — Anfiteatro Pa2, Pavilhão de Matemática
Bernold Fiedler, Freie Universität Berlin

Sturm global attractors: the three-dimensional balls

Sturm attractors $\mathcal{A}$ are global attractors of dissipative PDEs \[ u_t = u_{xx}+f(x,u,u_x), \] for $0 \lt x \lt 1$, , say with Neumann boundary conditions and hyperbolic equilibria $v$. The collection of unstable manifolds $c_v=W^u(v)$ forms a regular complex of cells $c_v \in \mathcal{C}$, in the sense of algebraic topology. Caveat: this is true for Sturm attractors, but totally wrong in general! We call $\mathcal{C}$ the Sturm complex.

We describe all $3$-ball Sturm complexes, i.e., all $\mathcal{C}$ which arise from the closure of a single $3$-cell $c_0$. We relate our description to the associated Fusco-Rocha meanders, and to the braid $x\mapsto(x,v(x),v_x(x))$ of equilibrium spaghetti. In particular we explain why, against all intuition, the top and bottom equilibria must be chosen as adjacent corners, in the octahedral $3$-cell complex, rather than antipodally opposite.

All results hold, equally, for the Jacobi systems studied by Waldyr M. Oliva. This is joint work with Carlos Rocha, and contains artwork by Anna Karnauhova.

See also http://dynamics.mi.fu-berlin.de/.