Carlos Rocha

Orbit equivalence of global attractors of semilinear parabolic differential equations

We consider global attractors Af of dissipative parabolic equations ut = uxx + f(x, u, ux) on the unit interval 0 ≤ x ≤ 1 with Neumann boundary conditions. A permutation πf is defined by the two orderings of the set of (hyperbolic) equilibrium solutions ut ≡ 0 according to their respective values at the two boundary points x = 0 and x = 1. We prove that two global attractors, Af and Ag, are globally C0 orbit equivalent, if their equilibrium permutations πf and πg coincide. In other words, some discrete information on the ordinary differential equation boundary value problem ut ≡ 0 characterizes the attractor of the above partial differential equation, globally, up to orbit preserving homeomorphisms.

Properties of the attractor of a scalar parabolic PDE

We consider a general scalar one-dimensional semilinear parabolic partial differential equation generating a semiflow with an attractor in an adequate state space. Generalizing known results, it is shown that this attractor is the graph of a function over a compact subset of a finite-dimensional subspace of the state space. In addition, we construct an example with a special interest for the geometric or bifurcation theory of this type of parabolic equations.

Connectivity and Design of Planar Global Attractors of Sturm Type. I: Bipolar Orientations and Hamiltonian Paths

Abstract Based on a Morse-Smale structure we study planar global attractors of the scalar reaction-advection-diffusion equation ut = uxx + ƒ(x, u, ux ) in one space dimension. We assume Neumann boundary conditions on the unit interval, dissipativeness of ƒ, and hyperbolicity of equilibria. We call Sturm attractor because our results strongly rely on nonlinear nodal properties of Sturm type. The planar Sturm attractor consists of equilibria of Morse index 0, 1, or 2, and their heteroclinic connecting orbits. The unique heteroclinic orbits between adjacent Morse levels define a plane graph which we call the connection graph. Its 1-skeleton consists of the unstable manifolds (separatrices) of the index-1 Morse saddles. We present two results which completely characterize the connection graphs and their 1-skeletons , in purely graph theoretical terms. Connection graphs are characterized by the existence of pairs of Hamiltonian paths with certain chiral restrictions on face passages. Their 1-skeletons are characterized by the existence of cycle-free orientations with only one maximum and only one minimum. Such orientations are called bipolar in [de Fraysseix, de Mendez, Rosenstiehl, Discr. Appl. Math. 56: 157–179, 1995]. In the present paper we show the equivalence of the two characterizations. Moreover we show that connection graphs of Sturm attractors indeed satisfy the required properties. In [Fiedler and Rocha, J. Diff. Equ. 244: 1255–1286, 2008] we show, conversely, how to design a planar Sturm attractor with prescribed plane connection graph or 1-skeleton of the required properties. In [Fiedler and Rocha, Connectivity and design of planar global attractors of Sturm type, III: Small and Platonic examples, 2008] we describe all planar Sturm attractors with up to 11 equilibria. We also design planar Sturm attractors with prescribed Platonic 1-skeletons.

An explicit Lyapunov function for reflection symmetric parabolic partial differential equations on the circle

We construct an explicit Lyapunov function for scalar parabolic reaction-advection-diffusion equations under periodic boundary conditions. We assume the nonlinearity is even in the advection term. We follow a method originally suggested by Matano and Zelenyak for, and limited to, separated boundary conditions.

Sturm global attractors for

We consider a semilinear parabolic equation of the form

S^1

u_t = u_{xx} + f(u,u_x)

-equivariant parabolic equations

defined on the circle

Sturm 3-Ball Global Attractors 2: Design of Thom–Smale Complexes

x ∈ S^1=\mathbb{R}/2\pi\mathbb{Z}

Estratigrafia linguística da hidrotoponímia de Portugal continental / Linguistic Stratigraphy of Mainland Portugal’s Hydrotoponymy

. For a dissipative nonlinearity

Examples of attractors in Scalar reaction-diffusion equations

f