Abstract: In this course we will build computer-aided representations for the global attractors of reaction-diffusion equations in one-dimension. Attractors are objects in a given space that significantly simplify the long-term dynamics of an evolution system. Knowing them is something extremely useful for a certain problem at hand, but there are few examples where it is possible to describe them precisely.

We will present tools that allow us to write the attractor for the one-dimensional reaction-diffusion problem as fixed points in space (known as equilibria) and connections between these points. We shall combine the existing theory on the subject with the use of computer programs to assist in this task. The course is aimed at a wide audience, not requiring much prior knowledge and it is thought for a 3-day course, approaching the following topics:

Day 1 – Introduction to the scalar reaction-diffusion equation: Existence of Lyapunov function, global attractor and structure of the attractor given as equilibria and connections between them.

Day 2- By using the Hamiltonian structure of the associated elliptic problem, we define the Time-map associated to the system and search for the equilibria for the equation.

Day 3 – Once the equilibria are determined, nodal properties of those solutions are used to establish connections between them.

Some of the references used to build this course are:

[1] Brunovsky, P., and Chow, S.-N. Generic properties of stationary state solutions of reaction-diffusion equations. J. Differential Equations 53, 1 (1984), 1-23.

[2] Fiedler, B., and Rocha, C. Heteroclinic orbits of semilinear parabolic equations. J. Differential Equations 125, 1 (1996), 239-281.

[3] Rocha, C. Realization of period maps of planar hamiltonian systems. Journal of Dynamics and Differential Equations, 19(3):571-591, 2007.