# Dynamical Systems

**Members:** Pablo Aguirre, Isabel Flores

Dynamical Systems theory deals with structures that change in time by means of a certain deterministic rule (the flow of a vector field, a set of differential equations or the iterations of a map). With many examples describing real world phenomena, the motivation to understand the nature of a given dynamical system has led to a growing symbiosis between the theoretical, the applied, and the experimental aspects of the subject.

Dynamical Systems concepts and tools appear naturally in many areas of mathematics and science. For instance, in the study of several kinds of PDEs (radial solutions, traveling waves, pattern formation, etc) and in the search for strategies to control a given system. On the other hand, subareas such as bifurcation and chaos have emerged as the underlying mathematical mechanisms to explain many complicated behaviours ranging from theoretical physics to neuroscience and many avant-garde areas in science and engineering.

Our general aim is to give a qualitative and quantitative description of these systems: existence and stability of equilibria and periodic solutions, homoclinic and heteroclinic connections, bifurcations and routes to chaos, etc. Our methods draw upon topological techniques, linear algebra, invariant manifold reductions, normal forms, and numerical continuation methods, among other techniques.

In AM2V, our research is focused in the following topics: Global bifurcations and invariant manifolds; Phase plane analysis of radial solutions of PDEs; Slow-fast dynamical systems; Numerics for dynamical systems; Applications in biology and engineering.