Main research lines:

• Qualitative theory of ordinary differential equations (ODE) and Piecewise Dynamical Systems (PWS). Continuous Dynamical Systems.
• Qualitative theory of difference equations. Discrete Dynamical Systems.
• Dynamical systems applications to neuroscience.

Framework of our research:

• Qualitative study of dynamical systems defined by differential equations, with applications in neural models.

Areas inthe framework: Our research projects address problems in three clearly non-disjointed. The advantage that the three areas share varied intersections makes the knowledge and progress in each of them reinforce knowledge and progress in the rest. Specifically, we propose to address the following three areas:

• periodic behaviors (centers, limit cycles, canard cycles in singularly perturbed systems, etc.) in low-dimension differential systems
• piecewise systems of dimension two and greater than two
• applications to neuroscience, among other areas.

The techniques we use: in general we use those of the qualitative theory of ordinary differential equations. More concretely,

• First, obtaining lower and upper bounds of the number of limit cycles of a differential equation is one of the objectives we propose. Among others, we study the number of limit cycles for families of Abel or polynomial differential equations. Likewise, we propose to obtain upper bounds criteria of the number of limit cycles in terms of the coefficients of the system, obtaining families verifying these criteria and to develop the tools that allow them to be obtained. For this we will be inspired by the algebraic tools used in the center-focus problem.
• Secondly, concerning piecewise linear differential equations (PWL), they are an example in which, through the concatenation of linear behaviors, a strongly non-linear and dynamic behavior is achieved. Related with singularly perturbed systems, our objective is twofold: first, to continue developing the elements of a theory of singular perturbations in the context of PWL systems, and on the other, to deepen the applications of these systems in the development of models of real systems, in particular in the field of Neuroscience.
• Finally, in the application of these results to Neuroscience we also propose the modeling of type I neurons (SNIF type bifurcation) and type II neurons (Hopf bifurcation) and, in addition, we will deal with problems applied in the field of: synaptic conductance estimation and problems in Neural networks. In the section on applications to the social sciences, we present a problem about how populations evolve to states of equilibrium and, more specifically, whether they tend to stabilize their growth or, on the contrary, tend to other states in an asymptotic way.

Coordinated reseach: Besides more specific projects, our global research project coordinates the research activity of two Spanish research groups from:

• Universidad de Extremadura
• Universitat de les Illes Balears

In this way we integrate and strength our research work, in the direction of breaking the trend towards fragmentation of research groups.

Keywords: Dynamical systems, Piecewise linear systems, Neuroscience, Abel equation, Singular perturbations.