# Research lines

## Complex Networks

Many aspects of the physical, biological and technological world are organized in systems of many elements with complicated connection patterns. The networks of protein interaction or of genetic regulation, social networks, networks of electrical distribution, the Internet or the WWW are examples of these systems which are known as *complex networks*. These can present a very big range of sizes and complexities, and are often characterized by obeying to external stimuli in a collective way so that they show the so-called emergent *estates*. The figure represents the food web of the species that inhabit a sea obstacle; **http://thecity.sfsu.edu/~wow/gallery_index.html**.

In recent years, data coming from these networks begin to be collected. Moreover, the study of these large amounts of data requires a computational capacity that has not been developed until recently. That is why the first mathematical models for the growth of these networks appeared at the end of the nineties. The study of the properties of the architecture of these networks according to their different growth mechanisms is one of the goals of this line of research in the group.

But networks are not only an object of study by themselves but also they define the setting where different processes take place: information transmission, propagation of energy, etc. One of the most interesting cases is that of the spread of epidemics in social networks or of virus in the Internet. In the classical models of epidemiology, it is assumed that everybody has contact with everybody when studying the spread of infectious diseases. This precept is nowadays being replaced by the fact of including the structure of the network as an ingredient of the epidemiological models. The so-called *epidemiology in networks* constitutes the second subject of study that has recently been started in our group. A clarifying example is offered by the outbreak of the SARSin China in November of 2002. This is a sample of how much important it is to take into account the pattern of contacts among individuals when making predictions about possible pandemics. In this case, the pessimistic initial predictions when the disease reached other countries like Canada -the OMS made a world-wide alert on March of 2003 - were not accomplished, as in July of 2003 the epidemic became** **controlled **(****http://es.wikipedia.org/wiki/SARS****).**

People working in this area are: Dr. A. Avinyó, Dr. D. Juher, Dr. J. Ripoll, Dr. J.L García-Domingo, Dr. J. Saldaña.

## Dynamics of structured populations and adaptative dynamics

When populations (animals, plants or even polymers in processes of industrial synthesis) are studied with a certain level of detail, one can observe that all their members are not equal. They show differences in age, size, social rank, etc. The description of the dynamics of these populations *structured* by one or more internal variables is generally made using Partial Differential Equations. The study of the properties of the solutions of these equations is one the lines of more tradition in the group. The thesis of three of our members has been on this subject.

On the other hand, knowing the properties of the equations that describe the dynamics of structured populations has very important applications in the formality of the so-called renewable natural resources as, for instance, the fisheries or the forest exploitation. In these fields, it becomes crucial for a sustainable management of the resources to know the implications that different types of exploitation can have about the future of the populations of interest.

Also, the fact that the individuals of biological populations are structured by the age (or by the size) allows to formulate in a very natural way questions of optimization as "which is the optimum age of maturation?”;" What it is better: doing A since arriving at a given size M and doing B afterwards, or doing the opposite? Or even doing both at the same time but investing less energy every time in one of both actions?”; "At which age it is better to start to vaccinate in order to prevent an infectious outbreak?” The theory of evolutionary games applied to this type of models constitutes the third great subject of research of this line. In this context, a strategy is given by the value that takes a certain feature or inheritable phenotypic characteristic, like for example the age of maturation. In particular, we are interested in the calculation of the so-called *evolutionary* *stable strategies *(or invincible when they are adopted by a majority of individuals) for different models of structured populations and phenotypic characteristics.

People working in this area are: Dr. J. Ripoll, J. Saldaña, Mr. J. Font.

External collaborators are: Dr. M.A. Greischar (Cornell University, US).

## Celestial mechanics

Another goal in the research carried out by members of the group is the analysis of the equations that describe the motion of the natural and artificial bodies under their mutual gravitational attractions, that is the equations of Celestial Mechanics. In spite that these equations are known since time ago, their solutions, which describe the trajectories in space of the different bodies, are not known when we consider at the same time the attraction of three or more bodies. We apply the analytical tools of Dynamical Systems, essentially, the study of existence of invariant objects and the existence of invariant manifolds, their behavior, intersections, etc. In particular, the models of Celestial Mechanics are of great interest in Astrodynamics applications, for example, for the design of missions with desired trajectories or with one or more fly-by (a close passage to a planet) in order to reduce the speed of approach to a body. In particular, the design of trajectories of low cost is another point of interest.

People working in this area are: Dr. E. Barrabés, in collaboration with the UPC Dynamical Systems Group and other researchers from Barcelona, México, Chile and USA.

Main external collaborators: G. Gómez (UB), M. Ollè, J.M. Cors, J. Soler (UPC), J.M. Mondelo, C. Pinyol (UAB), G.R. Hall (Boston University, MA, USA), M. Álvarez, M. Medina (Universidad Autónoma Metropolitana-Iztapalapa, México), C. Vidal, J. Andrade (Universidad del Bío-Bío, Chile), F. Borondo (UAM), D. Farrelly (Utah State University, USA), A. C. Fernandes (U. Federal de Itajubá, Brasil).

## Topological dynamics and discrete dynamical systems

The iteration of a continuous self-map f defined on a metric space X defines what is called a *discrete dynamical system*. For each point x in the space one can study the *orbit of x*, that is, the set of points obtained by mapping f on x over and over. This set can either be finite, in the sense that eventually one gets the initial value x (then we speak of a *periodic point* or a *periodic orbit*) or infinite, and in this case the orbit can either "fill" almost all the space X or stay inside a bounded subspace. When continuity is the only hypothesis on the map f, the study of the properties of such orbits of points is called *topological dynamics*.

Some members of the EDMA Group work on some problems in topological dynamics related to the set of periods of maps defined on 1-dimensional spaces (circles, intervals, trees and graphs) and their topological entropy. The *set of periods* is the set of the periods of all periodic orbits exhibited by the map. While the *topological entropy* is a measure of the degree of dynamic complexity of the system. A zero entropy map is "simple" and, for instance, it does not exhibit the phenomenon of *chaos*. On the other hand, the larger the entropy is, the more complex the mixing of points in the space becomes by iteration of f. The applicability of this branch of Mathematics relies on the fact that the map f can model a wide variety of empiric situations.

People working in this area are: Dr. D. Juher, Dr. D. Rojas.

Main external collaborators: Lluís Alsedà, Jérôme Los, Francesc Mañosas, Deborah King.

## Mechanical systems with dissipation

Motivated by its importance in Engineering, the mechanical suspension-damping systems have been object of research in the group. These systems can be found for example in vehicles, but also as part of the structure of buildings, especially of those placed in zones of high risk of seismic activity. The classical model for this type of systems (an Ordinary Differential Equation) does not take into account phenomena like possible internal differences in the deformation of the spring, the internal dissipation of the device, or possible external controls in order to regulate the state of the mechanism. This has brought us to a model of Partial Differential Equations for this type of systems. The asymptotic behaviour (at long time) of the solutions of this model and the comparison with the classical approach form part of this study.

People working in this area: Dr. M. Pellicer

## Spiral waves in oscillatory and excitable systems

There is a wide variety of physical, chemical and biological systems exhibiting spiral wave patterns. A rough classification of spiral waves would distinguish between the cases in which spiral waves propagate in media which are able to produce oscillations in homogeneous situations and the case in which the oscillation takes place only in the presence of the waves. Spirals in oscillatory media such as in the Belousov-Zhabotinski reaction, are very often modeled by the complex Ginzburg-Landau equation. The second type of spirals arise typically in the propagation of waves in the so-called excitable media, which are often studied with the Fitzhugh-Nagumo model. The mathematical theory that describes the waves in these two cases is fundamentally different although basic questions such us their stability or the origin of these singular solutions remain still unanswered.

People working in this area: Dr. M. Aguareles

Main external collaborators: Prof. S.J. Chapman, Prof. T. M-Seara, Dr. I. Baldomá.

## Mathematical modeling of industrial processes.

Many industrial processes can be described by means of basic laws of physics which, when combined together, can lead to complex systems of partial differential equations. These equations are typically analysed by means of perturbation methods and asymptotic analysis techniques, with which explicit approximate solutions can sometimes be obtained. These approximate solutions, combined with direct numerical computations, reveal the role of the different parameters providing a deep understanding of the relevant mechanisms. This deep information can be used to assess experiments, with special focus on upscaling, optimize the actual industrial process, suggest alternative operational regimes, etc. In fact, nowadays Industrial Mathematics is a rather loose term and it seems to cover almost any application of mathematics in a practical context.

The group is currently focused on applications in environmental sciences with special focus on:

**Contaminant removal from liquids or gases**

This topic comprises the analysis of sorption phenomena and photocatalysis. The physics involved are typically a combination of transport phenomena, energy transfer, fluid dynamics and electromagnetism. This research is performed in active collaboration with the experimentalists of the Laboratory of Chemical and Environmental Engineering (LEQUIA), which is a research group of the University of Girona devoted to the development of eco-innovative environmental solutions.

**Modeling of green roofs**

A green roof is a layer of vegetation planted over a waterproofing system that is installed on of a flat or slightly-sloped roof in order to improve the energy efficiency of buildings and to reduce the heat island effect. The main goal of this research topic is to analyse the energy storage capabilities of different configurations, with different vegetative coverage and/or different evapotranspiration conditions and to provide tools to manufacturers to easily decide on which configuration is best depending on the needs of the building and on the weather conditions in a given location.

People working in this area are: Dr. M. Aguareles, Dr. E. Barrabés.

Main external collaborators: Prof. T. Myers (CRM-UAB), Dr. F. Font (UPC), Dr. A. Valverde (CRM-UAB), Dr. A. Cabrera (LEQUIA-UdG), Dr. M. Calvo-Schwarzwalder.