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).
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.
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 when we consider at the same time the attraction of three or more bodies. Due to its beauty and geometric complexity, we can find a good example in the so-called hip-hop orbits for 2n bodies of equal masses. Another example are the p-q resonant orbits, in which a body of small mass carries out p turns around a second body (for example, a planet), while this second body carries out q turns around a third main body (the Sun). This type of orbits are of interest in Astrodynamics applications, for example, for the design of missions with one or more fly-by (a close passage to a planet) in order to reduce the speed of approach to a body.
Mostly, the research that is developed focuses on the context of the restricted three body problem (two bodies, the Sun and a planet for example, and a third body of no significant mass) and the description of the dynamics around the equilibrium points of the system. These points are of great importance because they are the natural places where a small object affected only by gravity can theoretically be stationary relative to two massive bodies, while they carry out tasks of data collection. In particular, the design of trajectories of low cost that allow to bring the satellites until these points and to retrieve them is another point of great interest.
Other subjects of study in Astrodynamics have to do with the determination of trajectories of sets of artificial satellites, called constellations of satellites, so that all of them follows a nominal trajectory while maintaining its relative position.
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. E. Barrabés.
Applications and modelling
In the former sections it has been shown the interest of the members of the group in the applications of its research. In this sense, it is also necessary to emphasize other very different areas where the research of the group EDMA has participated. Some examples are the mechanical systems, the medicine or, even, the dancesport.
- 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.
- Rhythmic quality in dancesport
The goal of this project is to obtain a method to measure the rhythmic quality in the dancesport, where the rhythm is one of the criteria used in the punctuation of the couples that are competing. In spite of its evident importance, the concept of rhythm is not as clear as it may seem and, therefore, leads to subjective interpretations. In this sense, the project aims at giving objectivity to this criterion: first, analyzing what is really understood by rhythm in the dancing and giving a mathematical representation of this concept; and second, designing a method that in practice allows measuring this rhythm and checking out the quality. All this, always in accordance with the opinion of the experts.
- Modelling of the wound healing
The process of wound healing is a very natural phenomenon, but also extremely complex from a biochemical point of view. That is why that, in spite of the great amount of research that they generate, these mechanisms are only partially understood and, sometimes, have only been analyzed from an experimental point of view. During the last years, however, the contribution of the mathematical modelling to the theoretical study of these processes has been significant. The models of the different phases that take place during the healing are useful for a better understanding of the usual mechanisms. But also to analyze other unknown aspects, as for instance the reasons for which some of these processes fail in patients with certain illnesses (as the diabetes or some infections), causing problems of healing, like chronic wounds. Moreover, the numeric simulation of these models also represents a great tool when analyzing the efficiency of new treatments.
- Cartograms and transformations with given Jacobian
The cartograms are maps where the representation of each territory is proportional to a datum different of its size (for example, the population). From the mathematical point of view, this problem is solved finding a function (change of variables) such that its Jacobian is given (a density). The algorithm of Moser allows to find this function. Our goal is to study the regularity of this algorithm when the density is a non regular function.
- 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.
- Power consumption of integrated circuits
A system-on-chip (SOC) is an integrated circuit in which all the electronic components are included on a single chip. The design of such a device is in general complex and costly, and one has to ensure that each element obtains the right amount of power to operate efficiently. To increase the speed of operation and general performance, large voltage drops between different parts of the circuit must be avoided. In particular, it is in the design of the interconnection between the semiconductor devices and the external circuitry where special care is needed. Electronic engineers have the need for good formulae to compute the maximum voltage drop, easy to handle and ready to be used in practical tasks of chip design. The derivation of such formulae is not a straightforward task since one needs to find first a convenient model which should be simple enough to allow some sort of analytical solution but complex enough to incorporate the most important phenomena.