FENOMEC (Proyecto Universitario de Fenómenos Nolineales y Mecánica) is the University Project of Nonlinear Phenomena and Mechanics and its purpose is to support research, teaching and dissemination activities, which tend to establish and strengthen collaborations between departments of the National Autonomous University of Mexico (UNAM). It is a flexible instrument that promotes the global and integrated school of thought in the field of Nonlinear Phenomena and Mechanics.

For more than two decades, a group of academics from different units of the UNAM, convinced of the importance of taking advantage of the existing structures and talents in our university, have given themselves the task of establishing and promoting an integrated school of thought; resolutely support research activities, teaching and dissemination of results, and promote the establishment and strengthening of collaboration between departments of the University and a mosaic of similar national and international institutions, in which there are prominent academic personalities with common interests and who are part of the project as members, advisers or scientific correspondents of the project.

In order to consolidate research, teaching and dissemination activities in the field of non-linear phenomena and contribute in an original way to its progress, FENOMEC actively participates in the training of students with the ability to integrate into the growing demands of this field and take advantage of an increasingly sophisticated global environment, characterised by the linking and interrelation of different scientific disciplines to address the solution of complex problems of great importance.

The formulation of unifying ideas from the mathematical point of view, allows us to understand, analyse and communicate the complex relationships underlying a huge range of non-linear phenomena and problems, characterised by presenting such richness and complexity that they require the focused assistance of specialists in various scientific disciplines, collaborating closely in a horizontal and inclusive way to achieve results that can effectively be translated into concrete solutions and applications.

Mathematics is the seed of all large structures. It is useful in the study of physics, economics, chemistry and in any phenomenon whose most important characteristics can be abstracted in a mathematical model, so that a result can be projected from the functions and behaviour of an equation.

However, not all models are linear, and although these non-linear ones were previously considered curiosities, their study is expanding rapidly today.

Proof of this is the Nonlinear and Mechanical Phenomena University Project (FENOMEC), directed by Dr Jorge Ize Lamache and made up of 30 members in 10 different UNAM units, all interested in applying this knowledge to their different areas of work.

To understand non-linear mathematical models, Dr Jorge Ize (born 1946) from UNAM's Applied Mathematics Research Institute (IIMA) explains how classical linear models work:-

If I apply a certain force to an object, I'm going to get a response; if I apply double or triple the force, I will cause the same response to double or triple. The result behaves linearly. The drawback is that these are approximations and not exact models.

One of the oldest examples showing the use of linear and non-linear models is the Malthusian model of population growth, which is linear and predicts that the population will grow exponentially until it reaches a population explosion. However, when the population grows so much that the resources to sustain it no longer suffice, a self-regulation mechanism begins to work, which is explained with a non-linear logistic model.

According to Dr Jorge Ize, nonlinearity is characterised by being very diverse, in such a way that even chaos forms a small part of nonlinear models; that is, it is not predictable if the nonlinear model is going to predict the result. However, the researcher emphasises, his interest is focused on predicting the results as accurately as possible. Jorge Ize explains:-

For example, there is a group specialising in marine sciences that works with a water pump which weighs liquid going from the sea to a lagoon and vice versa. The device already exists, it has been calibrated experimentally, but it is still necessary to know if the dimensions used are optimal or what is needed to make it work better. That's where the nonlinear mathematical modelling part comes in.

Ize Lamache adds that in the study of these models there are not always firm theories, so joint work by those who carry out the experiment and the theoretical part is required:-

No one can say: there are the equations and this is the result. It is an interdisciplinary work. The impact from linear modelling research has been significant, but these results are approximations. On the other hand, nonlinear ones can explain, for example, how light pulses - solitons - are transported in optical fibres. If we consider the technological impact of the development of optical fibres, we can consider the impact of non-linear models.

As coordinator of FENOMEC, Jorge Ize - with a master's degree in physics from the University of Lyon, France, and a master's degree in science and a doctorate in mathematics from New York University - explains that the purpose of the project is to avoid the tendency of scientists to isolate themselves, to work in their offices, to produce many articles and not see further:-

The ultimate goal is to build a new way of thinking about science. Approach from different angles with the collaboration of various disciplines. To have important advances, it is necessary to combine the various parts of a whole. Being a participant in a way other than the orthodox on equal terms. The idea is to create a more global science: a school of thought.