## Final Report Summary - DYNEURBRAZ (Dynamical complex systems)

The present research is theoretical research in mathematics. It concerns the mathematical viewpoint for chaos. The mathematical definition for chaos is a situation which presents extremely big sensibility to the initial conditions. The best image for this is the billiard: there is the theoretical play and the realisation of it. It may be very close to the theoretical ones but, any small difference will have very big consequences in a short future. Another well-known example is the butterfly effect, which actually means that it is impossible to predict the weather for a long period due to our impossible knowledge of all the parameters. Even if our knowledge is big, any small unknown thing (as the behaviour of one butterfly) may have big consequences: the butterfly in Sydney bay generates a hurricane in Europe.

The mathematical study of chaotic systems is usually done via the iterations of a map, which represents the time evolution. The goal is to obtain the maximum of information on the orbits of this map, that is the evolution of the system under time evolution.

An important point is that, even if the situation is chaotic (in the sense described above) the systems still get some regularity at a larger time scale. Most of the trajectories have the same behaviour if we let the system go for sufficiently long time. To use the image of the billiard, even if the real play is not the theoretical one, if the ball has sufficient energy, then statistically, it will eventually visit the whole border of the billiard with a deterministic law.

Some of the results obtained with support of DYNEURBRAZ involve the notion of phase transitions and the mathematical way to describe them. A phase transition is, in physics, for instance the boiling water. But recent researches in astronomy go in the direction that the big bang was not a big explosion, but a kind of phase transition. This illustrates what we means when we talk about 'larger time scale' and then, why we said the present researches are theoretical. As the research deal with the study of this kind of phenomenon, no direct consequence in the common life is a priori expected.

The programme was divided in 3 work packages (WPs):

The first concerned system with low disorder, that is quasi-periodic systems. A periodic system is for instance a crystal, or a regular tilling of the plane. A quasi-periodic system is not periodic but almost periodic. These kind of problems (see e.g. the well-known Penrose tilling) are related to number theory (for instance how numbers can be written in another basis than the usual 10-basis).

The principal result obtained via DYNEURBRAZ are then related to the classification of numbers with respect to their writing in different basis, and also some result describing orbits for systems (the 'contracting interval exchange') with low complexity / disorder.

The second WP deals with systems of higher complexity, that is systems with positive entropy. For most of them, the entropy is generated by a special behaviour called 'hyperbolic'. Then, two kinds of results were obtained via the DYNEURBRAZ project.

Firstly, the quantification of the number of systems that have this hyperbolic behaviour, or, somehow equivalently, the random pick of a system amongst all the possible systems with some restrictive conditions, the frequency with which we choose a hyperbolic one.

Secondly, a description of such a system. We said above that 'statistically' all the trajectories have the same behaviour. The goal was to define what 'statistically' means. There are usually several ways to define it, and the aim was to describe these different ways and show that some are more relevant than others. To refer to phase transition mentioned above, if several ways to define statistically are relevant, then there is a phase transition: at 100 degrees of Celsius the water is either liquid or vapour, and both states yield different 'statistics'.

The last WP concerned bifurcation theory. The goal was to study regularity of the solutions of some ordinary differential equations (ODEs) or partial differential equations (PDEs). We remind that these equations involve a relation between a function and its derivatives as the velocity, the acceleration and so on. These equations depend on some parameters and, every small random change of these parameters yields completely different solutions. This is what we mean by 'bifurcation'. The project aimed to lay the foundations for the development of a bifurcation theory for random dynamical systems, which are highly relevant to applications but surprisingly almost unexplored.

Beyond this, results were obtained to develop a mathematical theory for the emergence of structured synchronisation in heterogeneous networks of interacting dynamical systems (to describe e.g. neurons interactions).

An important work was also done in view to classify the type of singularities / bifurcation, that is the set of parameters which are at the border between to different dynamics.

The mathematical study of chaotic systems is usually done via the iterations of a map, which represents the time evolution. The goal is to obtain the maximum of information on the orbits of this map, that is the evolution of the system under time evolution.

An important point is that, even if the situation is chaotic (in the sense described above) the systems still get some regularity at a larger time scale. Most of the trajectories have the same behaviour if we let the system go for sufficiently long time. To use the image of the billiard, even if the real play is not the theoretical one, if the ball has sufficient energy, then statistically, it will eventually visit the whole border of the billiard with a deterministic law.

Some of the results obtained with support of DYNEURBRAZ involve the notion of phase transitions and the mathematical way to describe them. A phase transition is, in physics, for instance the boiling water. But recent researches in astronomy go in the direction that the big bang was not a big explosion, but a kind of phase transition. This illustrates what we means when we talk about 'larger time scale' and then, why we said the present researches are theoretical. As the research deal with the study of this kind of phenomenon, no direct consequence in the common life is a priori expected.

The programme was divided in 3 work packages (WPs):

The first concerned system with low disorder, that is quasi-periodic systems. A periodic system is for instance a crystal, or a regular tilling of the plane. A quasi-periodic system is not periodic but almost periodic. These kind of problems (see e.g. the well-known Penrose tilling) are related to number theory (for instance how numbers can be written in another basis than the usual 10-basis).

The principal result obtained via DYNEURBRAZ are then related to the classification of numbers with respect to their writing in different basis, and also some result describing orbits for systems (the 'contracting interval exchange') with low complexity / disorder.

The second WP deals with systems of higher complexity, that is systems with positive entropy. For most of them, the entropy is generated by a special behaviour called 'hyperbolic'. Then, two kinds of results were obtained via the DYNEURBRAZ project.

Firstly, the quantification of the number of systems that have this hyperbolic behaviour, or, somehow equivalently, the random pick of a system amongst all the possible systems with some restrictive conditions, the frequency with which we choose a hyperbolic one.

Secondly, a description of such a system. We said above that 'statistically' all the trajectories have the same behaviour. The goal was to define what 'statistically' means. There are usually several ways to define it, and the aim was to describe these different ways and show that some are more relevant than others. To refer to phase transition mentioned above, if several ways to define statistically are relevant, then there is a phase transition: at 100 degrees of Celsius the water is either liquid or vapour, and both states yield different 'statistics'.

The last WP concerned bifurcation theory. The goal was to study regularity of the solutions of some ordinary differential equations (ODEs) or partial differential equations (PDEs). We remind that these equations involve a relation between a function and its derivatives as the velocity, the acceleration and so on. These equations depend on some parameters and, every small random change of these parameters yields completely different solutions. This is what we mean by 'bifurcation'. The project aimed to lay the foundations for the development of a bifurcation theory for random dynamical systems, which are highly relevant to applications but surprisingly almost unexplored.

Beyond this, results were obtained to develop a mathematical theory for the emergence of structured synchronisation in heterogeneous networks of interacting dynamical systems (to describe e.g. neurons interactions).

An important work was also done in view to classify the type of singularities / bifurcation, that is the set of parameters which are at the border between to different dynamics.