## Making mathematical sense out of chaotic situations

An EU-funded project has carried out theoretical research on the mathematical viewpoint of chaos. Theoretical insights from mathematics can help research in the physical and biological sciences.

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Chaos is mathematically defined as a situation that presents extremely great sensibility to the initial conditions. A well known example is the so-called butterfly effect, whereby any parameter, no matter its size, can bring about change, so the butterfly flapping its wings in Australia can generate a hurricane in Europe.

Dynamical systems represent the mathematical approach to studying chaos. The EU-funded 'Dynamical complex systems' (DYNEURBRAZ) project set out to investigate systems of low complexity or disorder, and of higher complexity — i.e. systems with positive entropy.

In addition, the project looked at systems where a perturbation can alter the dynamics radically. For example, some of DYNEURBRAZ's results involved the notion of phase transitions, such as when water boils to form steam.

However, the main outcome from the project was related to the classification of numbers and writing them in bases other than the normal base 10. In its study of more complex systems, the project quantified the frequency with which a hyperbolic system is chosen as the random choice among all the possible systems available.

Bifurcation theory is the mathematical study of changes, such as those in the solutions of a family of differential equations. The project has laid the foundations of a bifurcation theory for random dynamical systems, which are highly relevant to many applications.

In the now completed DYNEURBRAZ project, another mathematical theory was developed for the emergence of synchronisation in networks of dynamical systems in order to describe, for example, neuron interactions. This result could help explain some of the paradoxical phenomena that are observed in brain recordings.

Dynamical systems represent the mathematical approach to studying chaos. The EU-funded 'Dynamical complex systems' (DYNEURBRAZ) project set out to investigate systems of low complexity or disorder, and of higher complexity — i.e. systems with positive entropy.

In addition, the project looked at systems where a perturbation can alter the dynamics radically. For example, some of DYNEURBRAZ's results involved the notion of phase transitions, such as when water boils to form steam.

However, the main outcome from the project was related to the classification of numbers and writing them in bases other than the normal base 10. In its study of more complex systems, the project quantified the frequency with which a hyperbolic system is chosen as the random choice among all the possible systems available.

Bifurcation theory is the mathematical study of changes, such as those in the solutions of a family of differential equations. The project has laid the foundations of a bifurcation theory for random dynamical systems, which are highly relevant to many applications.

In the now completed DYNEURBRAZ project, another mathematical theory was developed for the emergence of synchronisation in networks of dynamical systems in order to describe, for example, neuron interactions. This result could help explain some of the paradoxical phenomena that are observed in brain recordings.