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Networks of Delay-Couples Oscillators: how delay and topology affect synchronisation and dynamical properties

Wednesday, 29 June, 2011 - 16:00
Campus: Brussels Humanities, Sciences & Engineering campus
Faculty: Science and Bio-engineering Sciences
Ottilde D'Huys
phd defence

Delayed dynamical systems arise in various scientific fields; delays occur in the reaction of a
driver in traffic dynamics, the incubation time of a disease, in axonal transmission in neural
networks, or in the propagation time of light between two coupled semiconductor lasers, to
name only a few examples. Such time delays can have dramatic influences which, as we show
here, can be common to different systems despite their very different nature. The field of
delayed complex systems is nowadays a quickly emerging field in multidisciplinary research.

In this thesis we study different aspects of the dynamics of delay-coupled networks, focusing
on the interplay between delayed coupling and the onset of spatio-temporal patterns in
networks. We approach this problem mainly analytically, complemented by numerical
simulations. Starting point were the synchronisation properties of delay-coupled
semiconductor lasers, but this thesis has addressed much more general questions: which
networks synchronise? Why do some networks synchronise while others do not? Which is the
role of a coupling delay, the role of the network topology? Which properties are specific for
semiconductor lasers? To address these questions, we have studied a hierarchy of models with
different levels of generality and complexity.

In networks of Kuramoto oscillators, the simplest model, in which an oscillator is only
described by its phase - we study the effect of a coupling delay on periodic orbits. In networks
of Stuart-Landau oscillators, which in addition allow for a variable amplitude, we investigate
the different symmetry transitions on the route to chaotic behaviour. Finally we compare the
correlation properties of networks chaotic deterministic units to those of coupled linear
stochastic units. Many of these results are shown to be universal for delayed systems,
independent of the model.

To complement the generic features, we also study some specific properties of delayed optical
systems, such as the desynchronisation dynamics in coupled semiconductor lasers or the
appearance of square waves in opto-electronic oscillators. These results are not only of
fundamental importance, but also affect proposed applications based on chaos

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