Synchronisation and stability in nonautonomous oscillatory systems

Many natural and artificial systems can be modelled by ensembles of coupled oscillators. These types of systems can exhibit various synchronisation phenomena, where the interaction between the oscillators leads them to some kind of coherent behaviour, despite heterogeneities in the system. Moreover, many such systems are subject to a timevariable environment which effectively drives them. Many examples can be found in living systems, e.g., the dynamics of a cell is strongly dependent on the ever-changing intra- and extra-cellular ionic concentrations. Motivated by these considerations, this thesis investigates the effect of time-varying parameters on synchronisation and stability in ensembles of coupled oscillators. Timevariability is a crucial ingredient of the dynamics of many real-life systems, and interest in it is only recently starting to grow. Such systems are in general described by nonautonomous equations, which are hard to treat in general. This present work aims at answering questions such as: Can time-variability be detrimental/beneficial to synchronisation? If so, under which conditions? Can time-variability seed new dynamical phenomena? How can one best treat nonautonomous systems? The systems studied can be divided into two categories. First, the effect of a driving oscillator with a time-varying frequency is investigated. It is shown that increasing the amplitude of the frequency modulation can increase the size of the stability region in parameter space, under general assumptions. Short-term dynamics and stability properties are also investigated, and their dynamics is shown to be of importance. Second, the effect of time-varying couplings between the oscillators is considered. This is shown to be able to make the synchronous state unstable and yield oscillation death. Overall, the thesis illustrates that time-variability can be either beneficial or detrimental to synchronous dynamics, and investigates in detail and gives insight about cases of both. It argues towards the general fact that short-term dynamics is often crucial to a physically relevant understanding of nonautonomous systems.