Abstract
This thesis concerns the study of collective synchronized behaviour in discrete stochastic dynamical systems consisting of spatially distributed individual elements. The participating units may represent the constituent elements of an extensive range of biological, chemical or physical systems. They can be assumed as prey and predators of an ecosystem or species in a food chain with cyclic domination. They can be dynamical species reacting on catalytic surfaces or spins in magnetic systems. Depending on the definition of the units’ local dynamics and the interactions between them, they may also be used to investigate collective behaviour like synchronous activation of coupled neuronal networks or even social networks with evolving structure. Specifically, transitions from local density fluctuations to global bulk oscillations and synchronization phenomena are investigated in three different models where individual elements can interact locally only with their immediate neighbours. In a ...
This thesis concerns the study of collective synchronized behaviour in discrete stochastic dynamical systems consisting of spatially distributed individual elements. The participating units may represent the constituent elements of an extensive range of biological, chemical or physical systems. They can be assumed as prey and predators of an ecosystem or species in a food chain with cyclic domination. They can be dynamical species reacting on catalytic surfaces or spins in magnetic systems. Depending on the definition of the units’ local dynamics and the interactions between them, they may also be used to investigate collective behaviour like synchronous activation of coupled neuronal networks or even social networks with evolving structure. Specifically, transitions from local density fluctuations to global bulk oscillations and synchronization phenomena are investigated in three different models where individual elements can interact locally only with their immediate neighbours. In addition, long distance or global interactions as well as diffusive motion are also assumed and these additional interactions are shown to play a crucial role in the system evolution, inducing phenomena such as rhythmic oscillations and synchronization. Although these models are based on fundamental behaviours observed in chemical and biological systems they aim to be used in studies of generalized mathematical and physical principles governing the competition between individual tendencies and cooperation. In this direction two novel population dynamics models consisting of multiple species with cyclic domination are introduced. At the mean field level they exhibit conservative dynamics or quasi-periodic and chaotic dynamics, respectively. Both systems have two integrals of motion and their linear stability as well as their phase space structure are investigated in detail. Going beyond the deterministic mean field description, spatial distribution of individuals and intrinsic randomness of their local reactions are taken into account. The stochastic dynamics now is described by a Master equation whose solution is approached via kinetic Monte Carlo simulations. Additionally to local reactions, a gradually mixing process is introduced with a certain probability. This process can be realised as mutual position exchange between two distant species, in a sense of diffusive motion or with distant or global reactions. The interplay of local reactions with these mixing processes finally induces synchronization of species which arises after a phase transition as a function of the control probability of the corresponding process. Synchronization is also studied in a large number of globally coupled discrete state elements. Particularly, a semi-Markovian two-state system constitutes an individual noisy medium which, subjected to global coupling, gives rise to a complex competition between individual dynamics and macroscopic cooperative behaviour. The globally coupled system shows monostability and bistability. It can also undergo Hopf bifurcations when the coupling affects individual units with a certain time delay. In this latter case the interplay of noise with time delay induces synchronization and bulk oscillations emerge in the population of coupled units. Synchronization analysis in the three models is based on the analytical signal approach, introduced by D. Gabor on 1946. An instantaneous phase is defined in terms of the Hilbert transformation of a measured signal, projecting the physical frequencies of the system. The distribution of these phases gives a qualitative overview of the synchronization scenario, while a synchronization index originating from the defined phase is calculated giving a quantitative estimation. Furthermore the mutual information is calculated, giving an assessment of statistical interdependence of measured signals and thus an indication of the synchronization. In conclusion, this analysis shows that in the three studied models, the spatially distributed individual elements exhibit a synchronous cooperation when an order parameter exceeds a critical threshold where the synchronization index and mutual information take their maximal values simultaneously.
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