Details of PhD

This thesis explores the occurrence of emergent phenomena in various high-dimensional dynamical systems. In particular, it investigates the phenomena of synchronization through Kuramoto oscillators as well as ageing transitions via coupled Rulkov neurons, emphasizing how the characteristics of the interactions between different system components influence its long-term dynamics. The first part of the thesis explores extensions of the Kuramoto model which include different combinations of unidirectional, adaptive, multilayered, and higher-ordered coupling schemes. For such models, analytical estimates for the critical transition points are obtained using the Ott-Antonsen ansatz and other techniques. The results also indicate different routes to synchrony including continuous, explosive, and tiered transitions, which are then verified through numerical simulations. The next part of the thesis considers a model consisting of coupled Rulkov neurons, the parameters of which are modified to induce dynamical ageing. In this setting, the interplay between the connectivity and coupling strength is investigated, showcasing their effects on ageing transitions. The effect of fluctuations are also examined and shown to reduce the occurrence of explosive ageing transitions in the network. Finally, the thesis also presents the computational developments implemented to increase the efficiency of large-scale simulations of dynamical systems. This is achieved through parallelization of the numerical algorithms used for integrating ordinary differential equations by utilizing multiple GPUs. The developments have been benchmarked and have been shown to result in a major speedup in execution and can be utilized for simulating a large class of other dynamical systems.