Executive Summary : | Synchronization is one of the fascinating manifestations of self-organization and adaptation that nature uses to orchestrate and regulate dynamical processes optimally. When systems undergo synchronization, various interacting oscillatory units adjust their rhythms to operate at a collective rhythm. Synchronization allows the constituents units to exchange information and energy optimally. Therefore, the questions about the onset synchrony or loss of synchrony and their characteristics are essential in understanding the physical, chemical, biological and technological context. The central problem is designing a robust optimization, control, and prediction system. Recently, we have studied the phenomenon of explosive synchronization in bipartite networks [U. S. Thounaojam (2021), Chaos, Solitons, and Fractals 152, 111435]. Many real-world complex networks share the characteristics of bipartite networks, so it is crucial to understand the dynamics of coupled heterogeneous oscillators on this network. The problem of dynamical stability of delay-coupled phase oscillators on a star network (bipartite network) was addressed in our earlier work [U. S. Thounaojam (2020), Eur. Phys. J. B 93: 136]. Furthermore, we have shown that delayed phase oscillators' dynamical regimes and transitions on the Cayley tree are similar to the star network's oscillator dynamics, thereby establishing the functional equivalence of different bipartite networks [U. S. Thounaojam (2021), Eur. Phys. J. B, Eur. Phys. J. B 94: 18]. A widely held idea is that network dynamics with similar components are more likely to synchronize than networks with different elements. However, there are several situations where heterogeneities, noise, and time delay in signal propagation can lead to more order in the system's dynamics. For instance, frequency heterogeneities give rise to explosive synchrony. Random heterogeneity in complex networks outperforms homogeneity in network synchronization in directed rings with time delay. Furthermore, the system's response becomes maximal when in the presence of noise. These studies strongly suggest that rather than eliminating imperfections, disorder, heterogeneities, noise, and time delays, one can instead take advantage of them to promote the synchronization and stability required for the system to function. Since the components of complex systems are heterogeneous, it is essential to understand how noise, disorder, randomness, and time delay in signal propagation impart useful functions to the systems. This project will aim to understand how a large number of coupled oscillators undergo a synchronization process on various complex networks, study the characteristics of synchronization and types of phase transitions, and formulate a theoretical framework to explain the observed phenomenon. |