Cluster synchronization in delayed and multiplex networks

Abstract

The thesis describes the cluster synchronization in undelayed and delayed 1-d lattice, random networks, scale-free, small-world, Cayley tree and complete bipartite networks. Synchronization is an emergent phenomenon where the coupled units adjust their trajectories in some similar manner. Our thoughts, action, motion, perceptions all are controlled by the synchronization of neurons in the brain. Additionally synchronization plays very important role in electric power systems, digital audio, video, inscription in telecommunication, metabolic systems etc and has motivated an intense research in these systems. Synchronization can be global as well as local. The local synchronization leads to the cluster formation which is desired in some cases such as in the neural networks and undesired in some cases such as power grid networks, and thus has drawn tremendous attention the last decade. Furthermore, the finite speed of information transmission leads to time delay, which plays a vital role in synchronization. Additionally, in real world networks, due to signal travelling different distances, rate of information transmission can be different for different units, which leads to the heterogeneity in delay values. A delay may give rise to many new phenomena in dynamical systems such as stabilizing periodic orbits, enhancement or suppression of synchronization, chimera state, etc. The heterogeneous delays being more realistic have shown to maximize the stability of the uniform flow, which is good for traffic dynamics, and show higher-order chaos which can be used to have a more secured communication in chaos based encryption systems. In case of the food web it has been shown that the homogeneous delays lead to destabilization of the system, which may lead to the extinction of a particular species, while the distributed delays are known to yield larger stability regimes, closely resembling to the undelayed systems. The earlier works on the coupled maps have shown two main mechanisms of cluster formation, (1) Driven(D) and (2) Self-organization (SO). SO synchronization refers to the state when clusters are formed due to intra-cluster couplings, and D synchronization refers to the state when clusters are formed due to inter-cluster couplings. However, none of the studies so far have focused on the impact of delay on cluster synchronization and mechanism behind the cluster synchronization. For the formulation of the facts to achieve the above goal following aims are clarified in this thesis.