Chimera state in multiplex networks

Abstract

The dynamical systems theory, which deals with the temporal evolution of the state of interacting constituent entities of a system, forms the foundation of this thesis. The study of dynamics originated in the 1600s when Newton and Leibniz developed calculus to study the trajectories of celestial bodies [4]. However, before the twentieth century, investigation of a dynamical system could only be achieved for smaller cases due to the requirement of sophisticated mathematical techniques to solve higher-order systems. Only recent advancements in the fast computing and easily available vast amount of information enabled us to investigate and solve large complex systems found in the real-world. Network science, which describes the interaction architecture among the constituent entities of a system, helped us in mathematically modeling complex systems in terms of networks. A network which is a collection of nodes (or the constituent entities) and edges (interactions or links among the entities), sketches the collective dynamics of the entire system in terms of coupled differential or difference equations. Starting from cognitive functions arising from dynamics of the neural networks to the formation of public opinions arising from dynamics of the social networks, the wide range and variety of applications demonstrate the baffling scale in which network science operates. Moreover, it shows the reason for the success of network science in providing a combined mathematical framework to investigate complex systems across fields spanning the entire spectrum of science and technology [9]. In the late twentieth century, the phenomenon of synchronization, which at heart narrates a correlated dynamics between the interacting entities, has emerged as one of the biggest success stories in the study of dynamics on networks, building on the works on dynamical systems theory, mean-field theory, manifold theory, and bifurcation theory [11]. However, at the dawn of the twenty-first century, an exotic partial synchronization state, termed as chimera, describing dynamical symmetry breaking in an identically coupled network has been reported, attracting considerable interest in the past two decades and forming the main focus of this thesis. A chimera state refers to a hybrid dynamics, which displays coexistence of coherence - incoherence in a network of identical entities coupled in a symmetric fashion. In 2002, Kuramoto and Battogtokh reported a peculiar coexistence of coherent and incoherent dynamics on an array of identical phase oscillators arranged in a regular network, under certain special conditions [2]. Later on, Abrams and Strogatz christened this dynamical state as a chimera and provided a firm understanding over its strange appearance [3]. In the last two decades, the chimera state has been investigated both theoretically and experimentally, providing new understanding and insights as well as applications [32] to various fields including neuroscience. However, the majority of the studies is primarily focused on exploring chimera states in various systems having diverse underlying dynamics. Minimal emphasis has been paid to incorporate the new-found multiplex structure of the networks in the investigation of the emergent chimera state. The recent multiplex approach to network science, incorporating the existence of various types of interactions (edges) between the same pair of entities (nodes) by segregating them in different layers, provided a more realistic portrayal of the complex systems [87]. Furthermore, the inclusion of multiplex framework presents a wide variety of dynamical behaviors, which may be impossible to capture through single-layer framework [53]. This thesis aims to provide a complete report on the emergence of the chimera state in multiplex networks adding a new dimension to the study of chimera state. We first report the appearance of chimera state in multiplex networks demonstrating required special conditions. We further demonstrate the impact of delay, inhibitions, and non-identicality of the layers on the collective dynamical behavior of the chimera state in multiplex networks. We also provide a recipe to engineer the chimera state. The findings obtained through our systematic investigation, demonstrate the role of several crucial factors, which controls the emergence of chimera state in the multiplex network, commonly found in real-world complex systems.