This research aims to achieve a fundamental understanding of synchronization and its interplay with the topology of complex networks. Synchronization is a ubiquitous phenomenon observed in different contexts in physics, chemistry, biology, medicine and engineering. Most prominently, synchronization takes place in the brain, where it is associated with several cognitive capacities but is - in abundance - a characteristic of neurological diseases. Besides zero-lag synchrony, group and cluster states are considered, enabling a description and study of complex synchronization patterns within the presented theory. Adaptive control methods are developed, which allow the control of synchronization in scenarios where parameters drift or are unknown. These methods are, therefore, of particular interest for experimental setups or technological applications. The theoretical framework is demonstrated on generic models, coupled chemical oscillators and several detailed examples of neural networks.

The focus of this thesis are synchronization phenomena in networks and their intrinsic control through time delay, which is ubiquitous in real-world systems ranging from physics and acoustics to neuroscience and engineering. We encounter synchronization everywhere and it can be either a helpful or a detrimental mechanism. In the first part, after a survey of complex nonlinear systems and networks, we show that a seemingly simple system of two organ pipes gives birth to complex bifurcation and synchronization scenarios. Going from a 2-oscillator system to a ring of oscillators, we encounter the intriguing phenomenon of chimera states which are partial synchrony patterns with coexisting domains of synchronized and desynchronized dynamics. For more than a decade scientist have tried to solve the puzzle of this spontaneous symmetry-breaking emerging in networks of identical elements. We provide an analysis of initial conditions and extend our model by the addition of time delay and fractal connectivities. In the second part, we investigate partial synchronization patterns in a neuronal network and explain dynamical asymmetry arising from the hemispheric structure of the human brain. A particular focus is on the novel scenario of partial relay synchronization in multiplex networks. Such networks allow for synchronization of the coherent domains of chimera states via a remote layer, whereas the incoherent domains remain desynchronized. The theoretical framework is demonstrated with different generic models.

This is the first book devoted to chimera states - peculiar partial synchronization patterns in networks. Providing an overview of the state of the art in research on this topic, it explores how these hybrid states, which are composed of spatially separated domains of synchronized and desynchronized behavior, arise surprisingly in networks of identical units and symmetric coupling topologies. The book not only describes various types of chimeras, but also discusses the role of time delay, stochasticity, and network topology for these synchronization-desynchronization patterns. Moreover, it addresses the question of robustness and control of chimera states, which have various applications in physics, biology, chemistry, and engineering. This book is intended for researchers with a background in physics, applied mathematics, or engineering. Of great interest to specialists working on related problems, it is also a valuable resource for newcomers to the field and other scientists working on the control of spatio-temporal patterns.

This monograph studies the synchronization control of Markovian complex neural networks with time-varying delays, and the structure of the book is summarized as follows. Chapter 1 introduces the system description and some background knowledges, and also addresses the motivations of this monograph. In Chapter 2, the stochastic synchronization issue of Markovian coupled neural networks with partially unknown transition rates and random coupling strengths is investigated. In Chapter 3, the local synchronization issue of Markovian neutral complex networks with partially information of transition rates is investigated. The new delay-dependent synchronization criteria in terms of LMIs are derived, which depends on the upper and lower bounds of the delays. In Chapter 4, the local synchronization issue of Markovian nonlinear coupled neural networks with uncertain and partially unknown transition rates is investigated. The less conservative local synchronization criteria containing the bounds of delay and delay derivative are obtained based on the novel augmented Lyapunov-Krasovskii functional and a new integral inequality. In Chapter 5, the sampled-data synchronization issue of delayed complex networks with aperiodic sampling interval is investigated based on enhanced input delay approach, which makes full use of the upper bound of the variable sampling interval and the sawtooth structure information of varying input delay. In Chapter 6, the sampled-data synchronization issue of Markovian coupled neural networks with mode-dependent interval time-varying delays and aperiodic sampling intervals is investigated based on an enhanced input delay approach. Furthermore, the mode-dependent sampled-data controllers are proposed based on the delay dependent synchronization criteria. In Chapter 7, the synchronization issue of inertial neural networks with time-varying delays and generally Markovian jumping is investigated. In Chapter 8, we conclude the monograph by briefly summarizing the main theoretical findings.

The interplay between synchronization and spatio-temporal pattern formation is central for a broad variety of phenomena in nature, such as the coordinated contraction of heart tissue, associative memory and learning in neural networks, and pathological synchronization during Parkinson disease or epilepsy. In this thesis, three open puzzles of fundametal research in Nonlinear Dynamics are tackled: How does spatial confinement affect the dynamics of three-dimensional vortex rings? What role do permutation symmetries play in the spreading of excitation waves on networks? Does the spiral wave chimera state really exist? All investigations combine a theoretical approach and experimental verification, which exploit an oscillatory chemical reaction. A novel experimental setup is developed that allows for studying networks with N > 1000 neuromorphic relaxation oscillators. It facilitates the free choice of network topology, coupling function as well as its strength, range and time delay, which can even be chosen as time-dependent. These experimental capabilities open the door to a broad range of future experimental inquiries into pattern formation and synchronization on large networks, which were previously out of reach.

This book is mainly focused on the global impulsive synchronization of complex dynamical networks with different types of couplings, such as general state coupling, nonlinear state coupling, time-varying delay coupling, derivative state coupling, proportional delay coupling and distributed delay coupling. Studies on impulsive synchronization of complex dynamical networks have attracted engineers and scientists from various disciplines, such as electrical engineering, mechanical engineering, mathematics, network science, system engineering. Pursuing a holistic approach, the book establishes a fundamental framework for this topic, while emphasizing the importance of network synchronization and the significant influence of impulsive control in the design and optimization of complex networks. The primary audience for the book would be the scholars and graduate students whose research topics including the network science, control theory, applied mathematics, system science and so on.