On Coexistence Patterns

This book is about coexistence patterns in ensembles of globally coupled nonlinear oscillators. Coexistence patterns in this respect are states of a dynamical system in which the dynamics in some parts of the system differ significantly from those in other parts, even though there is no underlying structural difference between the different parts. In other words, these asymmetric patterns emerge in a self-organized manner. As our main model, we use ensembles of various numbers of Stuart-Landau oscillators, all with the same natural frequency and all coupled equally strongly to each other. Among the most notable states previously observed in these ensembles are so-called chimera states, characterized by some of the oscillators moving in synchrony while the others are unordered. These states may be considered a special type of coexistence pattern. Moreover, phenomenologically similar states have been identified in a wide range of other contexts, both theoretical and experimental. Thus, the first part of this book deals with chimera states. It begins with a critical review of their exploration in various systems, and of how their implicit research-community definition has often changed when new results were added. Chimera states in dynamical systems similar to our own model are treated with additional detail. Our original research begins with the case of only N = 4 oscillators and explores the symmetry-breaking bifurcations by which minimal chimeras can emerge from simpler solutions. Then, we increase the size of the ensemble and address how the uncovered mechanisms translate to larger N. Central to the rest of the book is the observation that the types of bifurcations leading to chimeras can also create a variety of other states. With this in mind, we return to the case N = 4 and vary the equation parameters more exhaustively, embedding the minimal chimeras in a hierarchy of interrelated solutions. These solutions are classified according to their symmetries, which provide an important key to the question of how they are interrelated. Finally, we address coexistence patterns for ever larger ensembles, thereby uncovering an intricate type of stepwise transition from low to high complexity. The book includes a general background chapter on the concepts and methods employed to interpret the presented results. Moreover, all main chapters comprise a separate background section on relevant prior research. Towards the end of each chapter, its findings are reflected in the light of possibly related work and with regards to their broader applicability.