On Coexistence Patterns Hierarchies of Intricate Partially Symmetric Solutions to Stuart-Landau Oscillators with Nonlinear Global Coupling

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. Employing computer simulations, bifurcation analysis and symmetry considerations, we uncover the mechanism behind a wide range of complex patterns found in these ensembles. Our starting point is the creation of so-called chimeras, which are subsequently treated within a new and broader context of related states.

Outline.- General Background.- From Two-Cluster State to Chimera.- Coexistence Patterns of Four Oscillators.- A Hierarchy of Solutions for N = 2n.- Conclusion and Outlook.

Sindre W. Haugland first engaged with nonlinear dynamics in the context of his Master's studies in physics at the Technical University of Munich in 2012. From 2013 onward, he conducted extensive numerical research on coupled nonlinear oscillators and oscillatory media at the chair of Prof. Katharina Krischer, leading amongst other things to the discovery of one of the first known instances of a self-organized alternating chimera state in 2014. In 2015, he completed his Master's thesis on the complex Ginzburg-Landau equation with added nonlinear global coupling and on nonlinearly globally coupled ensembles of Stuart-Landau oscillators.

Fom this early work grew an increased awareness of and deepened interest in the formerly unnamed general class of intricate dynamics that he later proposed to call "coexistence patterns". Thus, after having secured funding in the form of a competitive doctoral scholarship from the "Studienstiftung des deutschen Volkes", he subsequentlyspent five years investigating how such states of disparate dynamics can arise in an underlying isotropic system – ultimately leading to the insights covered in this book.