Coupling functions: Universal insights into dynamical interaction mechanisms Tomislav Stankovski Faculty of Medicine, Ss Cyril and Methodius University, 50 Divizija 6, Skopje 1000, Macedonia and Department of Physics, Lancaster University, Lancaster, LA1 4YB, United Kingdom Tiago Pereira Department of Mathematics, Imperial College London, London SW7 2AZ, United Kingdom and Institute of Mathematical and Computer Sciences, University of S ˜ ao Paulo, S ˜ ao Carlos 13566-590, Brazil Peter V. E. McClintock and Aneta Stefanovska Department of Physics, Lancaster University, Lancaster, LA1 4YB, United Kingdom (published 6 November 2017) The dynamical systems found in nature are rarely isolated. Instead they interact and influence each other. The coupling functions that connect them contain detailed information about the functional mechanisms underlying the interactions and prescribe the physical rule specifying how an interaction occurs. A coherent and comprehensive review is presented encompassing the rapid progress made recently in the analysis, understanding, and applications of coupling functions. The basic concepts and characteristics of coupling functions are presented through demonstrative examples of different domains, revealing the mechanisms and emphasizing their multivariate nature. The theory of coupling functions is discussed through gradually increasing complexity from strong and weak interactions to globally coupled systems and networks. A variety of methods that have been developed for the detection and reconstruction of coupling functions from measured data is described. These methods are based on different statistical techniques for dynamical inference. Stemming from physics, such methods are being applied in diverse areas of science and technology, including chemistry, biology, physiology, neuroscience, social sciences, mechanics, and secure communications. This breadth of application illustrates the universality of coupling functions for studying the interaction mechanisms of coupled dynamical systems. DOI: 10.1103/RevModPhys.89.045001 CONTENTS I. Introduction 2 A. Coupling functions, their nature, and uses 2 B. Significance for interacting systems more generally 3 1. Physical effects of interactions: Synchronization, amplitude, and oscillation death 3 2. Coupling strength and directionality 4 3. Coupling functions in general interactions 5 II. Basic Concept of Coupling Functions 5 A. Principle meaning 5 1. Generic form of coupled systems 5 2. Coupling function definition 5 3. Example of coupling function and synchronization 6 B. History 6 C. Different domains and usage 8 1. Phase coupling functions 8 2. Amplitude coupling functions 9 3. Multivariate coupling functions 10 4. Generality of coupling functions 11 D. Coupling functions revealing mechanisms 11 E. Synchronization prediction with coupling functions 12 F. Unifying nomenclature 13 III. Theory 13 A. Strong interaction 14 1. Two coupled oscillators 14 2. Comparison between approaches 17 B. Weak regime 17 1. Stable periodic orbit and its phase 18 2. Coupling function and phase reduction 19 3. Synchronization with external forcing 19 4. Phase response curve 20 5. Examples of the phase sensitivity function 21 C. Globally coupled oscillators 21 1. Coupling functions leading to multistability 22 2. Designing coupling functions for cluster states and chimeras 22 3. Coupling functions with delay 23 4. Low-dimensional dynamics 23 5. Noise and nonautonomous effects 24 D. Networks of oscillators 24 1. Reduction to phase oscillators 24 2. Networks of chaotic oscillators 25 IV. Methods 26 A. Inferring coupling functions 26 REVIEWS OF MODERN PHYSICS, VOLUME 89, OCTOBER–DECEMBER 2017 0034-6861=2017=89(4)=045001(50) 045001-1 © 2017 American Physical Society