Phase Reduction of Limit Cycle Oscillators: A Tutorial Review with New Perspectives on Isochrons and an Outlook to Higher-Order Reductions

Abstract

The phase reduction technique is essential for studying rhythmic phenomena across various scientific fields. It allows the complex dynamics of high-dimensional oscillatory systems to be expressed by a single phase variable. This paper provides a detailed review and synthesis of phase reduction with two main goals. First, we develop a solid geometric framework for the theory by creating isochrons, which are the level sets of the asymptotic phase, using the Graph Transform theorem. We show that isochrons form an invariant, continuous structure of the basin of attraction of a stable limit cycle, helping to clarify the concept of the asymptotic phase. Second, we systematically explain how to derive the first-order phase reduction for weakly perturbed and coupled systems. In the end, we discuss the limitations of the first-order approach, particularly its restriction to very small perturbations and the issue of vanishing coupling terms in certain networks. We finish by outlining the framework and importance of higher-order phase reductions. This establishes a clear link from classical theory to modern developments and sets the stage for a more in-depth discussion in a future publication.