Search

Cluster synchronization is a fundamental phenomenon in systems of coupled oscillators. Here, we investigate clustering patterns that emerge in a unidirectional ring of four delay-coupled electrochemical oscillators. A voltage parameter in the experimental setup controls the onset of oscillations via a Hopf bifurcation. For a smaller voltage, the oscillators exhibit simple, so-called primary, clustering patterns, where all phase differences between each set of coupled oscillators are identical. However, upon increasing the voltage, secondary states, where phase differences differ, are detected, in addition to the primary states. Previous work on this system saw the development of a mathematical model that explained how the existence, stability, and common frequency of the experimentally observed cluster states could be accurately controlled by the delay time of the coupling. In this study, we revisit the mathematical model of the electrochemical oscillators in order to address open questions by means of bifurcation analysis. Our analysis reveals how the stable cluster states, corresponding to experimental observations, lose their stability via an assortment of bifurcation types. The analysis further reveals complex interconnectedness between branches of different cluster types. We find that each secondary state provides a continuous transition between certain primary states. These connections are explained by studying the phase space and parameter symmetries of the respective states. Furthermore, we show that it is only for a larger value of the voltage parameter that the branches of secondary states develop intervals of stability. For a smaller voltage, all the branches of secondary states are completely unstable and are, therefore, hidden to experimentalists. [ABSTRACT FROM AUTHOR]

All online versions of the article were corrected on 26 June 2023. This article was originally published online on 12 June 2023 with the incorrect affiliation for Philipp Hövel. [Extracted from the article]

Phase models are a powerful method to quantify the coupled dynamics of nonlinear oscillators from measured data. We use two phase modeling methods to quantify the dynamics of pairs of coupled electrochemical oscillators, based on the phases of the two oscillators independently and the phase difference, respectively. We discuss the benefits of the two-dimensional approach relative to the one-dimensional approach using phase difference. We quantify the dependence of the coupling functions on the coupling magnitude and coupling time delay. We show differences in synchronization predictions of the two models using a toy model. We show that the two-dimensional approach reveals behavior not detected by the one-dimensional model in a driven experimental oscillator. This approach is broadly applicable to quantify interactions between nonlinear oscillators, especially where intrinsic oscillator sensitivity and coupling evolve with time. [ABSTRACT FROM AUTHOR]