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Abstract

Many complex systems exhibit periodic oscillations comprising slow–fast timescales. In such slow–fast systems, the slow and fast timescales compete to determine the dynamics. In this study, we perform a recurrence analysis on simulated signals from paradigmatic model systems as well as signals obtained from experiments, each of which exhibit slow–fast oscillations. We find that slow–fast systems exhibit characteristic patterns along the diagonal lines in the corresponding recurrence plot (RP). We discern that the hairpin trajectories in the phase space lead to the formation of line segments perpendicular to the diagonal line in the RP for a periodic signal. Next, we compute the recurrence networks (RNs) of these slow–fast systems and uncover that they contain additional features such as clustering and protrusions on top of the closed-ring structure. We show that slow–fast systems and single timescale systems can be distinguished by computing the distance between consecutive state points on the phase space trajectory and the degree of the nodes in the RNs. Such a recurrence analysis substantially strengthens our understanding of slow–fast systems, which do not have any accepted functional forms. Slow–fast oscillations are observed in numerous applications ranging from neuroscience and earth sciences to engineering. In this study, we perform a recurrence analysis of prototypical signals derived from well-established models, namely, the Van der Pol model, a modified form of the Izhikevich model, and the Hodgkin–Huxley model. First, we show a potential pitfall of phase space reconstruction, as the number of slow–fast regions could be exaggerated when the phase space is reconstructed by time delay embedding. We observe that the recurrence network, which represents the high-dimensional phase space of the underlying system, is clustered in certain regions and also exhibits protrusions, in addition to the closed-loop structure typically seen for periodic signals. We argue that these clustering and protrusions effects in the recurrence network arise due to the presence of slow and fast timescales in the system. Additionally, we can detect such features in micro-patterns along the diagonal lines of the corresponding recurrence plot. Finally, we observe similar features on the recurrence plots and recurrence networks of time series of signals acquired from experiments performed on a sub-scale liquid rocket combustor and a model gas turbine combustor during the state of thermoacoustic instability.