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Dynamics of minimal networks of limit cycle oscillators

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Abstract

The framework of mutually coupled oscillators on a network has served as a convenient tool for investigating the impact of various parameters on the dynamics of real-world systems. Compared to large networks of oscillators, minimal networks are more susceptible to changes in coupling parameters, number of oscillators, and network topologies. In this study, we systematically explore the influence of these parameters on the dynamics of a minimal network comprising Stuart–Landau oscillators coupled with a distance-dependent time delay. We examine three network topologies: ring, chain, and star. Specifically, for ring networks, we study the effects of increasing nonlocality from local to global coupling on the overall dynamics of the system. Our findings reveal the existence of various synchronized states, including splay and cluster states, a partially synchronized state such as chimera with quasiperiodic oscillations, and an oscillation quenching state such as amplitude death in these networks. Through an analysis of long-lived transients, we discover novel amplitude-modulated in-phase and amplitude-modulated 2-cluster states within ring networks. Interestingly, we observe that increasing nonlocality diminishes the influence of the number of oscillators on the overall behavior in these networks. Furthermore, we note that oscillators in chain networks exhibit clustering in both amplitude and phase, while star networks demonstrate remote synchronization. The insights from this study deepen our understanding of the dynamics of minimal networks and have implications for various fields, ranging from biology to engineering

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Data availability

The datasets generated and/or analyzed during the current study are available from the corresponding author on reasonable request.

Code availability

The codes used in the current study are available from the corresponding author on reasonable request.

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Acknowledgements

A.E.B. thanks Prof. Oded Gottlieb (Technion-Israel Institute of Technology) and Dr. Ankit Sahay (Indian Institute of Technology Madras) for useful discussions during the preparation of this manuscript.

Funding

The authors acknowledge the financial support from the J. C. Bose Fellowship (ASE/18-19/169/SERB/RISU) from the Science and Engineering Research (SERB) of the Department of Science and Technology (DST) of the Government of India, and from the Institute of Eminence (IOE) initiative (No. SB/2021/0845/AE/MHRD/002696) from the Ministry of Education (MoE) of the Government of India.

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All authors contributed to the conceptualization and methodology. Formal analysis was performed by Andrea Elizabeth Biju and Sneha Srikanth with support from Krishna Manoj on the software. The results were validated by Andrea Elizabeth Biju, Sneha Srikanth, and Samadhan A. Pawar. The work was supervised by Samadhan A. Pawar and R. I. Sujith. The first draft of the manuscript was written by Andrea Elizabeth Biju and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript. Resources and funding were managed by R. I. Sujith.

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Correspondence to R. I. Sujith.

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Appendices

Appendix

Computational process

The computational process behind the numerical simulations described in the paper involves solving delay differential equations corresponding to the chosen network configuration in MATLAB. Once the coupled equations are solved, we classify the behavior observed based on the relative frequency, phase, and amplitudes of oscillators. The equations used for the nonlocal, local, and globally coupled ring topologies are given in Eqs.  (2), (4), and (6), respectively. For the linear chain and star networks, the equations to be used are given in Eq. (4) and Eqs. (7)–(8). The flowchart in Fig. 13 summarizes these steps.

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Biju, A.E., Srikanth, S., Manoj, K. et al. Dynamics of minimal networks of limit cycle oscillators. Nonlinear Dyn 112, 11329–11348 (2024). https://doi.org/10.1007/s11071-024-09641-5

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