Cyclops States in Repulsive Kuramoto Networks: The Role of Higher-Order Coupling

Vyacheslav O. Munyayev, Maxim I. Bolotov, Lev A. Smirnov, Grigory V. Osipov, and Igor Belykh

Phys. Rev. Lett. 130, 107201 – Published 7 March 2023

Abstract

Repulsive oscillator networks can exhibit multiple cooperative rhythms, including chimera and cluster splay states. Yet, understanding which rhythm prevails remains challenging. Here, we address this fundamental question in the context of Kuramoto-Sakaguchi networks of rotators with higher-order Fourier modes in the coupling. Through analysis and numerics, we show that three-cluster splay states with two distinct coherent clusters and a solitary oscillator are the prevalent rhythms in networks with an odd number of units. We denote such tripod patterns cyclops states with the solitary oscillator reminiscent of the Cyclops’ eye. As their mythological counterparts, the cyclops states are giants that dominate the system’s phase space in weakly repulsive networks with first-order coupling. Astonishingly, the addition of the second or third harmonics to the Kuramoto coupling function makes the cyclops states global attractors practically across the full range of coupling’s repulsion. Beyond the Kuramoto oscillators, we show that this effect is robustly present in networks of canonical theta neurons with adaptive coupling. At a more general level, our results suggest clues for finding dominant rhythms in repulsive physical and biological networks.

Received 6 October 2022
Accepted 9 February 2023

DOI:https://doi.org/10.1103/PhysRevLett.130.107201

Physics Subject Headings (PhySH)

Coupled oscillators Dynamics of networks Synchronization

Nonlinear Dynamics

Authors & Affiliations

Vyacheslav O. Munyayev ¹, Maxim I. Bolotov ¹, Lev A. Smirnov ¹, Grigory V. Osipov ¹, and Igor Belykh ²

¹Department of Control Theory, Lobachevsky State University of Nizhny Novgorod, 23 Gagarin Avenue, Nizhny Novgorod 603022, Russia
²Department of Mathematics and Statistics and Neuroscience Institute, Georgia State University, P.O. Box 4110, Atlanta, Georgia 30302-410, USA

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Vol. 130, Iss. 10 — 10 March 2023

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Images

Figure 1
(a) Generalized splay state with a maximum $r_{2}$ in networks with even and odd $N$ : a two-cluster splay state for $N = 10$ (top) and a three-cluster, cyclops state for $N = 11$ . The angle of the incoming arrow indicates the oscillator’s phases; oscillators depicted by the same color have the same phase. The unit length of each arrow corresponds to $| z_{k} | = 1$ . (b). Local stability diagram $(α, m)$ for the two-cluster splay state with $r_{2} = 1$ . The blue (red) hatched area corresponds to the stability condition (3) for two-cluster splay state with $r_{2} = 1$ (generalized splay states with $r_{2} = 0$ ) with the blue (red) dashed line plotting the equality condition in (3) with $r_{2} = 1$ ( $r_{2} = 0$ ). The double hatched area is the region of stable coexistence of all generalized splay states with $0 < r_{2} < 1$ . (c). Diagram similar to (b) but calculated for the three-cluster, cyclops state with $r_{2} = (N - 3) / (N - 1)$ . The circles show the phase distributions $θ_{k}$ for the two-cluster state (b) and for the cyclops state (c). Phase angle $γ = \arccos [1 / (1 - N)]$ .
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Figure 2
(a) The onset of a symmetric cyclops state from randomly chosen initial conditions. The colors depict $\sin [θ_{n} (t) - θ_{6} (t)]$ , where the $6 t h$ element is a solitary oscillator. (b)–(e). Histograms for a numerically calculated probability density function (PDF) of the $r_{2}$ distribution for the established rhythms. Remarkably, all these rhythms are generalized splay states with $r_{1} = 0$ . The number of trials: 50000 from randomly generated initial conditions for $θ_{n}$ and ${\dot{θ}}_{n}$ , $n = 1, \dots, 11$ . The PDF is normalized over 50 bins. The circles show the phase distributions $θ_{j}$ for the most probable $r_{2}$ (indicated by the arrow above the bins). (b) The dominant cyclops state from (a) with $α = 1.78$ . (c) $α = 1.84$ . (d) $α = 1.96$ . (e) $α = 3.10$ . Other parameters: $N = 11$ , $m = 1.0$ , and $ω = 1.0$ .
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Figure 3
The role of the second (a),(c),(d) and third (b),(e),(f) coupling harmonics. Histograms for a numerically calculated PDF of the $r_{1}$ , $r_{2}$ distribution for the established rhythms. The number of trials: 50000 from randomly generated initial conditions. Symmetric and asymmetric cyclops states become the dominant rhythms in both cases of weak repulsive [(a),(c) and (b),(e) with $α = 1.96$ ] and strong repulsive coupling [(d),(f) with $α = 3.10$ ]. Time series (a) and (b) correspond to the cyclops states in (c) and (e), respectively. Other parameters: $N = 11$ , $m = 1.0$ , $ω = 1.7$ , and $K_{2} = 0.05$ , $α_{2} = 0.3$ , $K_{3} = 0.0$ , $α_{3} = 0.0$ (c),(d); $K_{2} = 0.05$ , $α_{2} = 0.3$ , $K_{3} = 0.1$ , $α_{3} = 1$ (e),(f).
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Figure 4
The role of the second harmonic in stabilizing a cyclops state in system (1) with $N = 101$ , $m = 1.0$ , $ω = 1.7$ , $α = 3.1$ . The system with only the first-order coupling ( $K_{2} = 0$ ) evolves into a generalized splay state with $r_{1} = 0$ from random initial conditions for $0 < t < 500$ . Switching on the second harmonic with $K_{2} = 0.002$ and $α_{2} = 0.2$ induces a stable cyclops state ( $500 < t < 2000$ ). (a) Colors indicate $\sin [θ_{n} (t) - θ_{51} (t)]$ . (b) The corresponding values of $r_{1}$ and $r_{2}$ .
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