Chimera dynamics of generalized Kuramoto–Sakaguchi oscillators in two-population networks

Abstract

Chimera dynamics, an intriguing phenomenon of coupled oscillators, is characterized by the coexistence of coherence and incoherence, arising from a symmetry-breaking mechanism. Extensive research has been performed in various systems, focusing on a system of Kuramoto–Sakaguchi (KS) phase oscillators. In recent developments, the system has been extended to the so-called generalized Kuramoto model, wherein an oscillator is situated on the surface of an M-dimensional unit sphere, rather than being confined to a unit circle. In this paper, we exploit the model introduced in Tanaka (2014 New. J. Phys. 16 023016) where the macroscopic dynamics of the system was studied using the extended Watanabe–Strogatz transformation both for real and complex spaces. Considering two-population networks of the generalized KS oscillators in 2D complex spaces, we demonstrate the existence of chimera states and elucidate different motions of the order parameter vectors depending on the strength of intra-population coupling. Similar to the KS model on the unit circle, stationary and breathing chimeras are observed for comparatively strong intra-population coupling. Here, the breathing chimera changes their motion upon decreasing intra-population coupling strength via a global bifurcation involving the completely incoherent state. Beyond that, the system exhibits periodic alternation of the two order parameters with weaker coupling strength. Moreover, we observe that the chimera state transitions into a componentwise aperiodic dynamics when the coupling strength weakens even further. The aperiodic chimera dynamics emerges due to the breaking of conserved quantities that are preserved in the stationary, breathing and alternating chimera states. We provide a detailed explanation of this scenario in both the thermodynamic limit and for finite-sized ensembles. Furthermore, we note that an ensemble in 4D real spaces demonstrates similar behavior.

Volcano transition in a system of generalized Kuramoto oscillators with random frustrated interactions

“…which is a function of the symmetry parameter η. Comparing equation (30) to equation ( 6), we observe that both have the same form except for g(0) = g(0) due to the higher-dimensional nature of our vector oscillators. Moreover, for D = 2, we regain the critical coupling strength for the Abelian Kuramoto model in equation ( 6).…”

“…where S D := 2π D/2 Γ(D/2) as pointed out in [30,34]. Hence, in the higher-dimensional OA manifold, the local field vector defined in equation (14) reads…”

“…where ρ = ρ ∥ρ∥ is a normalized OA vector and r = ηρ + n with ρ, n = 0 [21,30]. Finally, substituting equations ( 17) and ( 18) into the continuity equation ( 15), one can derive the higherdimensional OA equation [34]:…”

“…Parallel to the studies mentioned above, the collective dynamics of ensembles of so-called generalized Kuramoto oscillators have attracted considerable attention [21][22][23][24][25][26][27][28][29][30]. In particular, Lohe detailed the dynamics of generalized Kuramoto models on a compact classical Lie group and provided insights into their notable characteristics [31,32].…”

Conformists and contrarians on spheres

“…Bifurcation analysis of this system demonstrates that all phenomena arising in the classical model also appear in higher dimensions, although for different values of parameters. At this point we mention a very recent study [15] which investigates the existence of the chimera state in Kuramoto models on spheres, based on the same reduction to dynamical system for global variables.…”