Interlayer Synchronisation of Time-Varying Multiplex Kuramoto--Sakaguchi Networks in the Chimera Regime

Abstract

We study interlayer synchronisation in a duplex network of $N=300$ nonlocally coupled Kuramoto--Sakaguchi oscillators, with each layer operating in the chimera regime. The interlayer coupling is weak ($\ddot{I}{ƒ}_{12}=0.01$), sparse, and time-varying: a fixed number $N_{IL}$ of replica-node pairs are coupled symmetrically, and the active links are randomly redistributed every $T_{swt}$ time units. We characterise synchronisation by the time-averaged interlayer order parameter $Z$, the master stability function $\hat{I}¨(\ddot{I}{ƒ}_{12},T_{swt})$, and the finite-time transverse Lyapunov exponent $\hat{I}{»}_{\perp }$. In the static case, full synchronisation ($Z=1$, $\hat{I}¨<0$) requires all-to-all interlayer coupling ($N_{IL}=N$). Under temporal switching with $T_{swt}\le 25$, near-complete synchronisation is achieved with as few as $N_{IL}\approx N/3$ links, while the intralayer chimera structure is preserved. The master stability function confirms that short switching periods render the transverse dynamics stable at link densities where static coupling fails, and the transverse Lyapunov exponent heatmap delineates the critical link number as a joint function of $N_{IL}$ and $T_{swt}$. These results demonstrate that temporal redistribution of sparse interlayer connections can stabilise replica-node coherence in networks with spatially heterogeneous intralayer dynamics.

11 pages, 13 figures