Dynamic link switching induces stable synchronized states in sparse networks

Abstract

The flow of information in networked systems composed of multiple interacting elements strongly depends on the level of connectivity among these elements. Sparse connectivity often hinders the emergence of states in which information is globally shared, such as fully synchronized states. In this context, dynamically switching existing network links among system elements can facilitate the onset of synchronization. Here, we address this problem in a double-layer network of FitzHugh-Nagumo oscillators with sparse inter-layer connectivity at fixed density. We show that dynamically switching the existing cross-layer links induces inter-layer synchronization, with a clear dependence on the switching time. In agreement with intuition, shorter switching times suppress large deviations between temporally connected oscillators and more effectively promote synchronization; crucially, this effect persists even when each isolated layer is chaotic. Chaos at the layer level is verified by a strictly positive largest Lyapunov exponent, confirming that synchrony is induced by switching rather than by periodic dynamics. For a minimal double-layer system, we emulate switching using smooth square waves and compute the master stability function (MSF), which is in agreement with direct numerical simulations and delineates the stability regions in parameter space.

Figures (9)

• Figure 1: (a) Schematic of a double-layer network illustrating the two distinct populations of FHN oscillators in its nodes, represented in blue and gold. The constants $\sigma_1$ and $\sigma_2$ denote the intra-layer coupling intensities, while $\sigma_{12}$ represents the inter-layer coupling intensity. (b) Spatiotemporal dynamics of the activator variable $u_{1j}$ for the FHN oscillators in layer $1$. (c) Spatiotemporal dynamics of the activator variable $u_{2j}$ for the FHN oscillators in layer $2$. Parameters are $\varepsilon = 0.05$, $a_{ij}=0.87$ (even $j$), $a_{ij}=0.97$ (odd $j$), $\phi = \frac{\pi}{2} - 0.1$, $\sigma_{12} = 0$, $\sigma_1 = \sigma_2 = 0.1$, and $N=200$.
• Figure 2: The largest Lyapunov exponent $\hat{\lambda}_{\max}(t)$ as a function of time for an isolated layer ($\sigma_{12}=0$), computed via the Benettin method (with periodic renormalization of tangent vectors), converging to $\lambda_{\max} \approx 0.04$. The simulation parameters are $\varepsilon = 0.05$, $a_{\text{odd}} = 0.87$, $a_{\text{even}} = 0.97$, $\phi = \frac{\pi}{2} - 0.1$, $\sigma_1 = \sigma_2 = 0.1$, and $N = 200$, with local coupling radius $r = 1$. A positive $\lambda_{\max}$ confirms the presence of chaotic dynamics within each layer.
• Figure 3: Time evolution of the Euclidean distance $E_{35}$ between mirror nodes number $35$ is marked in red for the switching time a) $T_{swt}=120$ and b) $T_{swt}=23$. The time intervals marked in gray indicate the periods in which mirror nodes 35 are connected, while in white are time intervals in which these nodes are disconnected. In c) and d) blue curve shows the time evolution of the interlayer synchronization error $E^{12}(t)$ for $T_{swt}=120$ and $T_{swt}=23$, respectively. Additionally to the given parameters, $\sigma_{12}=0.1$ and $N=400.$
• Figure 4: Duration of the transient time, $T_{sync}$, before the onset of complete inter-layer synchronization as a function of the layer size $N$. The ratio $N_{IL}/N=0.25$ is kept constant while changing the population size $N$. The interlayer links are switched randomly. Exemplarily,the transient times to synchronization for $T_{swt}=5,10,15,23,25,35,50$, for the above-mentioned parameters. We note the time where the $E^{12}<10^{-4}$.
• Figure 5: This figure illustrates the maximum switching time, where the system still synchronizes under interlayer switching as a function of the layer population, using the abovementioned parameters. The ratio $N_{IL}/N=0.25$ is kept constant while changing the size of the population. The interlayer links are switched randomly.