Persistent synchronization of heterogeneous networks with time-dependent linear diffusive coupling

Abstract

We study synchronization for linearly coupled temporal networks of heterogeneous time-dependent nonlinear agents via the convergence of attracting trajectories of each node. The results are obtained by constructing and studying the stability of a suitable linear nonautonomous problem bounding the evolution of the synchronization errors. Both, the case of the entire network and only a cluster, are addressed and the persistence of the obtained synchronization against perturbation is also discussed. Furthermore, a sufficient condition for the existence of attracting trajectories of each node is given. In all cases, the considered dependence on time requires only local integrability, which is a very mild regularity assumption. Moreover, our results mainly depend on the network structure and its properties, and achieve synchronization up to a constant in finite time. Hence they are quite suitable for applications. The applicability of the results is showcased via several examples: coupled van-der-Pol/FitzHugh-Nagumo oscillators, weighted/signed opinion dynamics, and coupled Lorenz systems.

Figures (9)

• Figure 1: A temporal network of $N=5$ heterogeneous van der Pol oscillators, and update time $\Delta t=50$, as described above. Each column corresponds to a particular value of $c$ in \ref{['eq:vdp']}. In the first row, we plot the projection to the $(u,v)$-plane of the solutions. Since \ref{['eq:vdp']} is nonautonomous, the apparent intersections are just due to the projection. The second and third rows show the maximum of pairwise errors for each component. According to Corollary \ref{['cor:global_coupling']}, there is a large enough $c$ leading to synchronization. This is verified as one goes from left to right in the plots. We notice that the attractor, for example in the right most column, does not appear regular due to the time-dependent random switching of the network topology.
• Figure 1: A representative simulation for $N=15$ nodes. As described in the main text, within the black/red shaded time intervals, neighboring neurons synchronize with the dashed black/red neuron. This is because during such time-frames, the conditions of Theorem \ref{['thm:sync_cluster']} hold. Indeed one can particularly observe that, since $k=12$ is a neighbor of $l=9$, the $k$-th neuron (dashed red) has larger amplitude according to its own parameters during the red time-frames, but synchronizes with the smaller amplitude oscillator $l=9$ (dashed black) during the black time-frames. We further notice that during the overlap of the time-frames, some trajectories also seem to synchronize. This, however, is not characterized in the example, and may very well depend on further connectivity properties. Nevertheless, notice that all neighbors of $k=12$, except for $1$ and $15$, are also neighbors of $l$. During the overlap of the time-frames we hence see a common cluster that does not include the aforementioned neighbors.
• Figure 1: A representation of the star network setup used in Example \ref{['ex:starnetwork']}. Node $x_1$ is set as the hub and it has directed outgoing edges with positive weight $a_{i1}=a>0$, $i=2,\ldots,N$. Every other node $x_j$, for $j=2,\ldots,N$ has only one directed outgoing edge towards $x_1$, which has weight $a_{1j}=b<0$.
• Figure 2: Error plots analogous to those in Figure \ref{['fig:vdp1']}, corresponding to a simulation with $N=100$ and $\Delta t= 5$. Observe that, again, as $c$ increases the synchronization error decreases, as guaranteed by Corollary \ref{['cor:global_coupling']}.
• Figure 2: Neurons that are not neighbors of the $l$ nor the $k$ neurons.

Theorems & Definitions (50)

• Definition 2.1: Lipschitz Carathéodory functions
• Remark 2.2
• Definition 2.3
• Definition 2.4: Exponential dichotomy and dichotomy spectrum
• Remark 2.5
• Proposition 3.1
• Proof 1
• Remark 3.2
• Lemma 3.3
• Proof 2