TWO CASE STUDIES

Remark 17. In the corollary, we do not assume that the perturbation is symmetric. However, if it is we can apply the result repeatedly as long as the spectral gap is simple.

To conclude this section, we have seen that in undirected networks, im- proving the connection structure in the sense of making it denser and stronger, indeed yields higher synchronizability. Of course, the natural question arises, if we have similar results for the case of directed networks. Before investigating the general case we want to motivate it with some numerical examples.

2.4. Two case studies

Here, we present results of numerical simulations which show that adding links or increasing weights can destabilize the synchronous state in directed networks. The underlying digraph for the first two examples is shown on the top left of Fig. (2.4.1). It consists of two components which are strongly con- nected, visualized by the grey dotted lines. Again, this means every node is reachable from every other node through a directed path inside these compon- ents. Here, the smaller component does not influence the larger component, as there are no links from the smaller to the larger component. Nonetheless, the network still supports stable synchronous dynamics. Now, introducing a new link pointing from node 4 to node 1 improves the connection structure significantly in the sense that the whole network is now strongly connected: any two nodes in the network are connected by a directed path. However, this structural improvement has a surprising consequence for the dynamics: the synchronous state becomes unstable.

In Fig. (2.4.1) a) the local dynamics is given by the Hindmarsh-Rose model, a three dimensional ordinary differential equation which models the membrane potential of a neuron. Neurons are known to exhibit a wide range of dynamical behaviour, such as subthreshold oscillations, regular and chaotic spiking and bursting. Depending on the parameter settings, the Hindmarsh-Rose model exhibits spiking and bursting behaviour [89]. The local dynamics f is given by

˙x = y + a1x2− x3− z + I

˙y = 1 − 5x2− y

˙z = a2(s(x − xR) − z).

Here, I is a constant input current. We choose the parameters as follows: a1 = 3.01, a2 = 0.006, s = 4, I = 3.2 and xR = −1.6. As can be seen in the

inset of Fig. 2.4.1 a), in this regime each node exhibits chaotic bursting [13]. We consider the electrical synaptic interaction between neurons given by

H =   1 0 0 0 0 0 0 0 0  ,

20 2. SYNCHRONIZATION LOSS IN DIRECTED NETWORKS

Figure 2.4.1. The figures show simulation results for the net- works on top. All links in blue have weight one. In the main plots we show the difference of the first component of x₁ and the first component of x₅. In a) the node dynamics is given by Hindmarsh-Rose (HR) neurons, and in b) by Rössler dynamics. The global couplingα is chosen such that the nodes synchron- ize chaotically for the original network. This can be seen in the main plots for times until t = 2000 in blue. The introduc- tion of the new link 4 → 1 with weight 0.4 at time t = 2000 leads to a destabilization, displayed in red. The insets show the time series of a single node. For the HR neurons we consider a chaotic bursting mode and for the Rössler dynamics a chaotic state.

so the local coupling is only in thex-component, known as membrane potential. In this case, the assumption A3 for Theorem 10 is not satisfied. However, the stability conditionα<(λ2) > αc can be verified via a master stability function

approach [136]. All nonzero weightsWij in the network on top left are set to

one and in order to achieve stable synchronized motion for the whole network we fixed α = 0.96. In the main plot of Fig. 2.4.1 a) we show the difference of the first component of nodes 1 and 5. We observe synchronous dynamics for times t < 2000 (in blue). At time t = 2000 we add the new link 4 → 1 with a weight ofW14= 0.4, which leads to the strongly connected network on

2.4. TWO CASE STUDIES 21

the top right. As can be seen for times t > 2000 (in red) this destabilizes the synchronous state.

In the second example we consider the same network topology, now en- dowed with Rössler oscillators as local dynamics [153]

˙x = −y − z ˙y = x + a1y

˙z = a2+ z(x − a3).

Here we chose a1 = 0.2, a2 = 0.2 and a3 = 9. For these parameters, the

isolated nodes exhibit chaotic dynamics, as can be seen in the inset of Fig. 2.4.1 b). We consider the interaction in all variables

H =   1 0 0 0 1 0 0 0 1  .

In this case, we can apply Theorem 10 and we obtain the stability condition α<(λ2) > αc as in the previous case. Again, all nonzero weights Wij in the

network are set to one and in order to achieve stable synchronization we fixed α = 0.092. As in the previous example, we observe synchronization for times t < 2000, see Fig. 2.4.1 b). The introduction of a new link at time t = 2000, however, leads to a desynchronized state.

We remark that these findings are not restricted to networks consisting of strong components connected by a cutset, which corresponds to a so-called master-slave configuration. To illustrate this, we carried out simulations on a strongly connected network with Rössler systems as local dynamics as shown in Fig. 2.4.2. Here, we use the interaction

H =   1 0 0 0 0 0 0 0 0  

which corresponds to a resistive coupling in the first variable between the oscillators. We remark that taking the identity as interaction function yields similar results though. In a) we choose the global coupling α such that the nodes synchronize chaotically for the network shown on the right. Here again, all weights are set to one. At time t = 2000 we increase the weight on the link 2 → 3 shown in red. This decreases the spectral gap, which in turn destabilizes the synchronous state. In b), the global coupling is chosen slightly smaller, such that the nodes do not synchronize. Now, introducing the new link 1 → 2, shown as a dotted line, increases the spectral gap. And this in turn stabilizes the synchronous state, as can be seen for times t > 2000 (in blue). We emphasize that there are different possibilities which stabilize the synchronous state. For instance, increasing the weights on either of the links 2 → 1, 2 → 4 or 3 → 1 yields a similar result.