Stochastic master stability function for noisy complex networks

In this paper, we broaden the master stability function approach to study the stability of the synchronization manifold in complex networks of stochastic dynamical systems. We provide necessary and sufficient conditions for exponential stability that allow us to discriminate the impact of noise. We observe that noise can be beneficial for synchronization when it diffuses evenly in the network. On the contrary, an excessively large amount of noise only acting on a subset of the node state variables might have disruptive effects on the network synchronizability. To demonstrate our findings, we complement our theoretical derivations with extensive simulations on paradigmatic examples of networks of noisy systems.

Figure 1

Stochastic master stability function of the network of Van der Pol oscillators (22) for selected values of the noise intensity ${\sigma }^{*}$.

Figure 2

Simulation of $N=2$ uncoupled stochastic Van der Pol systems (22) when ${\sigma }^{*}=0.25$. Phase-portrait of node 1 (left panel) and time evolution of the norm of the synchronization error (right panel).

Figure 3

Stochastic master stability function of the network of stochastic Chua's circuits (6)–(23) for ${\sigma }^{*}\in [0,5]$. The white line is the locus $\eta -{\left({\sigma }^{*}\right)}^2/2=k_f$. The stars identify the points $(\sigma {\lambda }_2\left(\mathcal{L}\right),0)$ and $(\sigma {\lambda }_2\left(\mathcal{L}\right),1)$ for the network depicted in Fig. 4.

Figure 4

Simulation of a network of $N=100$ stochastic Chua's circuits (6)–(23) when $g$ and $h$ are both the identity function, and $\sigma =0.1$. The left panel depicts the network topology, generated through the Watts-Strogatz model [46], while the norm of the error with respect to the average trajectory $\overline{x}\left(t\right)={\mathbb{1}}_N\otimes {\sum }_{i=1}^N{x}_i\left(t\right)/N$ is plotted in the right panel when ${\sigma }^{*}=1$ (black line) and in the absence of noise (blue line).

Figure 5

Stochastic master stability function of the network of stochastic Rössler systems (6)–(25), with $h(x_i,t)=g(x_i,t)=x_i$, for ${\sigma }^{*}\in [0,1]$.

Figure 6

Stochastic master stability function of the network of stochastic Rössler systems (6)–(25), with $h(x_i,t)=g(x_i,t)=x_i$, for selected values of the noise intensity ${\sigma }^{*}$.

Figure 7

Stochastic master stability function of the network of stochastic Rossler systems (6)–(25) with $h(x_i,t)=g(x_i,t)={[x_{i1},0,0]}^T$ for ${\sigma }^{*}\in [0,10]$.

Figure 8

Stochastic master stability function of the network of stochastic Rossler systems (6)–(25) with $h(x_i,t)=g(x_i,t)={[x_{i1},0,0]}^T$ for selected values of the noise intensity ${\sigma }^{*}$.