Adaptive control of dynamical synchronization on evolving networks with noise disturbances

In real-world networked systems, the underlying structure is often affected by external and internal unforeseen factors, making its evolution typically inaccessible. An adaptive strategy was introduced for maintaining synchronization on unpredictably evolving networks [Sorrentino and Ott, Phys. Rev. Lett. 100, 114101 (2008)], which yet does not consider the noise disturbances widely existing in networks' environments. We provide here strategies to control dynamical synchronization on slowly and unpredictably evolving networks subjected to noise disturbances which are observed at the node and at the communication channel level. With our strategy, the nodes' coupling strength is adaptively adjusted with the aim of controlling synchronization, and according only to their received signal and noise disturbances. We first provide a theoretical analysis of the control scheme by introducing an error potential function to seek for the minimization of the synchronization error. Then, we show numerical experiments which verify our theoretical results. In particular, it is found that our adaptive strategy is effective even for the case in which the dynamics of the uncontrolled network would be explosive (i.e., the states of all the nodes would diverge to infinity).

Figure 1

(a) Time evolution of the $x$ variable of eight randomly chosen nodes and (b) synchronization error in a random network with the Rössler model as the node dynamics, by using the adaptive strategy defined in Eq. (15). For comparison, panels (c) and (d) show the same quantities of panels (a) and (b), respectively, but calculated in the case of the fully nonadaptive strategy [i.e, ${\sigma }_i\left(t\right)\equiv \gamma /V_i\left(0\right)]$. Parameter values are the following: $N=50$, $\langle k\rangle =10$, $\gamma =2.0$, $D=0.5$, $C=0.5$, $B_{1i}\left(0\right)={10}^4$, $B_{5i}\left(0\right)=0$, $B_{4i}\left(0\right)=B_{1i}\left(0\right)/V_i\left(0\right)$, and $\nu =1/1.4$.

Figure 2

(a), (b) Time average synchronization error ${\overline{E}}_x$, and (c), (d) the adaptation tracking error ${\overline{E}}_{\sigma }$ as a function of the noise intensity $D$ (a), (c) and the structure change strength $C$ (b), (d) for three different adaptive strategies: the method proposed in our paper, the SO strategy in Ref. [15], and a fully nonadaptive strategy (see legends for color codes). Each point is an average of 20 network realizations and initial conditions. The parameters are given by $C=0.5$ in (a), (c) and $D=0.5$ in (b), (d). Other parameters are the same as in the caption of Fig. 1.

Figure 3

Time average synchronization error ${\overline{E}}_x$ as a function of (a) the dispersion $c$ in the initial conditions for two different initial values of $B_{1i}\left(0\right)$, (b) the number of controlled nodes, and (c) the average degree of the nodes being controlled. In panels (a) and (b) the network topology is random with a homogeneous degree distribution ($N=100$ and $\langle k\rangle =20$), and in panel (c) it is a BA scale-free network ($N=100$ and $\langle k\rangle =12$). Each point is an average over 20 network realizations and initial conditions. Other parameters are the same as in Fig. 1, except for the two different values $B_{1i}\left(0\right)=1$ and $B_{1i}\left(0\right)=3\times {10}^4$ in panel (a).

Figure 4

(a) Time evolution of the $x$ variable of eight randomly chosen nodes and (b) synchronization error in the presence of noise, for the Foodweb model. For comparison, panels (c) and (d) show the same quantities of panels (a) and (b), respectively, but calculated in the case of a fully nonadaptive strategy. In panel (c) it is shown that the network is explosive for the case of a nonadaptive strategy and the states of all the nodes tend to infinity. Parameter values are $N=50$, $\langle k\rangle =10$, $\gamma =2.0$, $D=0.5$, $C=0.5$, $B_{1i}\left(0\right)={10}^4$, $B_{5i}\left(0\right)=0$, $B_{4i}\left(0\right)=B_{1i}\left(0\right)/V_i\left(0\right)$, and $\nu =1/1.4$.

Figure 5

(a), (b) Time average synchronization error ${\overline{E}}_x$ and (c), (d) the adaptation tracking error ${\overline{E}}_{\sigma }$ as a function of the noise intensity $D$ (a), (c) and the structure change strength $C$ (b), (d) for the Foodweb model and three different adaptive strategies: the method proposed in our paper, the SO strategy, and a fully nonadaptive strategy (see legends for color codes). Each point is an average of 20 network realizations and initial conditions. $C=0.35$ in (a), (c) and $D=0.5$ in (b), (d). Other parameters as in the caption of Fig. 4.

Figure 6

Same quantities as in the caption of Fig. 4, but for Lorenz model. $\gamma =20.0$, while the other parameters of the control method are the same as in the caption of Fig. 4.

Figure 7

Same quantities as in the caption of Fig. 5, but for Lorenz model. Parameter values are the same as in Fig. 5, except for $\gamma =20.0$ and $C=0.5$ in (a), (c).

Figure 8

Same quantities as in the caption of Fig. 4, but for Mackey-Glass chaotic dynamics. Panel (c) shows that the network is explosive for a nonadaptive control strategy. The parameter values are the same as in Fig. 4.

Figure 9

Same quantities as in the caption of Fig. 5, but for Mackey-Glass chaotic dynamics. Here the case of nonadaptation is not shown, as the network is explosive there [see Fig. 8]. Parameter values are the same as in Fig. 5, except for $C=0.5$ in (a), (c).

Figure 10

Same quantities as in the caption of Fig. 4, but for Rössler oscillators networked with a coupling function $H=[100;010;000])$. Panel (c) shows that the network is explosive for the nonadaptive case. Parameter values are the same as in Fig. 4.

Figure 11

Same quantities as in the caption of Fig. 5, but for Rössler oscillators networked with a coupling function $H=[100;010;000])$. The nonadaptive case is not shown, because the network is explosive there [see Fig. 10]. Parameter values are the same as in Fig. 5, except for $C=0.2$ in (a), (c) and $D=2.0$ in (b), (d).

Figure 12

Same quantities as in the caption of Fig. 4, but for Rössler oscillators networked with a coupling function $H=[100;010;001])$. Panel (c) shows that the network is explosive for the nonadaptive case. Parameter values are the same as in Fig. 4.

Figure 13

Same quantities as in the caption of Fig. 5, but for Rössler oscillators networked with a coupling function $H=[100;010;001])$. The nonadaptive case is not shown, because the network is explosive there [see Fig. 12]. Parameter values are the same as in Fig. 5, except for $C=0.5$ in (a), (c).