Emergence of synchronization in multiplex networks of mobile Rössler oscillators

Different aspects of synchronization emerging in networks of coupled oscillators have been examined prominently in the last decades. Nevertheless, little attention has been paid on the emergence of this imperative collective phenomenon in networks displaying temporal changes in the connectivity patterns. However, there are numerous practical examples where interactions are present only at certain points of time owing to physical proximity. In this work, we concentrate on exploring the emergence of interlayer and intralayer synchronization states in a multiplex dynamical network comprising of layers having mobile nodes performing two-dimensional lattice random walk. We thoroughly illustrate the impacts of the network parameters, in particular, the vision range $\varphi$ and the step size $u$ together with the inter- and intralayer coupling strengths $\epsilon$ and $k$ on these synchronous states arising in coupled Rössler systems. The presented numerical results are very well validated by analytically derived necessary conditions for the emergence and stability of the synchronous states. Furthermore, the robustness of the states of synchrony is studied under both structural and dynamical perturbations. We find interesting results on interlayer synchronization for a continuous removal of the interlayer links as well as for progressively created static nodes. We demonstrate that the mobility parameters responsible for intralayer movement of the nodes can retrieve interlayer synchrony under such structural perturbations. For further analysis of survivability of interlayer synchrony against dynamical perturbations, we proceed through the investigation of single-node basin stability, where again the intralayer mobility properties have noticeable impacts. We also discuss the scenarios related mainly to effects of the mobility parameters in cases of varying lattice size and percolation of the whole network.

Figure 1

(a) Typical random walk movements of three particular nodes (denoted by circles) starting from random positions with step size $u=20$ over a finite time interval in a $P\times Q$ two-dimensional grid where $P=Q=300$. (b) The interactional picture at a particular instant of time with a view of the vision sizes (having area ${\varphi }^2$) for particularly two nodes (red circles) moving towards positive $x$ and positive $y$ directions (blue arrows). For clarity of the figures, only a few moving nodes in a single layer are shown for each case.

Figure 2

(a) Interlayer synchronization error $E_1$ and (b) intralayer synchronization error $E_2$ in the $k-\epsilon$ parameter plane. A magnified version of a specified region in (a) is shown in (c).

Figure 3

(a) Interlayer and (b) intralayer synchronization errors $E_1$ and $E_2$, respectively; maximum Lyapunov exponents (c) ${\mathrm{\Lambda }}_1$ and (d) ${\mathrm{\Lambda }}_2$, respectively, for inter- and intralayer synchronization, all with respect to the parameter $k$ where $\epsilon =0.50$. The other parameters are $\varphi =10$ and $u=6$.

Figure 4

(a, d) Interlayer and (b, e) intralayer synchronization errors $E_1$ and $E_2$ together with (c, f) the normalized size $G$ of the giant connected component against $\varphi$ (the left panel) with $u=6$ fixed and against $u$ (the right panel) having $\varphi =10$ fixed. Here $\epsilon =0.20$. The vertical lines in the two panels correspond to the values $\varphi =10$ and $u=6$ which are used in Fig. 2.

Figure 5

(a), (c) Interlayer synchronization error $E_1$ and (b), (d) intralayer synchronization error $E_2$ for simultaneous variation of $k$ and $\epsilon$. Here $\varphi =20,u=10$ for (a), (b) and $\varphi =30,u=15$ for (c), (d).

Figure 6

Intralayer synchronization error $E_2$ (in color bar) in the $k_1-k_2$ parameter plane with (a) $\epsilon =0.01$, (b) $\epsilon =0.05$, (c) $\epsilon =0.20$, (d) $\epsilon =0.40$. Other parameters are $\varphi =10$ and $u=6$.

Figure 7

Interlayer synchronization error $E_1$ as function of the number of demultiplexed nodes $\gamma$ for three different values of $k=3.0,10.0$, and 20.0 in black, blue, and red, respectively, with (a) $\varphi =10$, (b) $\varphi =20$, and (c) $\varphi =30$. Here $u=6$.

Figure 8

Normalized size $G$ of the giant connected component with respect to the number of demultiplexed nodes $\gamma$ for three different values of the vision range $\varphi =10, \varphi =20,$ and $\varphi =30$.

Figure 9

Interlayer synchronization error $E_1$ with respect to the number of static nodes $\delta$ with three different values of $k=3.0,6.0$ and 10.0 in black, blue and red respectively, for (a) $\varphi =10$, (b) $\varphi =20$, and (c) $\varphi =30$. Here $u=6$ is fixed.

Figure 10

Normalized size $G$ of the giant connected component as function of the number of static nodes $\delta$ for three different values of the vision range $\varphi =10, \varphi =20$, and $\varphi =30$.

Figure 11

Single-node basin stability $\langle \text{BS}\left(i\right)\rangle$ of the interlayer synchronization for all the $i\mathrm{th}$ pairs of nodes: (a) $\varphi =10$ (magenta), $\varphi =20$ (deep green), and $\varphi =30$ (brown). The corresponding frequencies of $\langle \text{BS}\left(i\right)\rangle$ are shown in (b). Other parameters are $k=3.0, \epsilon =0.20$, and $u=6$.

Figure 12

The critical density ${\rho }_c$ for the emergence of interlayer synchrony as a function of the vision range $\varphi$, for different values of $k$. The other parameters are fixed as $N=100, \epsilon =0.20$, and $u=6$.