Master stability functions for complete, intralayer, and interlayer synchronization in multiplex networks of coupled Rössler oscillators

Synchronization phenomena are of broad interest across disciplines and increasingly of interest in a multiplex network setting. For the multiplex network of coupled Rössler oscillators, here we show how the master stability function, a celebrated framework for analyzing synchronization on a single network, can be extended to certain classes of multiplex networks with different intralayer and interlayer coupling functions. We derive three master stability equations that determine, respectively, the necessary regions of complete synchronization, intralayer synchronization, and interlayer synchronization. We calculate these three regions explicitly for the case of a two-layer network of Rössler oscillators and show that the overlap of the regions determines the type of synchronization achieved. In particular, if the interlayer or intralayer coupling function is such that the interlayer or intralayer synchronization region is empty, complete synchronization cannot be achieved regardless of the coupling strength. Furthermore, for any network structure, the occurrence of intralayer and interlayer synchronization depends mainly on the coupling functions of nodes within a layer and across layers, respectively. Our mathematical analysis requires that the intralayer and interlayer supra-Laplacians commute. But, we show this is only a sufficient, and not necessary, condition and that the results can be applied more generally.

Figure 1

Schematic representation of (a) intralayer synchronization and (b) interlayer synchronization, in a multiplex network of two layers.

Figure 2

The synchronized regions with respect to $\alpha$ and $\beta , R_{\alpha ,\beta }$ painted with green (gray) color, $R_{\alpha ,\beta }^{\mathrm{Intra}}$ enclosed by the dashed-dotted blue lines, and $R_{\alpha ,\beta }^{\mathrm{Inter}}$ enclosed by the dashed red lines. Here the Rössler oscillator is taken as nodal dynamics, and the intralayer coupling matrix $H$ and the interlayer coupling matrix $\mathrm{\Gamma }$ are chosen as follows: (a) $H=I_{11}, \mathrm{\Gamma }=I_{11}$, (b) $H=I_{11}, \mathrm{\Gamma }=I_{13}$, (c) $H=I_{11}, \mathrm{\Gamma }=I_{22}$, (d) $H=I_{13}, \mathrm{\Gamma }=I_{22}$.

Figure 3

Network synchronized regions for $H=I_{11}$ and $\mathrm{\Gamma }=I_{11}$. (a) The synchronized interval of the independent intralayer and interlayer Rössler network with respect to $\alpha$ and $\beta$; (b) the synchronized region with respect to $\alpha$ and $\beta$ for Rössler networks; (c) the synchronized region with respect to couplings $c$ and $d$ for a Rössler duplex consisting of two star layers with one-to-one inter-layer connections; (d) numerical synchronization areas with respect to couplings $c$ and $d$, in which the maroon (deep gray) region represents complete synchronization area, the yellow (medium gray) is for intralayer synchronization, and the cyan (light gray) is interlayer synchronization, and the white region represents nonsynchronization.

Figure 4

Network synchronized regions for $H=I_{11}$ and $\mathrm{\Gamma }=I_{13}$. (a) The synchronized interval of the independent intralayer and interlayer Rössler network with respect to $\alpha$ and $\beta$; (b) the synchronized region with respect to $\alpha$ and $\beta$ for Rössler networks; (c) the synchronized region with respect to couplings $c$ and $d$ for a Rössler duplex consisting of two star layers with one-to-one inter-layer connections; (d) numerical synchronization areas with respect to couplings $c$ and $d$, in which the maroon (deep gray) region represents complete synchronization area, the yellow (medium gray) is for intralayer synchronization, and the cyan (light gray) is interlayer synchronization, and the white region represents nonsynchronization.

Figure 5

Network synchronized regions for $H=I_{11}$ and $\mathrm{\Gamma }=I_{22}$. (a) The synchronized interval of the independent intralayer and interlayer Rössler network with respect to $\alpha$ and $\beta$; (b) the synchronized region with respect to $\alpha$ and $\beta$ for Rössler networks; (c) the synchronized region with respect to couplings $c$ and $d$ for a Rössler duplex consisting of two star layers with one-to-one inter-layer connections; (d) numerical synchronization areas with respect to couplings $c$ and $d$, in which the maroon (deep gray) region represents complete synchronization area, the yellow (medium gray) is for intralayer synchronization, and the cyan (light gray) is interlayer synchronization, and the white region represents nonsynchronization.

Figure 6

Network synchronized regions for $H=I_{13}$ and $\mathrm{\Gamma }=I_{22}$. (a) The synchronized interval of the independent intralayer and interlayer Rössler network with respect to $\alpha$ and $\beta$; (b) the synchronized region with respect to $\alpha$ and $\beta$ for Rössler networks; (c) the synchronized region with respect to couplings $c$ and $d$ for a Rössler duplex consisting of two star layers with one-to-one inter-layer connections; (d) numerical synchronization areas with respect to couplings $c$ and $d$, in which the maroon (deep gray) region represents complete synchronization area, the yellow (medium gray) is for intralayer synchronization, and the cyan (light gray) is interlayer synchronization, and the white region represents nonsynchronization.

Figure 7

Network synchronized regions for Rössler networks composed of two single-layer fully connected networks with different $H$ and $\mathrm{\Gamma }, H=I_{11},\mathrm{\Gamma }=I_{11}$ for (a) and (b), and $H=I_{11},\mathrm{\Gamma }=I_{22}$ (c) and (d). Panels (a) and (c) are the synchronized regions about $c$ and $d$; (b) and (d) are, respectively, corresponding numerical synchronization areas, in which the maroon (deep gray) region represents complete synchronization area, the yellow (medium gray) is for intralayer synchronization, and the cyan (light gray) is interlayer synchronization, and the white region represents nonsynchronization.

Figure 8

Network synchronized regions for Rössler networks composed of two single-layer fully connected networks with different $H$ and $\mathrm{\Gamma }, H=I_{11},\mathrm{\Gamma }=I_{13}$ for (a) and (b), and $H=I_{13},\mathrm{\Gamma }=I_{22}$ (c) and (d). Panels (a) and (c) are the synchronized regions about $c$ and $d$; (b) and (d) are respectively corresponding numerical synchronization areas, in which the maroon (deep gray) region means mere complete synchronization, the yellow (medium gray) region means mere intralayer synchronization, the cyan (light gray) region means interlayer synchronization, and the blue region means nonsynchronization.

Figure 9

The case of noncommutative supra-Laplacian matrices with different intralayer topologies and identical interlayer coupling weights. Network synchronized regions calculated from master stability equations (a) and the numerical synchronization areas (b)–(d) for $H=I_{11}$ and $\mathrm{\Gamma }=I_{11}$. The first layer of the duplex network is the star type, the second layer is the one generated from the star type with 1 (b), 2 (c), and 3 (d) additional edges, respectively. The one-to-one coupling between layers is identical.

Figure 10

The case of noncommutative supra-Laplacian matrices with different intralayer topologies and identical interlayer coupling weights. Network synchronized regions calculated from master stability equations (a) and the numerical synchronization areas (b)–(d) for $H=I_{11}$ and $\mathrm{\Gamma }=I_{22}$. The first layer of the duplex network is the star type, the second layer is the one generated from the star type with 1 (b), 2 (c), and 3 (d) additional edges, respectively. The one-to-one coupling between layers is identical.

Figure 11

The case of noncommutative supra-Laplacian matrices. Network synchronized regions from master stability equations (left) and numerical synchronization areas (right) for Rössler networks with identical star-type intralayer topologies and nonidentical one-to-one coupling weights between layers. Here, the interlayer supra-Laplacian matrix ${\mathcal{L}}^I=[1-1;-11]\otimes \text{diag}\{2,1,1,1,1\}, H=I_{11}$ and $\mathrm{\Gamma }=I_{11}$ for (a), (b), and $H=I_{11}$ and $\mathrm{\Gamma }=I_{22}$ for (c), (d).

Figure 12

The case of noncommutative supra-Laplacian matrices. Network synchronized regions from master stability equations (left) and numerical synchronization areas (right) for Rössler networks with identical fully connected intralayer topologies and nonidentical one-to-one coupling weights between layers. Here, the interlayer supra-Laplacian matrix ${\mathcal{L}}^I=[1-1;-11]\otimes \text{diag}\{2,1,1,1,1\}, H=I_{11}$ and $\mathrm{\Gamma }=I_{11}$ for (a), (b), and $H=I_{11}$ and $\mathrm{\Gamma }=I_{22}$ for (c), (d).

Figure 13

The case of a three-layer fully connected network, the interlayer linking is a link (layer I–layer II–layer III). Network synchronized regions from master stability equations (left) and numerical synchronization areas (right) for Rössler networks with different combinations of $H$ and $\mathrm{\Gamma }$. (a), (c) $H=I_{11}$ and $\mathrm{\Gamma }=I_{11}$, (b), (d) $H=I_{11}$ and $\mathrm{\Gamma }=I_{22}$.

Figure 14

The case of a three-layer fully connected network, the interlayer linking is a link (layer I–layer II–layer III). Network synchronized regions from master stability equations (left) and numerical synchronization areas (right) for Rössler networks with different combinations of $H$ and $\mathrm{\Gamma }$. (a), (c) $H=I_{11}$ and $\mathrm{\Gamma }=I_{13}$, (b), (d) $H=I_{13}$ and $\mathrm{\Gamma }=I_{22}$.

Figure 15

A diagram of CORS systems in China's BeiDou Navigation Satellite System. The below layer is the physical layer composed of some GNSS receivers, and the top layer is the corresponding data-processing layer.