Chaos Synchronization in Time-Dependent Duplex Networks

Though the complete chaos synchronization on single-layer networks has been well understood, it is still a challenge on multiplex networks. In this work, we study the complete chaos synchronization on time-dependent duplex networks in which interaction pattern among oscillators alternates periodically between two single-layer networks. The alternations between two layers are characterized by the offset strength A and the switching frequency ω. We find that there are two dynamical regimes depending on ω. For high ω, the critical A for the stable complete synchronization is independent of ω and the fast-switching approximation suggests that the time-dependent duplex networks can be approximated by the time-independent duplex networks with effective coupling strength. For low ω, the critical A depends on ω nonmonotonically. At extremely low ω, the estimation of the critical A can be obtained by a single-mode approximation taking one dominant transversal networkmode to complete synchronization into considerations.


Introduction
Collective behaviors are ubiquitous in natural systems and human society.Among these, synchronization, one of the most interesting collective behaviors, has been investigated since the early days in natural science and has drawn great attention in the last decades [1][2][3][4].There are various forms of synchronization, such as complete synchronization (CS) [5], cluster synchronization [6], remote synchronization [7], partial synchronization [8], phase synchronization [9], generalized synchronization [10], and lag synchronization [11].Among these various types of synchronization, CS is the simplest one and its stability has been well analyzed by using the master stability function method [12,13].When complex networks are involved, it has been found that both the coupling function and the network structure can affect the stability of CS.
Previous studies on network science focused on singlelayer networks.However, there are many complex systems ranging from social systems to technological systems which have to be modeled as multilayer networks other than singlelayer ones [14][15][16][17].For example, in social networks people may interact with each other through different channels such as WeChat, Twitter, and Facebook [18].Each channel consists of an independent layer on which people interact with the coupling function of their own layer and all layers supported by different channels are integrated into a multilayer network.Among multilayer networks, multiplex networks refer to the ones in which different layers share the same group of nodes and each layer of the networks owns its unique interaction pattern [19][20][21].Recently, CS has been investigated in multilayer networks [14,[22][23][24].Different from singlelayer networks on which the onset of the instability of CS is induced by the most unstable transversal network modes, it is still a challenge to figure out the mechanism for CS on multiplex networks to lose its stability.Except for some peculiar multiplex networks, there are no rigorous results for the stability of CS though some approximation methods have been proposed [25].
Interaction patterns may be time-dependent in many situations such as power transmission system [26], consensus problem [27], and person to person communications [28].In modeling, the time-varying interaction patterns may be realized in several ways, for example, by rewiring connections in networks in the course of time [29], by assuming oscillators to move in space [30], and by allowing the switch of network topologies among several given interaction patterns [31].Synchronization on time-varying networks has been 2 Complexity investigated when oscillators may move in space [32,33].It has been shown that synchronization time may depend nonmonotonically on the mobility of oscillators and the mechanism driving synchronization is different for different dynamical regimes [34][35][36][37].However, in these literatures concerning with synchronization on time-varying networks, the coupling function is always same.In this paper, we construct a time-varying duplex networks by switching the interactions pattern and coupling function among oscillators between two single-layer networks with different coupling functions and study CS among chaotic Lorenz oscillators on them.
The paper is organized as follows.In Section 2, we present the model.The time-varying duplex networks are characterized by the parameters, the switching frequency  and the offset strength .In Section 3, we first consider two single-layer networks and study CS on each of them.Then we consider the time-varying duplex networks constructed from these two single-layer ones and investigate the dependence of CS on the switch frequency and the offset strength.The numerical results and some discussions are presented.In Section 4, a summary is made.

Numerical Simulations
Before investigating CS on time-dependent duplex networks, we take a look at CS on single-layer networks.For simplicity, we let  = 6.We generate two random networks  (1) and  (2) which are illustrated in the insets in Figures 1(a) and 1(b).Using the master stability function method, the stability of CS on  (1) or  (2) can be analyzed by  decoupled linear differential equations with  = 1, 2.  ()  ( = 0, 1, . . .,  − 1) are the eigenvalues of the Laplacian matrix constructed from  () and we suppose Except for the eigenvalue  () 0 = 0 accounting for the synchronous network mode, other  − 1 nonzero  ()  represent the transversal network modes ] ()  to CS. DF(s) and DH(s) are the Jacobian matrices of the corresponding functions evaluated at CS.The stability of CS requires that the largest transversal Lyapunov exponent, Λ ()  , for each nonzero  ()  to be negative.Figures 1(a) and 1(b) show Λ ()   for  − 1 transversal network modes for the singlelayer networks  (1) and  (2) , respectively.Clearly, the network  (1) supports stable CS for  1 > 3.3 while CS is always unstable on  (2) .For an arbitrary perturbation to CS, it can always be expanded as a linear combination of all transversal network modes.With time elapsing, it is the network mode with the largest transversal Lyapunov exponent that takes responsibility for determining the stability of CS.According to Figures 1(a) and 1(b), we may find that, for the layer  (1) , the network mode ] (1)  1 is the dominant one which always has the largest Λ.On the other hand, the dominant network mode depends on the coupling strength for layer  (2) .For example, the dominant mode is ] (2)  1 for  2 < 3.4 while it changes to ] (2)  5 for  2 > 3.4.Furthermore, we calculate the Lyapunov exponent spectrum for the model dynamics equation (1) on the networks  (1) and  (2) .The first four largest Lyapunov exponents against the coupling strength  1 and  2 are presented in Figures 1(c) and 1(d), respectively.Beyond the critical coupling strength  1 ≃ 3.3, CS is established for the network  (1) since all Lyapunov exponents are negative except for the largest Lyapunov exponent taking the value close to that at  1 = 0 and the second largest Lyapunov exponent to be zero.On the other hand, CS is impossible for the network  (2) and the model dynamics may be either chaotic or time-independent for sufficiently strong  2 .
Then, we consider the time-dependent duplex networks constructed from the networks  (1) and  (2) in Figure 1 and focus on the effects of the parameters  and  on the model dynamics in (1).To measure the synchronization in the model, we consider the synchronization error Δ, which is given by Figure 1: The transversal Lyapunov exponents for different transversal network modes are plotted against the coupling strength in (a) for the layer  (1) and in (b) for the layer  (2) .The insets in these two plots show the network structures of  (1) in (a) and of  (2) in (b).The first four largest Lyapunov exponents are plotted against the coupling strength in (c) for the layer  (1) and in (d) for the layer  (2) . = 6.
refers to the Euclidean norm between oscillators  and  and ⟨⋅⟩  stands for the time average.For each parameter combination, the synchronization error is averaged over 10 3 time units after a transient time around 10 4 time units.We first set  1 = 5 where CS is an attractor on the layer  (1)  and  2 = 4 where chaotic desynchronous state is realized on the layer  (2) .Increasing  means that oscillators spend more time on the layer  (2) , which tends to disfavor CS on the timedependent duplex network.Compatible with the intuition, Figure 2(a) shows that Δ always increases with .There seems to exist a critical   for each  above which Δ becomes nonzero and, to be stressed,   increases rapidly at low  and seems to saturate to a value at high .Figure 2(b) presents Δ against  for different .For  close to 1, Δ keeps nonzero and CS is impossible.On the other hand, Δ = 0 is always possible for high  when  is not close to 1. Next, we set  2 = 5 where equilibrium is the attractor on the layer  (2) .The dependence of Δ against  and  are presented in Figures 2(c) and 2(d) and we observe the similar behaviors to those at  2 = 4. Thus, CS on the time-dependent duplex network is insensitive to chaotic or regular dynamics on the isolated single-layer networks.To be mentioned, Figure 1(d) suggests a quenched state at  2 = 5 on the network  (2) where the maximum Lyapunov exponent is negative.Figure 2(b) further suggests that the quenched state is actually an inhomogeneous one according to the nonzero Δ at  = 1.
To get an overview on CS in the model (1), we present Δ in the plane of  and .Figures 3(a) and 3(b) display the results for the parameter combinations ( 1 = 5,  2 = 4) and ( 1 = 5,  2 = 5), respectively.It is clear that CS may be realized in a large range of domain in the parameter plane, for example, high  and  not close to 1. Actually, there exist two dynamical regimes depending on .For high , the critical   separating CS from desynchronous states is roughly independent of , which is around 0.8 for ( 1 = 5,  2 = 4) and 0.6 for ( 1 = 5,  2 = 5).In contrast, for low , the parameter regimes supporting CS are narrow.The enlargement of the parameter plane in Figures 3(c) and 3(d) shows irregular boundary between CS and desynchronous states at low .
The results at high  can be understood as follows.For high , the interaction pattern among oscillators switches between the layers  (1) and  (2) rapidly.In this case, the fastswitching approximation [34,38], which averages out the effect of alternation between  (1) and  (2) , may be applied.Consequently, the interaction pattern among oscillators may be approximated by a time-independent duplex network with  (1) and  (2) being its two layers and with the effective coupling strengths  1, =  1 / and  2, =  2 ( − )/ where  = 2/ and  is the time duration that oscillators interact with each other in the time-dependent network through  (1) .Just like social networks where each person may simultaneously communicate with others via Facebook network and Twitter network, in this static duplex network as an approximation, each oscillator may simultaneously interact with others through two different networks assisted by the coupling functions H (1) and H (2) , respectively.Since  is determined by sin  −  > 0, it is unchanged with  for a given , which means that  1, and  2, are determined by  uniquely.We present the phase diagram in the plane of  1 and  2 for the time-independent duplex network with  (1) () =  (1) and  (2) () =  (2) in Figure 3(e) to show the CS region.Then we present the effective coupling strengths  1, and  2, acquired from the coupling strengths  1 and  2 on the time-dependent duplex network by changing  from −1 to 1 with the increment 0.1.The green dots in Figure 3(e) are for the set ( 1 = 5,  2 = 4) while the red dots for the set ( 1 = 5,  2 = 5).According to the plot, we find the critical  is around 0.8 for  2 = 4 and 0.6 for  2 = 5, which are in agreement with Figures 3(a) and 3(b).
On the other hand, the dynamics is quite complicated at low .However, for the extremely low , we could present some plausible explanations.For the extremely low , the system has to spend sufficiently long time on one layer before switching to the other one.Consequently, the transversal network modes ] (1)  on the layer  (1) and ] (2)    on the layer  (2) will have different effects on model dynamics.To sort out these effects from different network modes, we firstly take the transversal network modes with the largest transversal Lyapunov exponent on each layer into considerations, for example, ] (1)  1 on the layer  (1) and, on the layer  (2) , ] (2) 1 for  2 < 3.4 and ] (2)  5 for  2 > 3.4.Supposing the transversal Lyapunov exponent of the network mode in the layer  (1)  to be Λ (1) and in the layer  (2) to be Λ (2) , any perturbation away from CS would evolve in the form  Λ (1) +Λ (2) (−) .For  (1) and  (2) .The shaded area corresponds to the CS region.The dot lines denote the effective  1, and  2, acquired by the fast-switching approximation.(f) The critical  are plotted against  2 with  * 1 in black,  * 2 in blue, and   acquired from the synchronization error in red.In addition,   against  2 in the time-dependent duplex network with  (2) replaced by  (3) is plotted in green. 1 = 5 for the solid symbols and  1 = 6 for the open symbols.The inset in (f) shows the single-layer network  (3) .the perturbation to decay in one period ,  should be larger than Λ (2) /(Λ (2) − Λ (1) ).That is, it requires  <  * 1 with  * 1 obeying * 1 against  2 is presented in Figure 3(f) for  1 = 5 and  1 = 6.Clearly,  * 1 at  1 = 6 is higher than that at  1 = 5 since Λ (1) at  1 = 5 is closer to 0. Then, we numerically calculate Δ against  for different  2 and the critical   for the onset of desynchronization acquired in this way is also presented in Figure 3(f).There is strong discrepancy between   and  * 1 in the range of  2 ∈ (1.5, 3.5) where   may reach a much higher value than  * 1 .The discrepancy lies at the inappropriate approximation that only the network mode with the largest Λ 1 in each layer is considered.Actually, all transversal network modes collectively determine how the perturbation away from synchronous chaos evolves.Considering the fact that the inner products, ⟨] (1)   | ] (2)  ⟩, of network modes on different layers determine the redistribution of the perturbation on ] (1)  (] (2)  ) over ] (2)   (] (1)  ) at the time when the two layers are switching, we think that, in the approximation of one network mode, which network mode in one layer should be taken into considerations depends on its inner products with the network modes on the other layer and its transversal Lyapunov exponents.Along this line, the network mode ] (1)  1 is the one in the layer  (1) crucial to   .On the layer  (2) , though ] (2)  1 has the largest Lyapunov exponent for  2 < 3.4, the product ⟨] (1) 1 ⟩ ≃ 0.15 is much smaller than ⟨] (1)  1 | ] (2) 2 ⟩ = 0.59 and ⟨] (1) 1 | ] (2) 5 ⟩ ≃ 0.77.Therefore, the dominant network mode on the layer  (2) is the mode ] (2)  2 for  2 < 2.9 and ] (2) 5 otherwise according to Figure 1(d).Substituting Λ (2) with the transversal Lyapunov exponent with the corresponding dominant mode in (4), we obtain a new critical  (denoted as  * 2 ).As Figure 3(f) shows,  * 2 may reach a much higher value in the range of  2 ∈ (1.5, 3.5).
Actually, the above analysis can be further confirmed by replacing  (2) with another network, the layer  (3) [shown in the inset of Figure 3(f)], which keeps the overall interaction pattern unchanged but with rearranged oscillators on the layer.The Laplacian of  (3) has the same eigenvalues with  (2)  and, therefore, the stability of CS on  (3) is exactly the same as that on  (2) .Different from  (2) , the product between ] (3) 1 and ] (1) 1 is around 0.66, which assures that the network mode ] (3)  1 is the dominant one in determining   .Consequently,   , acquired according to Δ, in the range of  2 ∈ (1.5, 3.5) is greatly lowered and the discrepancy between   and  * 1 is greatly reduced, as shown in Figure 3(f).
We have found that CS is possibly established on a timedependent duplex network when one of the layers does not support CS on its own.Actually, CS can be realized on timedependent duplex networks even when two layers do not allow for the stable CS on their own.To illustrate it, we consider the coupling functions H 1 (x  − x  ) = [  −   , 0, 0] and H 2 (x  − x  ) = [  −   , 0, 0] where CS is unstable on either  (1) or  (2) .We present the synchronization error Δ in the plane of  and  in Figure 4

Conclusion
In this work, we constructed time-dependent duplex networks in which the network topology alternates periodically between two single-layer networks.The temporal properties of the networks are characterized by the offset strength  and the switching frequency .We studied the complete chaos synchronization on this type of time-dependent duplex networks.By monitoring the synchronization error, we find that there are two dynamical regimes depending on the switching frequency .For high  where the fast-switching approximation is valid, the time-dependent duplex networks can be approximated by time-independent duplex network with effective coupling strength.The critical  for the onset of stable CS is independent of .For low , the critical  strongly depends on  and the coupling strengths on the two layers.At extremely low , we find that the dependence of the critical  on the coupling strengths can be roughly explained based on the picture in which only one dominant network mode in each layer is taken into considerations.We found that which network mode is the dominant one is determined by its transversal Lyapunov exponent and its inner product with the dominant one on the other layer.

Figure 3 :
Figure 3: The contour plot of the synchronization error Δ in the plane of  and  in (a) for  1 = 5 and  2 = 4 and in (b) for  1 = 5 and  2 = 5.(c) and (d) The enlargement view of (a) and (b) in the range of  ∈ (0, 0.08), respectively.(e) The phase diagram in the plane of  1 and  2 for the time-independent duplex network with two layers represented by (1) and (2) .The shaded area corresponds to the CS region.The dot lines denote the effective  1, and  2, acquired by the fast-switching approximation.(f) The critical  are plotted against  2 with  * 1 in black,  * 2 in blue, and   acquired from the synchronization error in red.In addition,   against  2 in the time-dependent duplex network with (2) replaced by (3) is plotted in green. 1 = 5 for the solid symbols and  1 = 6 for the open symbols.The inset in (f) shows the single-layer network (3) .

Figure 4 :
Figure 4: The contour plot of the synchronization error Δ in the plane of  and  in (a) for  1 = 5 and  2 = 5 and in (b) for  1 = 8 and  2 = 4. (c) The phase diagram showing the CS region in the plane of  1 and  2 for the time-independent duplex network with two layers represented by  1 and  2 .The dot line denotes the effective  1, and  2, acquired by the fast-switching approximation.H 1 (x  − x  ) = [  −   , 0, 0] and H 2 (x  − x  ) = [  −   , 0, 0].
(a) with  1 =  2 = 5 and Figure 4(b) with  1 = 8 and  2 = 4. Except for the parameter regimes close to  = −1 and  = 1, CS can always be produced at high  where the critical  for CS is insensitive to .
Figure 4(c) shows the stability diagram in the plane of  1 and  2 for the time-independent duplex network.The critical  in Figures 4(a) and 4(b) can be acquired by plotting the effective coupling strengths  1, and  2, at different  in Figure 4(c).