Synchronization of wave structures in a heterogeneous multiplex network of 2D vdP lattices with attractive and repulsive intra-layer coupling
SSynchronization of wave structures in aheterogeneous multiplex network of 2D vdPlattices with attractive and repulsive intra-layercoupling
I.A. Shepelev ∗ , S.S. Muni † , T.E. Vadivasova ∗ January 20, 2021
Abstract
We explore numerically the synchronization effects in a heterogeneoustwo-layer network of two-dimensional (2D) lattices of van der Pol oscilla-tors.The inter-layer coupling of the multiplex network has an attractivecharacter. One layer of 2D lattices is characterized by attractive couplingof oscillators and demonstrates a spiral wave regime for both the localand nonlocal interaction. The oscillators in the second layer are coupledthrough active elements and the interaction between them has the re-pulsive character. We show that the lattice with the repulsive type ofcoupling demonstrates complex spatiotemporal cluster structures, whichcan be called as labyrinth-like structures. We show for the first timethat this multiplex network with fundamentally different types of intra-layer coupling demonstrates the mutual synchronization and a competi-tion between two types of structures. Our numerical study indicates thatthe synchronization threshold and the type of spatiotemporal patterns inboth the layers strongly depend on the ratio of the intra-layer couplingstrength of the two lattices. We also analyze the impact of intra-layercoupling ranges on the synchronization effects. ∗ Department of Physics, Saratov State University, 83 Astrakhanskaya Street, Saratov,410012, Russia † School of Fundamental Sciences, Massey University, Palmerston North, New ZealandKeywords: Synchronization, multiplex networks, spatiotemporal patterns, repulsive cou-pling, van der Pol oscillator, spiral waveE-mail addresses: I.A. Shepelev ( [email protected] ), S.S. Muni( [email protected] ),T.E. Vadivasova ( [email protected] ) a r X i v : . [ n li n . PS ] J a n xtended abstract A lot of objects in nature and technical systems represent complex network withensembles of nonlinear elements interacting between each other through differentcoupling. They can be networks of neurons, populations of living organisms,ensembles of interacting quantum oscillators, computer networks, power gridsand many more. In addition, different multicomponent system (the networksand ensembles) can interact between each other and form a complex multilayernetwork. Exploring such systems with different topology is currently one of themost relevant directions in nonlinear dynamics.The fundamental phenomenon of synchronization plays an important role inthe dynamical behavior of complex multicomponent systems. It leads to partialor complete oscillation coherence of individual elements. The synchronizationin spatially distributed ensembles and networks is reflected by the formation ofspatiotemporal patterns with different complexity. When layers (sub-ensembles)of the multilayer network interact, there is not only the synchronization of thetemporal dynamics, but also the partial or complete synchronization of spa-tiotemporal structures and wave processes. The degree and nature of hetero-geneity of interacting layers also plays a very important role.The heterogeneity is inherent in most real multicomponent systems. Mul-tilayer networks can be used for a simulation of real complex systems, whichinclude different subsystems. In this case, the layers can consist of elements ofdifferent types. For example, the layers can correspond to groups of neuronswith various characteristics, populations of diverse living organisms, differenttypes of transport coupled in a network, certain energy sources included in asingle energy system, etc. But even if all the elements of the network are identi-cal, the layers of the network can significantly differ in the intra-layer coupling.For example, elements of one network layer can be coupled locally or not atall, while elements of another layer are characterized by nonlocal interaction.Such models are often used in neurodynamics. However, it is not only thetopology of the links that is important, but also their nature. In a number ofworks, attractive and repulsive types of coupling are considered. They lead tosignificant differences in the behavior of networks of oscillators. The repulsivecoupling is especially interesting as the neuron interaction in certain cases isrepulsive and can be efficient in modeling neuronal networks.. Networks withthe attractive coupling demonstrate the effects of partial or complete in-phasesynchronization, wave regimes, chimera states. On the contrary, the repulsivecoupling impedes the in-phase synchronization and often leads to the effect ofoscillation death. In general, the dynamics of the network becomes simpler.The question arises, if a multilayer network consists of heterogeneous layers, inwhich the heterogeneity is associated with the attractive and repulsive characterof the intra-layer coupling, then what will be dynamics of the interacting layersand what structures will be predominant?The present work aims to give an answer to this question by using an ex-ample of a two-layer network, which consists of two 2D lattices of self-sustainedoscillators with attractive and repulsive intra-layer coupling. We consider a2ossibility of the mutual synchronization of the lattices and its features. Wealso analyze what structure will prevail depending on the ratio of intra-layercoupling coefficients.
The phenomenon of synchronization is one of the most important dynamicaleffects in nature. It plays a crucial role in the collective dynamics of complexmulti-component systems and networks [1, 2, 3, 4, 5, 6, 7, 8]. The emergingfield of synchronization has a great importance in the study of applied issues,for example, in neuronal networks [9, 10, 11], social networks [12], transportsystems [13], technological networks [14, 15, 16, 17], etc. Real networks ingeneral are characterized by a very complex topology. For this reason, numericalsimulation of these systems is sufficiently complicated and often impossible.However, the study of simplified models enables us to find out the main featuresin the network dynamics, in particular the synchronization effects. Recently,exploring the dynamics of multilayer networks has become one of the actualdirections in nonlinear dynamics [18, 19, 20, 21]. The latter better describeand simulate the real world systems. For example, neural systems of the brainare characterized by a complex multilayer topology [22, 23], namely neuronsinteracts through electrical and chemical synaptic connection. This leads to thecoexistence of domains with spatially coherent and incoherent behavior [24, 25].In multilayer topology, a network consists of several layers (in each layer nodescan interact with each other in different ways) coupled with various types ofinter-layer coupling.Different types of synchronization have been explored in multiplex networks,namely inter-layer synchronization [26, 27], generalized synchronization, exter-nal and mutual synchronization [28, 29], adaptive synchronization [30], explo-sive synchronization [31, 32, 33] and remote synchronization [34, 35]. Thereare a number of research works [36, 37, 38, 39] devoted to the synchronizationof complex spatiotemporal patterns in multiplex networks with non-identicallayers. These works have explored the influence of different factors of bothintra-layer and inter-layer interactions on the synchronization of structures inboth layers of the network. These factors can be time delays in the coupling[37, 38], different topology of the interaction [39] and proportion of various typesof coupling [40] within one of the layer. Indeed, different types of intra-layercoupling can significantly change the dynamics of isolated layers and can alsolead to noticeable changes in the synchronization effects. Despite of a largenumber of publications studying the interaction of layers in multiplex networkswith different nature of intra-layer coupling, there are still many issues in thisdirection which are not yet explored. One of these issues is about the interac-tion between layers with the attractive and repulsive coupling. This case hasbeen considered in [40] for two coupled 1D ring of the phase oscillators. It hasbeen shown that the repulsive links in the second layer significantly affects thedynamical behavior of the first ring. In particular, they contribute to the for-3ation of chimera states. This leads to many unanswered interesting questionssuch as a) the dynamical behavior of similar multiplex networks with complexnonlinear elements (for example, van der Pol oscillator) b) the possibility ofmutual synchronization of layers with different types of intra-layer coupling c)is full synchronization possible in this multiplex network and, if possible, whichsynchronous regime will be the most typical?. It should be noted that the at-tractive and repulsive interaction leads to essentially different dynamics of theisolated layers. The attractively coupled layer demonstrate complete synchro-nization and different wave regimes such as traveling waves [41, 42, 43], spiraland target waves [41, 44, 45] as well as various chimera states [46, 47, 48], whichrepresents the spatiotemporal patterns with coexisting coherence and incoher-ence domains.The repulsive coupling in turn generally impedes the synchronization andleads to the amplitude or oscillatory death[49, 50, 51, 52]. It has also been shownthat this coupling induces oscillations in various excitable systems [53], and evenleads to the emergence of traveling waves and a regime of partial synchronization[54, 52]. Interest in the study of systems with repulsive interaction is associatedwith the use of negative (inhibitory) coupling in the model of neurodynamics[55, 56, 57].Note that in most of the previous studies of multilayer networks, the lat-ter consisted of interacting one-dimensional ensembles of nonlinear systems. Itcan be quite interesting to study synchronization in a multiplex network whichincludes two-dimensional (2D) lattices of oscillators. It is well known that 2Densembles with the attractive coupling can demonstrate wave spatiotemporalregimes, such as spiral and target waves [41, 44, 45] and spiral and target wavechimeras [58, 59, 60, 61]. This leads to an unexplored question about the dy-namics of a 2D lattice with repulsive coupling.The purpose of this work is to reveal and study synchronization in a hetero-geneous two-layer multiplex network consisting of coupled 2D lattices of van derPol oscillators. In the first lattice, the intra-layer coupling is repulsive (throughan element with the negative differential resistance), while in the second latticeelements are coupled through resistance (attractively). The inter-layer couplingis bidirectional, pairwise and has the same attractive character as the intra-layercoupling in the second layer. A regime under study in the 2D lattice with theattractive coupling represents well known spiral waves. The lattice with repul-sive coupling demonstrates only standing waves with a complex spatial profiles.We observe the in-phase synchronization between the layers and competitionbetween the structure. We show that a type of structure in the synchronizedlattice strongly depends on the value of the intra-layer coupling strength. Be-sides, we show that the synchronization is possible only within certain ratios ofvalues of the intra-layer coupling strength in the layers. We also analyze howthe nonlocality degree of the intra-layer interaction affects the synchronizationfeatures. 4
The model
We study a multiplex network which consists of two layers and is schematicallyshown in Fig. 1(a). Each layer represents a two-dimensional (2D) lattice ofcoupled nonlinear systems and includes N × N = 50 × nodes. The layersare pairwise and bidirectionally coupled with each other. The local dynamicsof each element are governed by the van der Pol (vdP) oscillator: ˙ x = y, ˙ y = ε (1 − x ) y − ω x, (1)where ( x, y ) are dynamical variables, ε denotes the nonlinearity level, and ω isthe natural frequency of self-sustained oscillations. We fix ε = 2 . and ω = 2 . ensuring the regime of relaxation oscillations in the individual elements.The multiplex network is described by the following system of equations: ˙ x li,j = x li,j , ˙ y li,j = y li,j + J l σ l Q l (cid:80) m l ,n l (cid:0) y lm l ,n l − y li,j (cid:1) + (cid:80) k =1 γ kl (cid:0) y ki,j − y li,j (cid:1) , (2)where ( x li,j , y li,j ) are dynamical variables, l labels the layer, l = 1 , , double lowindices ( i, j ) , where i, j = 1 , . . . , N = 50 , indicates the position of the elementon the two-dimensional lattice. The coefficient γ kl is the inter-layer couplingstrength, while parameter σ l corresponds to the intra-layer coupling strength inthe l th layer, l, k = 1 , .The intra-layer coupling is introduced in the second equation in Eq. (2) ofeach lattice. Parameter J l determines the type of the intra-layer coupling andis equal to − for the first layer and +1 for the second one. This way of theintroducing intra-layer links corresponds to repulsive active coupling in the firstlattice and to attractive resistive coupling in the second layer. σ l is the intra-layer coupling strength, and P l denotes the sizes of coupling ranges within eachlayer. The parameters P l determines the number Q l of all intra-layer links inboth directions for each node of the l th layer. The quantity Q l for each node ofthe l th layer represents a combination of all the links with the indices m l and n l according to the following relations corresponding to the no-flux boundaryconditions: (cid:40) max(1 , i − P l ) (cid:54) m l (cid:54) min( N, i + P l ) , max(1 , j − P l ) (cid:54) n l (cid:54) min( N, j + P l ) , (3)This type of the boundary conditions is illustrated in Fig. 1(b)-(d) for an isolatedlattice and for different locations of the selected oscillator when the intra-layercoupling range is equal to 2. In particular, accordingly to Fig.1(b) a centralelement is coupled with Q l = (2 P l + 1) − neighbors.5 l= l= σ σ γ (a)(b) (c) (d)Figure 1: (Color online) (a) Scheme of a two-layer multiplex network of coupled2D lattices which are characterized by the intra-layer coupling strength σ l andthe inter-layer coupling γ kl ( k, l = 1 , ). (b)-(d) Schemes of the intra-layer cou-pling in an isolated 2D lattice for the case of no-flux boundary conditions Eq. (3)for different locations of the selected node (marked in red): (b) in the latticecenter, (c) in the middle of the right edge, and (d) in the right upper corner.Oscillators coupled with the selected ( i, j ) th node are depicted in blue, and therest uncoupled nodes are shown in white. The intra-layer coupling range P l = 2 .The pairwise bidirectional coupling between the layers is introduced in thesecond equation of each layer and described by the third term in the sys-tem Eq. (2). γ kl is the inter-layer coupling strength and is defined as follows: γ kl = (cid:18) γ γ (cid:19) (4)where the row and column numbers are related to the corresponding layers. Inour numerical simulation γ = γ = γ . The coupling range of the inter-layerinteraction corresponds to the common multiplex network, namely each ( i, j ) thoscillator of the first layer is linked only with ( i, j ) th element of the second layer.The initial conditions for all the dynamical variables of the network Eq. (2)are chosen to be random and uniformly distributed within the interval [ − , .The model equations (2) are integrated using the Runge-Kutta 4th ordermethod with the time step dt = 0 . . The transient time T tr is equal to T tr = 10000 time units for all the cases under study.6 .2 Dynamics of isolated lattices We explore the dynamical regimes which can be obtained in the 2D latticesof coupled vdP oscillators when the interaction between the two layers in thenetwork Eq. (2) is absent ( γ = 0 ). At first, we consider the dynamics of the lat-tice with attractive coupling (second layer, J = +1 in Eq. (2)). This couplingcorresponds to the case when elements interact through a resistance. Threedynamical regimes are observed with the variation of the intra-layer couplingparameters σ and P . When the coupling strength σ is very weak the os-cillators behave incoherently with each other. An increase in σ leads to theformation of a wave structure, namely the spiral waves. A different number ofspiral waves can coexist in the lattice for various set of initial conditions. Westudy the case when there is only one spiral wave in the lattice. The spiralwaves are observed only for the local P = 1 and short nonlocal P = 2 cou-pling. It should be noted that for the local interaction, the wave structure isformed within significantly longer range of the coupling strength than for thecase of nonlocal coupling P = 2 . The third regime is the partial or completesynchronization of element oscillations. It is considered when P ≥ and forthe case of strong coupling strength and P = 1 , (the spiral waves disappear).In the present work we study the lattice in the regime of spiral waves. They areexemplified in Fig. 2(a) for P = 1 and in Fig. 2(b) for P = 2 . The spatiotem-poral dynamics is illustrated in a space time plot of the j = 12 cross-sectionpassing through the wave center in Fig. 2(c). It is seen that the wave front ro-tates around the wave center. Note, that this spiral wave cannot be transformedinto a spiral wave chimera since the wave is destroyed when the coupling rangebecomes greater than P ≥ .The repulsive type of coupling between vdP oscillators presents an interac-tion through an active element with the linear negative differential resistance( J = − in Eq.2). Our study shows that this coupling leads to the appearance j i -1012 x i,j (a) j i -1012 x i,j (b) t i -1012 x i,j (c)Figure 2: (Color online) Spiral waves in the second isolated layer (lattice) withattractive coupling of the network Eq. (2) at σ = 0 . . Snapshots of the systemstate (a) P = 1 (local coupling) and (b) P = 2 (nonlocal coupling). (c) is aspace-time plot of the cross-section j = 12 for the local coupling. Parameters: J = 1 , ε = 2 , ω = 2 , N = 50 . 7f a completely different spatiotemporal regime in the lattice and that the spiralwaves cannot be obtained for any initial conditions. Examples of the typicalregimes for the local ( P = 1 ) and nonlocal ( P = 2 ) coupling are exemplifiedin Fig. 3(a) and (b), respectively. The spatiotemporal structure presents alter- j i -1012 x i,j (a) -2-1010 5 10 15 20 25 x t (c) j i -1012 x i,j (b) t i -1012 x i,j (d)Figure 3: (Color online) Labyrinth-like structures in the first isolated layer (lat-tice) with repulsive coupling of the network Eq. (2) for σ = 0 . . Snapshots ofthe system state (a) P = 1 (local coupling) and (b) P = 2 (nonlocal coupling).(c) time realizations for the (10 , th (black line) and (11 , th oscillators, (d)a space-time plot of the cross-section j = 30 for the local coupling. Parameters: J = − , ε = 2 , ω = 2 , N = 50 .nation of “strips”, in each of which all elements oscillate with the very similarphase. However, in the two adjacent strips the oscillation phases are shiftedby the half period, i.e. the oscillations are in anti-phase. These strips havedifferent length and can be located in either vertical or horizontal direction.Similar spatiotemporal patterns have been discovered in [62] in the lattice ofstrongly coupled vdP oscillators. These structures have been called “labyrinth-like structures”. Despite of the qualitative similarity between the state in thesetwo systems, there are significant changes between them. At first, in [62] a rea-son behind the formation of such structures is the emergence of two coexistingchaotic attractors in the phase space of individual oscillators induced by thestrong coupling. In contrast, for system under study, there is no bistability andoscillations in the neighboring strips differ by only the instantaneous phases.8his feature is illustrated in a plot of the time series for two selected oscillatorsfrom adjacent strips in Fig. 3(c). The spatiotemporal dynamics is depicted bya space-time plot of the j = 30 th cross-section in Fig. 3(d). The elongation ofthe coupling range P leads to a simplification of the spatiotemporal structure(see Fig. 3(b) for P = 2 ). The strips with similar phases become wider andthe number of strip intersections decreases. For the long coupling ranges thestructures become regular and represent an alternation of strips or squares ofoscillators with certain values of the instantaneous phases. At the same time,the quantitative features of the regime for both short and long coupling rangesremain very similar. This enables us to assume that the same regime with dif-ferent spatial topology is observed within a wide interval of the coupling rangevalues. Since the spiral waves in the second lattice exist only for the couplingrange P = 1 and P = 2 , we will study structures in the first lattice for thesame values of P .It should be noted that all the structures under study presented above in Fig-ures 2 and 3 are sufficiently robust and are not qualitatively and quantitativelychanged when the intra-layer coupling strength are varied within the interval σ , ∈ [0 . , . . For this reason we will study the inter-layer interaction γ forthe cases when value of the intra-layer coupling strength σ , is varied withinthe mentioned interval.The structures presented above are chosen to be the initial states in thefirst and second lattices. Besides, using these initial states we have obtaineda set of initial conditions for each values of the intra-layer coupling strength σ , ∈ [0 . , . and use them in studying the synchronization effects in themultiplex network in Eq. (2). To quantify and distinguish synchronization effects between the layers in the net-work Eq. (2), we calculate the correlation coefficient between the corresponding( i, j )th pairs of oscillators of the first and second layers as below: R i,j = ˜ x i,j · ˜ x i,j (cid:113) (˜ x i,j ) · (˜ x ki,j ) , ˜ x i,j = x i,j − x i,j , (5)where · · · means time-averaging over T av = 10000 time units. Using the cor-relation measure Eq. (5) for arbitrary oscillation modes, one can evaluate ef-fective synchronization between the layers when R i,j ≈ ± but R i,j (cid:54) = ± andcomplete synchronization when R i,j = ± . Besides, the correlation coefficientvalue enables one to diagnose the in-phase and anti-phase synchronization ofspatiotemporal structures in the regime of periodic oscillations of the latticenodes. It is +1 in the first case and − in the second one. Note that since ournetwork Eq. (2) is heterogeneous (the coupling in different layers has differentcharacter), the synchronization measure in Eq. (5) cannot be strictly equal to ± and synchronization effects must be understood in their effective sense. In9his case a synchronization criterion must be imposed. Synchronization is as-sumed to take place if | R i,j | ≥ . . It should be noted that this condition isarbitrary and any value of R i,j close to ± may be chosen as a threshold. We now introduce the bidirectional inter-layer coupling γ between the layers tothe y dynamical variable in the multiplex network Eq. (2) and explore synchro-nization effects between them. The inter-layer coupling has the same attrac-tive (dissipative) character as the intra-layer coupling in the second layer. Tohighlight the influence of the nonlocality of intra-layer coupling on the synchro-nization effects we study two cases of mutual synchronization, namely when theintra-layer coupling is local P , = 1 and nonlocal P , = 2 . We now study the synchronization effects for the case of an interaction of thetwo 2D lattices with the local intra-layer coupling. These lattices are in theregimes exemplified in Fig. 3(a) for the first layer and in Fig. 2(a) for the secondlattice. Our study shows that the synchronization feature of these structuresstrongly depends on values of the intra-layer coupling in the lattices. Besides, in-phase synchronization is observed within a wide range of the inter-layer couplingstrength γ . To describe and illustrate these effects in detail we fix one of theintra-layer coupling strength σ , = 0 . and vary the second one within a range σ , ∈ [0 . , . as well as the γ ∈ [0 , with sample steps ∆ σ , = 0 . and ∆ γ = 0 . .Synchronization effects are detected by evaluating and analyzing the corre-lation coefficient R i,j . To diagnose synchronization between the layers, we plottwo 2D diagrams of synchronous and desynchronized regimes in the ( σ , γ ) pa-rameter plane (for fixed σ = 0 . , Fig. 4(a)) and in the ( σ , γ ) plane (for fixed σ = 0 . , (Fig. 4(b)) by using the values of (cid:104) R (cid:105) , where (cid:104)· · · (cid:105) means that valuesof the correlation coefficients are averaged over all the oscillator pairs ( i, j ). Thecolor scheme in the diagrams corresponds to the ratio of a number of synchro-nized ( (cid:104) R (cid:105) ≥ . ) oscillator pairs N s to a whole number of elements in theeach lattice N . In order to get better insight into the effects of the synchro-nization in the network Eq. (2), we highlight the following region with differentdynamical regimes, namely region of desynchronization (denoted by DS), wherethe spatiotemporal behavior in the two lattices are not identical ( N s N < ), andthe two regions of synchronization (denoted by S A and S B ). Within region S A ,a spatiotemporal structure which is similar to the initial state in the first lattice(see Fig. 3(a)) is observed in both the layers, while the synchronized spiral wavesare formed in both of the layers in region S B . Boundaries of these regimes are10 B S A DSS A N σ /N γ s σ (a) DSDSS B S A N σ /N γ s σ (b)Figure 4: (Color online) 2D diagrams of a ratio N s /N of a number of synchro-nized i, j th oscillator pairs N s to whole number of pairs N and synchronous anddesynchronized regimes of the network Eq. (2) with local intra-layer coupling R , = 1 in (a) ( σ , γ ) parameter plane at σ = 0 . , and in (b) ( σ , γ ) parameterplane at σ = 0 . . Region DS corresponds to an absence of the structure syn-chronization, while in regions S A and S B the lattices are synchronized. In region S A the same labyrinth-like structures (initial state of the first layer) are formedin both lattice, and in region S B the identical spiral waves (initial regime of thesecond lattice) are observed in the layers. Boundaries between the regions areillustrated by dotted lines. Parameters: J = − , J = 1 , ε = 2 , ω = 2 , N = 50 .illustrated by dotted lines. These plots show that the synchronization effectshave a strong and complex dependence on the ratio of σ to σ . It is seen thatthe closeness of these values impedes the synchronization. The synchronizationis possible only when either σ ≥ σ (see Fig. 4(a)) or σ (cid:29) σ (Fig. 4(b)).Moreover, these results show that the labyrinth-like structure (initial state inthe first lattice with the repulsive interaction) is more stable than the spiral waveregime. The first one suppresses the spiral wave within the whole region DS aswell as within the synchronous region for the case when values of the inter-layercoupling strength γ are sufficiently small and σ > σ (region S A ). Examples ofstructures in the first and second lattices are represented in Fig. 5(a) and (b) forregion DS and in Fig. 5(d) and (e) for region S A . At first sight, it might seemthat the structures in the layers are identical for the both regimes. However,spatial distributions of the correlation coefficient R i,j illustrated in Fig. 5(c)(region DS) and Fig. 5(f) are noticeably different from each other. For regionDS, only a small part of oscillator pairs demonstrate the synchronous behavior(their R i,j ≥ . , while rest of the pairs have R i,j < . ). This means thatthe instantaneous phases and/or amplitudes are not the same, i.e. the phasesynchronization is absent. The values of correlation coefficients for region S A are R i,j ≥ . for all the i, j th oscillator pairs. Hence, the effective in-phasesynchronization of the whole system takes place. Note that in both the struc-tures there is a certain rotation center around which the wave front propagates11 B , γ = . S A , γ = . D S , γ = . first layer j i -1012 x i,j (a) j i -1012 x i,j (d) j i -1012 x i,j (g) second layer j i -1012 x i,j (b) j i -1012 x i,j (e) j i -1012 x i,j (h) R i,j j i R i,j (c) j i R i,j (f) j i R i,j (i)Figure 5: (Color online) Typical spatiotemporal structures in the region ofdiagrams represented in Fig. 4 at fixed values of σ = 0 . and σ = 0 . . Plotsin the left and middle columns demonstrate snapshots of the system state forthe first and second lattices, respectively. Spatial distribution of the correlationcoefficients R i,j are represented in the right column. The top line of plotscorresponds to region DS at γ = 0 . , the median line is for region S A at γ = 0 . ,and the bottom line shows plots for for region S B at γ = 0 . . Parameters: J = − , J = 1 , R , = 1 , ε = 2 , ω = 2 , N = 50 .and the spatial structure corresponds to the labyrinth-like one.Significant changes occurs in region S B shown in Fig. 4. The labyrinth-likestructure completely disappears and the spiral wave regime is formed in bothlayers after a certain transient process. This is illustrated by snapshots of thesystem state for both of the lattices in Fig. 5(g) and (h). In this case theoscillations are most synchronous in comparison with the other regimes. It is12ell seen in a spatial distribution of the correlation coefficients R i,j in Fig. 5(i).Most of the i, j th oscillator pairs oscillate in-phase and are characterized by R i,j ∼ = 1 and only the oscillators around the wave centers have values of R i,j (cid:47) .When studying the synchronization of locally coupled 2D lattices it has beendiscovered that the spiral wave regime in the attractively coupled lattice is lessstable than the labyrinth-like structure in the lattice with repulsive coupling.The spiral wave can suppress the state in the first lattice only when the intra-layer coupling strength σ is stronger than one in the first layer, i.e. σ >σ . In other cases, the labyrinth-like structure suppresses the spiral wave. Aquestion arises how does the non-locality of the intra-layer coupling affects thesynchronization effects? We now study the case when values of the intra-layer coupling ranges are equal S B DSS A N σ /N γ s σ (a) DSS B S A N σ /N γ s σ (b)Figure 6: (Color online) 2D diagrams of a ratio N s /N of a number of synchro-nized i, j th oscillator pairs N s to a whole number of pairs N and synchronousand desynchronized regimes of the network Eq. (2) with nonlocal intra-layercoupling R , = 2 in (a) ( σ , γ ) parameter plane at σ = 0 . , and in (b) ( σ , γ )parameter plane at σ = 0 . . Region DS corresponds to an absence of the struc-ture synchronization, while in regions S A and S B the lattices are synchronized.In region S A the same labyrinth-like structures (initial state of the first layer)are formed in both of the lattice, and in region S B the identical spiral waves(initial regime of the second lattice) are observed in the layers. Boundaries be-tween the regions are illustrated by dotted lines. Parameters: J = − , J = 1 , ε = 2 , ω = 2 , N = 50 .to P , = 2 . The initial states for these values of P , are depicted in Fig. 3(b)for the first repulsively coupled lattice and in Fig. 2(b) for the second latticewith the attractive coupling. Note that we consider only the case of P , = 2 asthe spiral wave exists only for P = 1 and 2.13o diagnose the synchronization between the layers we calculate and plotthe 2D diagrams of synchronous and desynchronous regimes in analogy withthe diagrams in Fig. 4. They are shown in Fig. 6.In general, the non-locality of intra-layer coupling does not lead to qualita-tive changes in the synchronization features. They have the same dynamicalregimes as of the previous case. The inter-layer synchronization is observedfor a significantly wider interval of the inter-layer coupling strength values γ ,especially when values of the intra-layer coupling strength σ is less than σ .Moreover, region S B extended within values of both γ and σ (see Fig. 6(a)).Examples of the structures for each region in the diagrams of regimes inFig. 6 are illustrated by snapshots of the system states for the first and secondlayers in Fig. 7 (the left and middle columns, respectively). Correspondingspatial distributions of the correlation coefficients are shown in Fig. 7 (the rightcolumn). It is seen that the synchronization effects for the case of P , =2 are very similar to the case of P , = 1 , despite the fact that the initialspatiotemporal structures (for γ = 0 ) in the layers are noticeably different thanin the case of local intra-layer coupling P , = 1 . We have explored numerically the synchronization effects in a multiplex networkconsisting of pairwise and bidirectionally coupled 2D lattices of van der Pol os-cillators. The intra-layer interaction between elements in the first lattice hasthe repulsive character, while the interaction in the second lattice is attractiveand dissipative. Both type of coupling have simple physical interpretation. Ourstudy has shown that these type of intra-layer coupling can lead to a signifi-cantly different dynamics in the isolated layers. The repulsively coupled latticedemonstrate the so-called labyrinth-like structures [62] within a wide range ofthe coupling parameters. A regime typical in the second layer is that of a spi-ral wave. It is obtained only for the cases of local and short nonlocal coupling.These spatiotemporal structures have been chosen as initial states in both of thelayers. Note that for any initial combinations, spiral waves are never observedin the lattice with the repulsive coupling, while the labyrinth-like structures areabsent in the attractively coupled layer.It has been shown for the first time that the multiplex network of coupled2D lattices with fundamentally different types of the intra-layer coupling candemonstrate mutual synchronization of spatiotemporal structures when the at-tractive and dissipative (resistive) inter-layer coupling γ is introduced betweenthe layers. The calculations have revealed that a ratio of the intra-layer couplingstrengths σ /σ plays a very important role in the synchronization features,namely the threshold level of intra-layer coupling strength when the effectivesynchronization occurs and a type of structures in the interacting layers. Aregime in the synchronized layers always corresponds to only one of the initialregime, and not a combination of these structures. Thus, a competition betweenthe initial regimes takes place at the mutual interaction between layers. When14 B , γ = . S A , γ = . D S , γ = . first layer j i -1012 x i,j (a) j i -1012 x i,j (d) j i -1012 x i,j (g) second layer j i -1012 x i,j (b) j i -1012 x i,j (e) j i -1012 x i,j (h) R i,j j i R i,j (c) j i R i,j (f) j i R i,j (i)Figure 7: (Color online) Typical spatiotemporal structures corresponding todifferent regions in the diagram of Fig. 6 for fixed values of σ = 0 . and σ = 0 . . Plots in the left and middle columns demonstrate snapshots of thesystem state for the first and second lattices, respectively. Spatial distributionof the correlation coefficients R i,j are represented in the right column. The topline of plots corresponds to region DS at γ = 0 . , the median line is for region S A at γ = 0 . , and the bottom line shows plots for for region S B at γ = 0 . .Parameters: J = − , J = 1 , R , = 2 , ε = 2 , ω = 2 , N = 50 .the effective synchronization between the layers is absent, the spiral wave inthe second layer is destroyed and the same labyrinth-like structures are formedin both of the lattices for any values of the intra-layer and inter-layer couplingstrengths. However, the instantaneous phases of oscillations of correspondingelements in the lattices are slightly shifted against each other. If a value of thecoupling strength in the first layer is greater than one in the second lattice then15he same labyrinth-like structures are formed in both of the lattices. When thecoupling strength in the first layer is greater, the synchronization becomes im-possible. On the other hand, if the coupling in the second layer is greater thanone in the second layer then labyrinth-like regime in the first lattice can be fullyreplaced with the spiral wave. In this case, the oscillations in the layers are themost synchronous. Thus, the introduction of bidirectional inter-layer couplingleads to in-phase synchronization and to the formation of the structures whichcannot be observed in the isolated lattices, namely the spiral wave in the layerwith the repulsive coupling and the labyrinth-like structure in the layer withthe attractive coupling.It has been shown that in the case of nonlocal coupling in both layers, thequalitative features of synchronization remains very similar to the case of lo-cal interaction. However, the synchronization of the spatiotemporal structuresbegins for lower values of the inter-layer coupling strength. Furthermore, theregion in which the synchronized spiral waves are observed in both of the latticesexpanded.In our work we have used the correlation coefficient Eq. (5) as a synchro-nization measure. This measure enables to diagnose phase relations, namelyin-phase or anti-phase synchronization of spatiotemporal structures. Unfortu-nately, we cannot compare our findings with the data of previously publishedworks. Note that other synchronization criterias used in [26, 36, 27] did notallow judging of the phase relations. Acknowlegements
The reported study has been funded by the Russian Science Foundation (projectNo. 20-12-00119). S.S.M acknowledges the use of New Zealand eScience Infras-tructure (NeSI) high performance computing facilities as part of this research.
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