Dynamics of a 2D lattice of van der Pol oscillators with nonlinear repulsive coupling
DDynamics of a 2D lattice of van der Poloscillators with nonlinear repulsive coupling
I.A. Shepelev ∗ , S.S. Muni † , T.E. Vadivasova ∗ September 22, 2020
Abstract
We describe spatiotemporal patterns in a network of identical van derPol oscillators coupled in a two-dimensional geometry. In this study, weshow that the system under study demonstrates a plethora of differentspatiotemporal structures including chimera states when the coupling pa-rameters are varied. Spiral wave chimeras are formed in the networkwhen the coupling strength is rather large and the coupling range is shortenough. Another type of chimeras is a target wave chimera. It is shownthat solitary states play a crucial role in forming an incoherence clusterof this chimera state. They can also spread within the coherence cluster.Furthermore, when the coupling range increases, the target wave chimeraevolves to the regime of solitary states which are randomly distributed inspace. Growing the coupling strength leads to the attraction of solitarystates to a certain spatial region, while the synchronous regime is set in theother part of the system. This spatiotemporal pattern represents a soli-tary state chimera, which is firstly found in the system of continuous-timeoscillators. We offer the explanation of these phenomena and describe theevolution of the regimes in detail.
Introduction
The study of the self-organization phenomena in complex multicomponent sys-tems in the form of oscillatory networks and ensembles is one of the most relevantdirections in the nonlinear dynamics and related disciplines [18, 28, 30, 4, 8].The multicomponent systems with different dynamics of individual elements are ∗ Department of Physics, Saratov State University, 83 Astrakhanskaya Street, Saratov,410012, Russia † School of Fundamental Sciences, Massey University, Palmerston North, New ZealandKeywords: Spatiotemporal pattern formation, chimera state, van der Pol oscillator, spiralwave, spiral wave chimera, target wave, target wave chimera, solitary state, solitary statechimera, nonlocal couplingE-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 . AO ] S e p odels of many real systems both in nature and technology. Their examplesrepresent neuronal network, population of living organisms, transport and com-puter networks and so on. The features of the element to element interactionsignificantly impacts the dynamics of the multicomponent systems (networksand ensembles), the synchronization effects, the formation of different types ofwaves and cluster structures. The coupling topology plays an important role[11, 7, 50, 5]. However, the features of the element to element coupling, namelya type of the coupling function, is very important too. Generally, the linear in-ertialess coupling is considered in most of the models of ensembles and complexnetworks. It increases the dissipation and, usually, is called dissipative coupling.In the case of identical systems, the dissipative coupling leads to a regime ofthe complete synchronization, when oscillations of the interacting systems arein-phase. At the same time, a type of the coupling in real complex systems canbe different. It can be dissipative and inertial [20], nonlinear [53, 32], memristive[47, 49] and with delayed feedback [52, 21, 39, 12]. A special type of couplingrepresents the repulsive interaction, when the coupling coefficient is negative.In general, the repulsive coupling impedes the emergence of in-phase oscilla-tions. An interest to the study of features of systems with repulsive interactionis explained by the fact that this type of coupling takes place in issues of bio-physics and neurodynamics [3, 48, 51, 33, 46]. There are a lot of works, wheredifferent types of repulsive and mixed coupling have been studied. Systems oftwo or three self-oscillators with the periodic or chaotic dynamics have beeninvestigated for the cases of repulsive and mixed coupling in [10, 15, 2, 56, 38].It has been shown that the repulsive coupling in self-oscillating systems usu-ally leads to the amplitude or oscillatory death, while this coupling can induceoscillations in excitable systems [51]. There are a number of works, in whichensembles of oscillators have been studied for the cases of both local [3, 46] andglobal [45] repulsive coupling, and for the case of mixed attractive and repulsivecoupling [16, 14, 27, 22]. Moreover, systems with the local and global repulsivecoupling with the delayed feedback has been considered in [48, 6]. The repulsivecoupling in the ensembles of self-oscillators leads to the amplitude and oscilla-tory death [46, 14, 27]. An ensemble of phase oscillators demonstrates differentsynchronous regimes [45], desynchronization, partial synchronization, travelingwaves [16]. Oscillatory ensembles with mixed attractive (dissipative) and repul-sive coupling show regimes with the complex spatiotemporal dynamics, namelythe solitary states [22] and chimera-like structures [26].Despite these works, there are still many issues concerning the dynamics ofsystems with repulsive coupling. The dynamics of self-oscillator ensembles isyet to be explored for the case of repulsive nonlocal interaction between theoscillators. It is known that the nonlocal coupling with a limited number ofcoupled neighbors in ensembles of identical oscillators of different types, fromthe phase oscillators to stochastic excitable systems, can lead to the formationof chimera states for certain values of parameters. These states presents thecluster structures, including patterns with the coherent and incoherent behav-ior [19, 1, 29, 13, 24, 55, 31, 36, 43, 54]. The question arises about the presenceof chimera states and other complex structures in ensembles with the nonlocal2epulsive coupling. A wide variety of spatiotemporal dynamics can be exceptedin a 2D lattice of the self-oscillators with the both local and nonlocal repulsivecoupling. Another unexplored issue is the influence of the nonlinear characterof repulsive coupling, which can be represented by a two-pole with negativeconductivity. This coupling element should have the current-voltage curve witha falling section and be an active element which adds the negative dissipationto the system. Furthermore, this element is essentially nonlinear. Hence, in-troducing the constant coupling coefficient is an approximate description of thecoupling function, which is valid only when the amplitude is low. Generally, itis necessary to take into account the influence of the nonlinearity of repulsivecoupling on the oscillator dynamics.In the present work, we simulate the dynamics of a 2D lattice of the identi-cal van der Pol (vdP) oscillators, which are coupled by the nonlocal nonlinearrepulsive coupling. The aim of this study is to establish the features of the for-mation of spatial structures in the lattice with the nonlocal nonlinear repulsiveinteraction when different coupling parameters are varied, namely the couplingcoefficient and the coupling range. We also show the role played by the couplingnonlinearity in the formation of different spatiotemporal dynamics of the lattice. At first we consider an electronic circuit of two self-oscillators coupled throughan active nonlinear element with the negative conductivity (for example a tunneldiode). An equivalent circuit design is illustrated in fig.1.Figure 1: Circuit diagram of two electronic oscillators coupled through a tunneldiode.Here g , , L , , C , and N , are linear conductivity, inductors, capacitors andnonlinear elements of the two self-oscillators, respectively. N coup is an elementof the nonlinear coupling. The currents i N , and i coup through the nonlinearelements N , and N coup are described by the following expression: i N , = − α , U , + β , U , ,i coup = − γU + δU , (1)where α , , β , , γ , δ are certain positive parameters. The chosen form of thecurrent-voltage characteristics of the nonlinear elements N , corresponds to van3er Pol oscillators. The Kirchhoff equations of this circuit take the followingform: C dU dt + i L + g U + i N + i coup = 0 C dU dt + i L + g U + i N − i coup = 0 ,di , dt = 1 L , U , , C , dU , dt = i C , . (2)Here t is real time. The following equations for the voltages U , can be ob-tained when the currents i L , , i C , , i N , and i coup are excluded: d U dt − (cid:18) α − g C − β C U (cid:19) dU dt + ω U == − (cid:18) γC − δC ( U − U ) (cid:19) (cid:18) dU dt − dU dt (cid:19) d U dt − (cid:18) α − g C − β C U (cid:19) dU dt + ω U == − (cid:18) γC − δC ( U − U ) (cid:19) (cid:18) dU dt − dU dt (cid:19) , (3)where ω = ( L C ) − / and ω = ( L C ) − / are the parameters specifyingfrequencies of the two self-oscillators. Let C = C = C, α = α = α, β = β = β and L (cid:54) = L . Using the substitution t = ω t , (cid:113) βcω U , = x , ,where ω is some fixed frequency, we obtain the following system of equationsin dimensionless variables: ¨ x − (cid:0) ε − x (cid:1) ˙ x + ω x = − (cid:0) k − m ( x − x ) (cid:1) ( ˙ x − ˙ x )¨ x − (cid:0) ε − x (cid:1) ˙ x + ω x = − (cid:0) k − m ( x − x ) (cid:1) ( ˙ x − ˙ x ) (4)where ε = α − gCω , ω = ω ω , ω = ω ω , k = γω C , m = δβ . We rewrite the modelunder study in the form of a system of differential equations of the first order: ˙ x = y , ˙ y = (cid:0) ε − x (cid:1) y − ω x − (cid:0) k − m ( x − x ) (cid:1) ( y − y ) , ˙ x = y , ˙ y = (cid:0) ε − x (cid:1) y − ω x − (cid:0) k − m ( x − x ) (cid:1) ( y − y . ) (5)If ω = ω then both the oscillators are completely identical.Similarly, we can write the equations for 2D lattice of N identical van der Poloscillators with the nonlocal repulsive nonlinear coupling, which we are going4o study: ˙ x i,j ( t ) = y i,j , ˙ y i,j ( t ) = ( ε − x i,j ) y i,j − ω x i,j − i + Pj + P (cid:80) k = i − Pp = j − P (cid:16) k − m ( x k,p − x i,j ) (cid:17) ( y k,p − y i,j ) ,i, j = 1 , ...N, (6)The double index of dynamical variables x i,j and y i,j with i, j = 1 , ..., N de-termines the position of an element in the two-dimensional lattice. All theoscillators are identical with respect to parameters and each of them is coupledwith all the lattice elements from a square with side (1 + 2 P ) in the center ofwhich this element is located. The integer P defines the nonlocal character ofcoupling and is called the coupling range. The case of P = 1 corresponds tothe local coupling, while P = N/ is the case of global coupling, when each el-ement interacts with the whole system. It determines the number of neighbors Q = (1 + 2 P ) − , which each element is coupled with. Parameter k is thecoefficient of the linear repulsive coupling term while m is the coefficient of theattractive nonlinear coupling term.Two components of coupling can be distinguished from the system (6): linearrepulsive and nonlinear attractive. With this in mind, the equations under studytake the form ˙ x i,j ( t ) = y i,j , ˙ y i,j ( t ) = ( ε − x i,j ) y i,j − ω x i,j − σQ i + Pj + P (cid:80) k = i − Pp = j − P ( y k,p − y i,j ) ++ m i + Pj + P (cid:80) k = i − Pp = j − P ( x k,p − x i,j ) ( y k,p − y i,j ) ,i, j = 1 , ...N. (7)Here the linear repulsive coupling coefficient σ is introduced, which is reduced tothe number of coupled neighbors Q , while the nonlinear attractive term does notdepend on the parameters P and σ . For simplicity, we will call the parameter σ as the coupling strength. It can be seen from the eq.(7) that the ratio betweenthe linear and nonlinear parts has noticeably changed when the coupling range P is varied. When the value of P is low, the linear part prevails over the nonlinearone. However the contribution of the nonlinear part increases intensively withthe elongation of the coupling range. Let us consider the following example.Let a value of the coupling strength is chosen as σ = 0 . . The value of k in theeq.(6) is k = σ/ ((2 P + 1) − . Thus, k = 0 . ( km = 5 ) when P = 1 and already k = 0 . ( km = 0 . ) when P = 4 . For this reason, we expect a significant changein the system dynamics with an elongation of the coupling range due to theincreasing influence of the coupling nonlinearity.We will assume that the boundary conditions for (7) are toroidal, i.e. peri-5dic in both directions: (cid:40) x ,j ( t ) = x N,j ( t ) ,x i, ( t ) = x i,N ( t ) , (cid:40) y ,j ( t ) = y N,j ( t ) ,y i, ( t ) = y i,N ( t ) , (8)Initial conditions for all considered cases are the random values of the variableswith an uniform distribution within x ∈ [ − , , y ∈ [ − , . The systemequations are integrated using the Runge-Kutta 4th order method with a timestep of dt = 0 . . All the regimes under study are obtained after the transientprocess of t trans = 6000 time units. Now we start to explore the dynamics of the lattice (7) when the parameters σ and P are varied and the control parameters are fixed as m = 0 . , ε = 2 . ,and ω = 2 . . Note that a value of ω does not play a sufficient role and can beexcluded by the variable replacement. Thus, the nonlinear part of the couplingis significantly lower than the linear part. At the same time, we show that evena small nonlinear addition into the coupling leads to a qualitative change in thesystem under study.We plot a regime diagram for the lattice (7) in the ( r, σ ) parameter planewithin the range σ ∈ [0 , and P ∈ [1 , as shown in Fig. 2(a). A sequenceof randomly distributed initial conditions within the intervals x i,j ∈ [ − , and y i,j ∈ [ − , are used to construct the regime diagram. The dashed regionscorrespond to the case of multistability in the lattice, when two different steadyregimes are observed for various sets of the initial conditions.An important feature of the system (7) with the repulsive coupling is theabsence of any type of propagating wave regimes. There are no traveling waves,spiral and target waves for any values of σ , P , and initial conditions. Onlystanding waves are realized in the lattice (7). Spatially homogeneous regimeswith the complete synchronization of oscillators are also not typical for thesystem (7).The repulsive coupling increases the total energy. This leads todissynchronization between the system elements, what is most noticeable whenthe coupling range is very short. This effect is observed in the region IRR in the regime diagram for the cases P = 1 (local coupling) and P = 2 . Thespatial structure becomes more and more complex with increase in the couplingstrength. Elongation of the coupling range significantly attenuates this effectand the appearance of the incoherent structures is observed only for large valuesof σ > . . When the coupling strength is sufficiently low and coupling rangeis short ( P < ) regular spatiotemporal states are observed in the lattice.These states are characterized by a piecewise smooth spatial profile and thevery similar periodic oscillations of all the lattice elements. They exist withinthe region RS in the regime diagram in fig.2. Unlike the case of local coupling,growth of σ leads to the formation of the incoherence only within a certainspatial region while the structure in the rest part of a lattice remains regular.6 P σ RSIRR CHCL BS
Figure 2: (Color online) Regime diagram for the lattice (7) in the (
P, σ ) pa-rameter plane when m = 0 . , ε = 2 . and ω = 2 . . Region IRR correspondsto the complete incoherence; region RS relates to the regime of regular states;CL is the region of existence of chimera-like structures; CH corresponds to thechimera states; TS is the region of two-state structures. The regions of coexis-tence of different regimes are shown by alternating strips of corresponding colors(tones).Thus, the state with coexisting coherence and incoherence domains appear inthe system what are typically the chimera states. However, we divide thesestates into two groups, namely the regime of chimera-like states ( CL region)and the regime of chimeras ( CH region). A ( P, σ ) tuple is classified as CL stateif the number of incoherent elements constitute less than of the lattice anddo not form the incoherence cluster. The regime diagram illustrates that theregime of chimera-like structures (region CL ) leads to the evolution of regularstructures with increase in the coupling strength. In turn, the region of chimerastates is observed for the longer coupling range than CL region and expandswith an increase in the values of σ . Further elongation of P leads to a switchof the system to "two-state" regime (region T S ), when all the oscillators areirregularly distributed between two characterized states. Only this regime isrealized for the long coupling range ( P (cid:62) ). Next we explore all the mainregimes in detail. We study the regime of regular spatiotemporal patterns realized in the region RS of the regime diagram in fig.2. Spatial profiles of the structures for thecases of local and nonlocal coupling are significantly different. An example ofthe structure for P = 1 is illustrated by a snapshot of the system state infig.3(a). The shape of this structure is sufficiently complex, however the spatial7rofile is always smooth and the instantaneous states of the adjacent oscillatorsare very similar. Its spatiotemporal dynamics is depicted in fig.3(b) by a set ofinstantaneous cross sections of the two-dimensional spatial profile shown afterevery the half period T / . It can be seen that the spatial profile repeats itselfafter each period T = 2 . time units. This indicates that a motion of thewavefront is absent and the regime represents the standing wave, and also thatoscillations of all the elements are periodic. The regular character of oscillations
10 20 30 40 j i − − − x i,j (a)
10 20 30 40 j i − − − x i,j (d) − − −
10 20 30 40 x i (b) − − −
10 20 30 40 x i (e) − − − − − − − y x (c) − − − − − − − y x (f)Figure 3: (Color online) Regular spatiotemporal states in the system (7) (region RS in the regime diagram in fig.2). The case of local coupling P = 1 with σ = 0 . is illustrated by panels (a)-(c), and the case of P = 2 and σ = 0 . corresponds to panels (d)-(f). (a), (d) snapshots of the system states; sets of 20instantaneous spatial cross-sections shown every the half period for j = 10 (b)and j = 31 (e); projections of the phase trajectories for the i = 7 , j = 20 th (solidblack line), i = 30 , j = 22 th (solid green line), i = 37 , j = 15 th (solid red line), i = 44 , j = 35 th (dotted black line) elements (c) and for the i = 17 , j = 31 th(solid black line), i = 25 , j = 31 th (solid green line), i = 26 , j = 31 th (solid redline), i = 41 , j = 20 th (dotted black line) elements (f). Parameters: m = 0 . , ε = 2 . , ω = 2 . , N = 50 .is also shown by projections of phase trajectories for different elements of thelattice in fig.3(c). Furthermore, all the oscillators are characterized by almostthe same attractors. It should be noted that here and further we do not usethe terms "an attractor" and "a limit cycle" in a sense of an attractor in thephase space of the multidimensional dynamical system (7), but in a sense of anattracting manifold in a plane of the variables of an individual oscillator. Hence,there is only one steady state in the system.8hen the coupling becomes nonlocal ( P = 2 ), the shape of the spatial profileundergoes a significant change. A typical spatial structure in the lattice (7) forthe nonlocal coupling is depicted by a snapshot of the system state in fig.3(d).The distinctive feature of this state is its square step-structure. This structurerepresents a step pyramid with a deep recess in the center. Apparently, thesquare shape of the steps is due to the coupling geometry, namely each oscillatoris coupled with all the oscillators from a square with the edge (2 P + 1) . A setof spatial cross-section intersecting the pyramid center is shown in fig.3(e). Thespatial cross sections are illustrated after every the half period T , which isequal to T = 2 . time units. It is seen that the spatial structure regularly"breathes" in time. Fig.3(f) demonstrates projections of the phase trajectoriesfor different oscillators of the lattice. The plot indicates that all the oscillatorsare characterized by the same projection in a form of the closed curve. Thus,the oscillators located on different steps are distinguished from each other byonly instantaneous phase, while the oscillators on the same step have the verysimilar instantaneous phase and amplitude. It should be noted, that this typeof spatiotemporal structure is unique for a lattice of the coupled van der Poloscillators and is never observed for the case of attractive coupling. Apparently,this is a result of the repulsive interaction between elements. An increase in the coupling strength leads to the complication of spatial struc-tures. This effect is most pronounced in the case of a very short coupling range P = 1 and P = 2 . Moreover, for the case of P = 2 this regime is observedfor significantly higher values of σ than when the coupling is local and is notrealized for the longer coupling range within σ ∈ [0 , . Examples of these statesare shown in fig.4(a) for P = 1 and in fig.4(b) for P = 2 . It can be seen that theshape of spatial profiles becomes significantly complex. At the same time therectangular geometry of patterns is preserved. Further growth of the couplingstrength leads to an increase in the irregularity of the spatiotemporal struc-tures. Our study of this regime shows that a growth in the strength of repulsivecoupling leads to qualitative and quantitative changes in the system dynamics,namely the lattice (7) becomes highly multistate. If for the previous case the allthe lattice elements are characterized by the same closed curve in the common x − y plane (which we conditionally call as a limit cycle ) then for the irregularstructures oscillations of different elements are already related to various limitcycles. This effect is clearly seen in the projections of the phase trajectories infig.4(c),(d). When the coupling is local, each chosen element is characterizedby unique limit cycle. For the case of P = 2 , this effect is attenuated but itis possible to detect at least three various limit cycles. Hence, growth of thestrength of repulsive coupling σ leads to the emergence of new stable solutionsand the system becomes multistable. Apparently, an increase in the contribu-tion of nonlinear attractive coupling with growth of P reduces a number of thestates of lattice elements. 9 = 1
10 20 30 40 j i − − − x i,j (a) − − − − − − − y x (c) P = 2
10 20 30 40 j i − − − x i,j (b) − − − − − − − y x (d)Figure 4: (Color online) Irregular structures in the system (7) (region IRR inthe regime diagram in fig.2). The case of local coupling P = 1 and σ = 0 . isillustrated by panels (a), (c), and the case of P = 2 and σ = 0 . correspondsto panels (b), (d). (a) and (b) snapshots of the system states; projections of thephase trajectories for (c) the i = 7 , j = 20 th (solid black line), i = 9 , j = 35 th(solid green line), i = 16 , j = 21 th (solid red line), i = 18 , j = 21 th (dottedblack line) elements and (d) shows projections for the i = 4 , j = 31 th (solidblack line), i = 11 , j = 17 th (solid green line), i = 16 , j = 11 th (solid red line), i = 31 , j = 21 th (dotted black line) elements. Parameters: m = 0 . , ε = 2 . , ω = 2 . , N = 50 . Now we discover the spatiotemporal behavior of the regime realized in region CL of the regime diagram in fig.2. For this regime contribution of the non-linear attractive coupling noticeably increases. The spatiotemporal structuresrepresents mixing of two previously described structures, namely the most partof the lattice in partly coherent state, while certain groups of elements oscil-late asynchronously with adjacent neighbors. Examples of these structures areillustrated in figs.5(a) and (d) for two different values of P . It is visible thatthe consequence of elongation of the coupling range is the disappearance of thesquare step-structure. There are a small number of elements which oscillateincoherently (the small incoherence clusters) and the rest part of system os-cillators, which form regular structures (the coherence cluster). The temporal10ynamics is regular and is presented by a sets of spatial cross-section throughthe incoherence clusters in figs.5(b) and (e) for P = 3 and P = 6 , accordingly.The system states are shown every half period T / . The instantaneous states ofoscillators in the incoherence domain are noticeably different from the states ofthe oscillators from the coherence cluster. Oscillations of all the oscillators areperiodic similar to the previous cases. As it has been shown above, an increase
10 20 30 40 j i − − − x i,j (a)
10 20 30 40 j i − . − − . . . x i,j (d) − − −
10 20 30 40 x i (b) − − −
10 20 30 40 x i (e) − − − − − − − y x (c) − − − − − − − y x (f)Figure 5: (Color online) Chimera-like states in the system (7) (region CL inthe regime diagram in fig.2). The case of P = 3 and σ = 0 . is illustrated bypanels (a)-(c), and the case of P = 6 and σ = 0 . corresponds to panels (d)-(f).(a), (d) snapshots of the system states; sets of 20 instantaneous spatial cut-offsshown every the half period for (b) j = 9 and (e) j = 25 ; projections of thephase trajectories for (c) the i = 15 , j = 9 th (solid black line), i = 20 , j = 9 th(solid green line), i = 21 , j = 21 th (solid red line), i = 27 , j = 27 th (dotted blackline) elements and for (f) the i = 14 , j = 14 th (solid black line), i = 17 , j = 24 th(solid green line), i = 20 , j = 24 th (solid red line), i = 40 , j = 40 th (dottedblack line) elements. Parameters: m = 0 . , ε = 2 . , ω = 2 . , N = 50 .in the coupling strength σ leads to the emergence of new states of individualoscillators of the system under study. At the same time, the elongation of thecoupling range P reinforces a contribution of the nonlinear attractive couplingterm. Apparently, the nonlinearity of coupling decreases the effect of birth ofthe new states of the oscillators. Fig.5(c) illustrates projections of the phasetrajectories for oscillators of both the coherence and incoherence domains forthe case of P = 3 and sufficiently high value of σ . It is clearly seen that theoscillations correspond to different limit cycles. This feature of the oscillatorsof incoherence domain is similar with the feature of solitary states, which are11bserved in different systems with attractive coupling [17, 23, 44, 42, 35, 40],including the lattice of van der Pol oscillators [41]. Thus, this regime can beconsidered as a type of solitary state. When the value of P is larger and thecoupling strength is weaker (see fig.5(d)), the limit cycles corresponding to oscil-lations in coherence and incoherence clusters becomes significantly more similar.They are depicted in fig.5(f). A further increase in the coupling range leads to the growth of a number ofoscillators with the asynchronous behavior. They tend to form an incoherencecluster of the large size, localized in certain places of the lattice. Thus, wecan consider this state as a chimera. Examples of the chimera states in thesystem (7) are illustrated in figs.6(a),(d) and (c) for different values of P . Allspatiotemporal structures contain two sufficiently large incoherence clusters. Asit has been mentioned above, the square topology of structures is already absentfor these values of P for initial conditions under study. In order to quantify thisregime we calculate the spatial distribution of the root-mean-square deviation(RMSD) by the following formula: ∆ i,j = (cid:112) (cid:104) ( x i,j − x i +1 ,j +1 ) (cid:105) , (9)where (cid:104)(cid:105) mean averaging in time.This characteristic shows statistical difference between states of adjacent os-cillators. If the instantaneous amplitude and phases of oscillations of adjacentelements are similar then the RMSD values are small. At the same time, valuesof ∆ i,j are large when the oscillation features of two adjacent elements are no-ticeably different. Thus, the coherence cluster is characterized by small valuesof the RMSD, and the incoherence cluster has high values of the ∆ i,j . Spatialdistributions of the RMSD for the structures under study are represented infigs.6(b),(e) and (h). The incoherence clusters for all the three are character-ized by the maximum values of the RMSD, while the coherence clusters havesufficiently low values of ∆ i,j . Besides, there are vertical lines in figs.6(b) and(e) with high values of ∆ i,j . Their existence is associated with a step-like shapeof the spatial structures of coherence clusters, which is seen from the snapshotsof system states in figs.6(a) and (d). The instantaneous states of adjacent oscil-lators in the boundaries of these "steps" are always significantly different andvalues of the RMSD for these oscillators are large. Interestingly, for the longcoupling range ( P = 8 ) the step-like shape disappears and a structure withinthe coherence cluster becomes smooth. As it has been shown above a reasonof the emergence of an incoherence in the system (7) is the appearance of newstable periodic states with growing the P and σ . We assume that the elongationof the coupling range leads to an expansion of the basin of attraction of the newperiodic regimes. Hence, more and more number of oscillators are inside thesebasins from randomly distributed initial conditions. Coexistence of differentlimit cycles are illustrated by the ( x i,j , y i,j ) projections of the phase trajectories12 = 5
10 20 30 40 j i − − x i,j (a)
10 20 30 40 j i . . . . . ∆ i,j (b) − − − − − − − y x (c) P = 6
10 20 30 40 j i − . − − . − − . . x i,j (d)
10 20 30 40 j i . . . . . . . . ∆ i,j (e) − − − − − − − y x (f) P = 8
10 20 30 40 j i − . − − . . . x i,j (g)
10 20 30 40 j i . . . . . . . . ∆ i,j (h) − − − − − − − y x (i)Figure 6: (Color online) Chimera states in the system (7) (region CH in theregime diagram in fig.2). The case of P = 5 and σ = 0 . is illustrated by panels(a)-(c), the case of P = 6 and σ = 0 . corresponds to panels (d)-(e), and thecase of P = 8 and σ = 0 . corresponds to panels (g)-(h). (a), (d) and (g)are snapshots of the system states; (b), (e), (h) are spatial distributions of theRMSD, projections of the phase trajectories for (c) the i = 10 , j = 22 th (solidblack line), i = 31 , j = 26 th (solid green line), i = 31 , j = 27 th (solid red line);for (f) the i = 26 , j = 18 th (solid black line), i = 27 , j = 18 th (solid green line), i = 39 , j = 13 th (solid red line); for (i) the i = 21 , j = 24 th (solid black line), i = 22 , j = 24 th (solid green line), i = 26 , j = 10 th (solid red line). Parameters: m = 0 . , ε = 2 . , ω = 2 . , N = 50 .for different oscillators in figs.6(c),(f) and (i). There are at least three differentlimit cycles of individual oscillators in the common ( x, y ) plane for the cases P = 5 and P = 6 . At the same time, difference between limit cycles when P = 6 is less visible than when P = 5 . Apparently, this is due to the fact that13he value of the coupling strength σ is significantly lower for the case of P = 6 .When the coupling range becomes longer ( P = 8 ), only two characterized pe-riodic attractors remains in the system (see fig. fig:CH(i)). We assume thatthis is a consequence of the strong influence of the coupling nonlinearity. Asit has been shown above, elongation of P leads to a significant decrease in thelinear repulsive coupling term and the reinforcement of the nonlinear attractivecoupling term. Hence, when the nonlinear attractive coupling term prevails overthe linear one, the system under study becomes two-state with two distinctivestable periodic attractors in the phase plane of individual oscillators. Thus, thechimera states in the lattice (7) are similar to solitary state chimeras, whichexist in a lattice of van der Pol oscillators with the attractive coupling [41] andother systems [37, 34, 25]. Now we discover the system dynamics when the coupling range is sufficientlylong. As it has been mentioned in the previous section, the system under studybecomes two state (there are only two different state of each oscillators) whenthe value of P is large, namely there are only two coexisting limit cycles. Aspecial spatiotemporal behavior is observed in region T S of the regime diagramin fig.2. This is the only possible regime when
P > . An example of this stateis depicted in fig.7(a). The structure represents irregularly distribution betweentwo the states with two close certain levels. The spatiotemporal dynamics isshown by a set of snapshots of the system states shown after every half period T / in fig.7(b). The features of this regime is similar to the case of completesynchronization. However, there are two peculiar states, and all the elements areirregularly distributed between them. These levels correspond to the two stableperiodic attractors of individual oscillators, which are shown in the projectionsof the phase trajectories for different selected oscillators in fig.7(c). Apparently,the increasing influence of the nonlinear attractive coupling term leads to thesimilarity of the sizes of the two basins of attraction. Hence, the oscillators areequiprobably located between them. In the previous sections, the coefficient of nonlinear attractive coupling in (7)has been fixed as m = 0 . and we have varied the parameters of the linearrepulsive coupling term. Now a question arises, how does an increase in thecoefficient m change the system dynamics? To answer this question, we study avariety of the main dynamical regimes when a value of the parameter m is fixedas m = 0 . . Fig.8 demonstrates a diagram of the main regimes in the ( P, σ )parameter plane. For this value of m there are the same dynamical regimes asfor the previous case of m = 0 . . At the same time, the regions of existence of14
10 20 30 40 j i − . − . x i,j (a) − − −
10 20 30 40 x i (b) − − − − − − − y x (c)Figure 7: (Color online) Two-state irregular structures (region T S in the regimediagram in fig.2) when P = 9 and σ = 0 . . (a) snapshot of the system state, (b)spatial crossection for j = 24 shown every half period T / , (c) phase portraitprojection for the i = 22 , j = 25 th (black line) and for the i = 23 , j = 25 th(green line). Parameters: m = 0 . , ε = 2 . , ω = 2 . , N = 50 . P σ BSRS CHCLIRR
Figure 8: (Color online) Diagram of the regimes for the lattice (7) in the (
P, σ )parameter plane when m = 0 . . Region IRR corresponds to complete incoher-ence; region RS relates to the regime of regular structures; CL is the region ofexistence of chimera-like structures; CH correspond to chimera states; TS is theregion of two-state structures. The regions of coexistence of different regimesare shown by alternating strips of the corresponding colors (tones). Parameters: ε = 2 . , ω = 2 . , N = 50 .the different regimes has noticeably changed. At first, the irregular structuresin the region IRR are observed only for the case of local coupling ( P = 1 ) andhigher values of the coupling strength σ than for the case of m = 0 . . Theregion of chimera states (region CH ) becomes more narrow. However, the maindifference is that the regime of two-state structures (region T S ) is now realizedfor the significantly shorter coupling range P . These changes can be explainedby the stronger influence of the nonlinear coupling term in the present case.15ence, all the effects associated with the coupling nonlinearity are realized forlower values of P and σ .Now we fix the value of the coupling strength as σ = 0 . and vary thecoupling range P and the nonlinear coupling coefficient m . All the main regimesare observed for the chosen value of σ . This study enables us to evaluate how thesystem dynamics has changed with an increase in the nonlinearity of coupling.Fig.9 demonstrate the parametric diagram of regimes in the ( P, m ) parameterplane. This diagram shows that the regime of irregular structures are possible P m RS BSCHCLIRR
Figure 9: (Color online) Phase-parametric diagram of the regimes for the lattice(7) in the (
P, m ) parameter plane for the fixed value of σ = 0 . . RegionIRR corresponds to complete incoherence; region RS relates to the regime ofregular structures; CL is the region of existence of chimera-like structures; CHcorrespond to chimera states; TS is the region of two-state structures. Theregions of coexistence of different regimes are shown by alternating strips of thecorresponding colors (tones). Parameters: m = 0 . , ε = 2 . , ω = 2 . , N = 50 .only for the minimum values of the nonlinear coupling coefficient m . The regionof regular structures (region RS ) is realized for the short coupling range andexpands with the growth of m . Elongation of P leads to the transition to theregion of chimera-like states, which, in turn, transform into chimera structureswith large incoherence clusters. Both the regions narrow down with an increasein the coupling nonlinearity, unlike region RS . A further increase in the valuesof the coupling range P leads to a switch of the system (7) to the regime of two-state structures (region T S ). Growth of the coefficient m is accompanied byalmost linear expansion of this region. Apparently, the increase in the coefficientof nonlinear attractive coupling m decreases the irregularity in the system andrestricts a number of coexisting periodic states of individual elements. For thisreason, chimera and chimera-like states are realized within a wide range of P values only when the coupling nonlinearity is sufficiently weak. The strongcoupling nonlinearity leads to the coexistence of two limit cycles of individual16lements, which are characterized by the similar sizes of the basins of attractions. Conclusions
In this work we have studied the dynamics of a two-dimensional lattice of non-locally interacting identical van der Pol oscillators with linear repulsive andnonlinear attractive coupling between the elements with toroidal boundary con-ditions. We derive the system equation from a radiophysical circuit of thecoupled van der Pol oscillators interacting through the nonlinear element witha current-voltage characteristic of the N type (for example, a tunnel diode).The behavior of the 2D lattice under study is discovered for both the local andnonlocal character of an oscillator interaction and for variation of the couplingparameters. The coupling function can be divided into two terms, namely thelinear repulsive coupling and nonlinear attractive coupling. Moreover, the coef-ficient of nonlinear coupling does not depend on values of coupling strength σ of linear coupling and of the coupling range P . This character of coupling ex-tremely changes the dynamics of the system and the dynamical regimes becomescompletely different from the case of a 2D lattice of van der Pol oscillators withcommon attractive (dissipative) coupling [9, 41]. For example, all the regimesrepresent standing waves for any randomly distributed initial conditions underconsideration. At the same time, there are no any regimes of traveling, spiraland target waves, and of spatio-uniform structures in the lattice with both thelocal and nonlocal interaction.We have constructed the diagrams of the regimes, that are realized in thelattice (7), in the ( P, σ ) parameter plane when the nonlinear coupling coefficientis fixed as m = 0 . and m = 0 . . Numerical simulation of the lattice dynamicsenables us to reveal the following features of its behavior. When the couplingstrength is sufficiently weak and the coupling range is short, different structureswith the regular temporal behavior form in the lattice. When the coupling islocal, the shape of the structures are very complex. The coupling nonlocalityleads to significant change in the structure shape, which becomes simpler andhas the square step-like shape, namely groups of oscillators forms rectangles ofa certain width, where all the instantaneous phases and amplitudes are verysimilar. At the same time oscillations of all the elements are characterized bythe same periodic attractor in the phase plane of an individual oscillator. Anincrease in the coupling strength induces the emergence of new stable periodicstates for individual oscillators in the system (7). However, the elongation ofthe coupling range restricts this phenomenon due to growth of the influence ofnonlinear attractive coupling. For this reason, this effect is mostly strong whenthe coupling range is equal P = 1 or P = 2 . Emergence of a large numberof coexisting stable limit cycles for different elements increases the spatiotem-poral incoherence and the spatial structures become irregular. As it has beenmentioned, elongation of the coupling range decreases the number of coexist-ing stable states. This leads to the appearance of new types of spatiotemporalstructures, namely the chimera-like and chimera states. They are characterized17y the coexistence of clusters of the synchronous (coherence) and asynchronous(incoherence) behavior of neighbor oscillators. The number of elements withincoherent behavior are very small for chimera-like structures and is sufficientlylarge for chimeras. An important feature of this structures is that oscillationsof all the coherence cluster elements corresponds to the same periodic attractorin the ( x, y ) plane, while elements of the incoherence cluster are characterizedby several other coexisting periodic attractors. Thus, these chimeras are simi-lar to solitary state chimeras, observed in systems with the attractive coupling[37, 34, 25, 41]. We explore that the chimera-like states evolve into the chimeraswith elongation of the coupling range. For this regime, all the oscillator areirregularly distributed between two stable states.Additionally, we study the behavior of the system when the nonlinear cou-pling coefficient m is varied and the coupling strength is fixed. We plot theregime diagram in the ( P, m ) parameter plane. The diagram shows that the re-gion of chimera and chimera-like structures decrease with the growth of m . Onthe contrary, the regions with simple spatial structures and two-state structuresincrease. Thus, the nonlinear attractive coupling leads to the simpler behaviourof the lattice under study. Acknowledgements
I.A.S. and T.E.V. thank for the financial support the Russian Science Founda-tion (grant 20-12-00119), S.S.M. acknowledges the use of New Zealand eScienceInfrastructure (NeSI) high performance computing facilities as part of this re-search.
References [1] D.M. Abrams and S.H. Strogatz. Chimera states for coupled oscillators.
Physical review letters , 93(17):174102, 2004.[2] S. Astakhov, A. Gulai, N. Fujiwara, and J. Kurths. The role of asymmet-rical and repulsive coupling in the dynamics of two coupled van der poloscillators.
Chaos , 26:023102, 2016.[3] G. Bal´azsi, A. Cornell-Bell, A.B. Neiman, and F. Moss. Synchronizationof hyperexcitable systems with phase-repulsive coupling.
Phys. Rev. E. ,64:041912, 2001.[4] A. Barrat, M. Barthelemy, and A. Vespignani.
Dynamical processes oncomplex networks . Cambridge university press, 2008.[5] I. Belykh, D. Carter, and R. Jeter. Synchronization in multilayer networks:When good links go bad.
SIAM Journal on Applied Dynamical Systems ,18(4):2267–2302, 2019. 186] B.K. Bera, C. Hens, and D. Ghosh. Emergence of amplitude death scenarioin a network of oscillators under repulsive delay interaction.
Phys. Lett. A.2016 , 380(31–32):2366–2373, 2016.[7] B.K. Bera, S. Majhi, D. Ghosh, and M. Perc. Chimera states: Effects ofdifferent coupling topologies.
EPL , 118:1, 2017.[8] S. Boccaletti, A. Pisarchik, C. Genio, and A. Amann.
Synchronization:From Coupled Systems to Complex Networks . CUP, 03 2018.[9] A. Bukh, G. Strelkova, and V. Anishchenko. Spiral wave patterns in a two-dimensional lattice of nonlocally coupled maps modeling neural activity.
Chaos, Solitons & Fractals , 2019. (in press).[10] Y. Chen, J. Xiao, W. Liu, L. Li, and Y. Yang. Dynamics of chaotic systemswith attractive and repulsive couplings.
Phys. Rev. E. , 80:046206, 2009.[11] C.I. del Genio, J. G´omez-Garde˜nes, I. Bonamassa, and S. Boccaletti. Syn-chronization in networks with multiple interaction layers.
Science Advances ,2(11), 2016.[12] T. Kau e, D. Peron, and F.A. Rodrigues. Explosive synchronization en-hanced by time-delayed coupling.
Phys. Rev. E. , 86:016102, 2012.[13] A.M. Hagerstrom, T.E. Murphy, R. Roy, P. H¨ovel, I. Omelchenko, andE. Sch¨oll. Experimental observation of chimeras in coupled-map lattices.
Nature Physics , 8:658–661, 2012.[14] C.R. Hens, O.I. Olusola, P. Pal, and S.K. Dana. Oscillation death indiffusively coupled oscillators by local repulsive link.
Phys. Rev. E. ,88(3):034902, 2013.[15] C.R. Hens, P. Pal, S.K. Bhowmick, P.K. Roy, A. Sen, and S.K. Dana.Diverse routes of transition from amplitude to oscillation death in coupledoscillators under additional repulsive links.
Phys. Rev. E. , 89(3):032901,2014.[16] H. Hong and S.H. Strogatz. Kuramoto model of coupled oscillators withpositive and negative coupling parameters: An example of conformist andcontrarian oscillators.
PRL , 106:054102, 2011.[17] P. Jaros, S. Brezetsky, R. Levchenko, D. Dudkowski, T. Kapitaniak, andY. Maistrenko. Solitary states for coupled oscillators with inertia.
Chaos:An Interdisciplinary Journal of Nonlinear Science , 28(1):011103, 2018.[18] Y. Kuramoto. Chemical oscillations, waves and turbulence.
Springer-Verlag , 1984.[19] Y Kuramoto and D Battogtokh. Coexistence of coherence and incoherencein nonlocally coupled phase oscillators.
Nonlinear Phenom. Complex Syst. ,5(4):380–385, 2002. 1920] A.P. Kuznetsov, N.V. Stankevich, and L.V. Turukina. Duffing oscilla-tors: phase dynamics and structure of synchronization tongues.
PhysicaD , 238:14:1203–1215, 2009.[21] X. Liu and T. Chen. Exponential synchronization of nonlinear coupled dy-namical networks with a delayed coupling.
Physica A: Statistical Mechanicsand its Applications , 381:82–92, 2007.[22] Y. Maistrenko, B. Penkovsky, and M. Rosenblum. Solitary state at theedge of synchrony in ensembles with attractive and repulsive interactions.
Phys. Rev. E. , 89:060901, 2014.[23] Y. Maistrenko, B. Penkovsky, and M. Rosenblum. Solitary state at theedge of synchrony in ensembles with attractive and repulsive interactions.
Physical Review E , 89(6):060901, 2014.[24] E.A. Martens, S. Thutupalli, A. Fourri`ere, and O. Hallatschek. Chimerastates in mechanical oscillator networks.
Proceedings of the NationalAcademy of Sciences , 110(26):10563–10567, Jun 2013.[25] M. Mikhaylenko, L. Ramlow, S. Jalan, and A. Zakharova. Weak multi-plexing in neural networks: Switching between chimera and solitary states.
Chaos: An Interdisciplinary Journal of Nonlinear Science , 29(2):023122,2019.[26] A. Mishra, C. Hens, M. Bose, P.K. Roy, and S.K. Dana. Chimeralike statesin a network of oscillators under attractive and repulsive global coupling.
Phys. Rev. E. , 92(6):062920, 2015.[27] M. Nandan, C.R. Hens, P. Pal, and S.K. Dana. Transition from amplitudeto oscillation death in a network of oscillators.
Chaos , 24:043103, 2014.[28] V.I. Nekorkin and M.G. Velarde.
Synergetic Phenomena in Active Lattices .Springer Series in Synergetics. Springer, Berlin, Heidelberg, 2002.[29] I. Omelchenko, Y. Maistrenko, P. H¨ovel, and E. Sch¨oll. Loss of coherencein dynamical networks: spatial chaos and chimera states.
Phys. Rev. Lett. ,106:234102, 2011.[30] G.V. Osipov, J. Kurths, and C. Zhou.
Synchronization in Oscillatory Net-works . Springer Series in Synergetics. Springer, Berlin, Heidelberg, 2007.[31] M.J. Panaggio and D.M. Abrams. Chimera states: coexistence of co-herence and incoherence in networks of coupled oscillators.
Nonlinearity ,28(3):R67–R87, Feb 2015.[32] J. Petereit and A. Pikovsky. Chaos synchronization by nonlinear coupling.
CNSNS , 44:344–351, 2017.[33] M.I. Rabinovich, P. Varona, A. I. Selverston, and H. D. I. Abarbanel. Dy-namical principles in neuroscience.
Rev. Mod. Phys. , 78:1213, 2006.2034] E. Rybalova, V.S. Anishchenko, G.I. Strelkova, and A. Zakharova. Solitarystates and solitary state chimera in neural networks.
Chaos: An Interdis-ciplinary Journal of Nonlinear Science , 29(7):071106, 2019.[35] E. Rybalova, N. Semenova, G. Strelkova, and V. Anishchenko. Transitionfrom complete synchronization to spatio-temporal chaos in coupled chaoticsystems with nonhyperbolic and hyperbolic attractors.
Eur. Phys. J. Spec.Top. , 226:1857–1866, Jun 2017.[36] E.V. Rybalova, D.Y. Klyushina, V.S. Anishchenko, and G.I. Strelkova. Im-pact of noise on the amplitude chimera lifetime in an ensemble of nonlocallycoupled chaotic maps.
Regular and Chaotic Dynamics , 24(4):432–445, 2019.[37] E.V. Rybalova, G.I. Strelkova, and V.S. Anishchenko. Mechanism of real-izing a solitary state chimera in a ring of nonlocally coupled chaotic maps.
Chaos, Solitons & Fractals , 115:300–305, 2018.[38] M. Dev Shrimali S. Dixit, A. Sharma. The dynamics of two coupled vander pol oscillators with attractive and repulsive coupling.
Physics LettersA. , 383:125930, 2019.[39] G.S. Schmidt, A. Papachristodoulou, U. M¨unz, and F. Allg¨uwer. Frequencysynchronization and phase agreement in kuramoto oscillator networks withdelays.
Automatica , 48:3008–3017, 2012.[40] N. Semenova, T. Vadivasova, and V.S. Anishchenko. Mechanism of solitarystate appearance in an ensemble of nonlocally coupled lozi maps.
TheEuropean Physical Journal Special Topics , 227(10-11):1173–1183, 2018.[41] I.A. Shepelev, A.V. Bukh, S.S. Muni, and V.S. Anishchenko. Role of soli-tary states in forming spatiotemporal patterns in a 2d lattice of van derpol oscillators.
Chaos, Solitons & Fractals , 135:109725, 2020.[42] I.A. Shepelev and T.E. Vadivasova. Solitary states in a 2d lattice of bistableelements with global and close to global interaction.
Nelineinaya Dinamika[Russian Journal of Nonlinear Dynamics] , 13(3):317–329, 2017.[43] I.A. Shepelev and T.E. Vadivasova. Variety of spatio-temporal regimes in a2d lattice of coupled bistable fitzhugh-nagumo oscillators. formation mech-anisms of spiral and double-well chimeras.
Communications in NonlinearScience and Numerical Simulation , 79:104925, 2019.[44] E. Teichmann and M. Rosenblum. Solitary states and partial synchronyin oscillatory ensembles with attractive and repulsive interactions.
Chaos:An Interdisciplinary Journal of Nonlinear Science , 29(9):093124, 2019.[45] L.S. Tsimring, N. F. Rulkov, M. L. Larsen, and M. Gabbay. Repulsivesynchronization in an array of phase oscillators.
Phys. Rev. Lett. , 95:014101,2005. 2146] E. Ullner, A. Zaikin, E.I. Volkov, and J.G. Ojalvo. Multistability andclustering in a population of synthetic genetic oscillators via phase-repulsivecell-to-cell communication.
Phys. Rev. Lett. , 99:148103, 2007.[47] M.A. Volosyuk, A.V. Volosyuk, and N.Y. Rokhmanov. The role of inter-stitial (crowdion) mass-transfer for crack high-temperature healing underuniaxial loading.
Functional materials , 2015.[48] Q. Wang, G. Chen, and M. Perc. Synchronous bursts on scale-free neuronalnetworks with attractive and repulsive coupling.
PLoS ONE , 6(1), 2001.[49] F. Xu, J. Zhang, M. Jin, Sh. Huang, and T. Fang. Chimera states and syn-chronization behavior in multilayer memristive neural networks.
NonlinearDynamics , 94(2):775–783, 2018.[50] H. Yamamoto, S. Kubota, F.A. Shimizu, A. Hirano-Iwata, and M. Niwano.Effective subnetwork topology for synchronizing interconnected networksof coupled phase oscillators.
Frontiers in Computational Neuroscience , 12,2018.[51] T. Yanagita, T. Ichinomiya, and Y. Oyama. Pair of excitable fitzhugh-nagumo elements: Synchronization, multistability.
Phys. Rev. E. ,72:056218, 2005.[52] M.K.S. Yeung and S.H. Strogatz. Time delay in the kuramoto model ofcoupled oscillators.
Phys. Rev.Lett. , 82:648–651, 1999.[53] Y. Yuan, T. Solis-Escalante, M. van de Ruit, F.C.T. van der Helm, and S.C.Alfred. Nonlinear coupling between cortical oscillations and muscle activityduring isotonic wrist flexion.
Frontiers in Computational Neuroscience , 10,2016.[54] A. Zakharova.
Chimera Patterns in Networks: Interplay between Dynamics,Structure, Noise, and Delay . Springer International Publishing, 2020.[55] A. Zakharova, M. Kapeller, and E. Sch¨oll. Chimera death: Symmetrybreaking in dynamical networks.
Physical review letters , 112(15):154101,2014.[56] N. Zhao, Z. Suna, and W. Xu. Amplitude death induced by mixed attractiveand repulsive coupling in the relay system.