Quantifying the transition from spiral waves to spiral wave chimeras in a lattice of self-sustained oscillators
aa r X i v : . [ n li n . AO ] J u l Quantifying the transition from spiral waves tospiral wave chimeras in a lattice of self-sustainedoscillators
I.A. Shepelev ∗ , A.V. Bukh ∗ , S.S. Muni † , V.S. Anishchenko ∗ July 21, 2020
Abstract
The present work is devoted to the detailed quantifying the transitionfrom spiral waves to spiral wave chimeras in a network of self-sustainedoscillators with two-dimensional geometry. The basic elements of the net-works are the van der Pol oscillator and the FitzHugh-Nagumo neuron.The both models are in the regime of relaxation oscillations. We analyzethe regime with using the indexes of local sensitivity which enables us toevaluate the sensitivity of each individual oscillator at finite time. Spi-ral waves are observed in the both lattices when the interaction betweenelements have the local character. The dynamics of all the elements isregular. There are no any expressed high-sensitive regions. We havediscovered that when the coupling becomes nonlocal the features of thesystems are significantly changed. The oscillation regime of the spiralwave center element switches to chaotic one. Beside this, a region withthe high sensitivity occurs around this oscillator. Moreover, we show thatthe latter expands in space with elongation of the coupling range. As aresult, an incoherence cluster of the spiral wave chimera is formed exactlywithin this high-sensitive area. Forming this cluster is accompanied by thesharp increase in values of the maximal Lyapunov exponent to the posi-tive region. Furthermore, we explore that the system can even switch tohyperchaotic regime, when several Lyapunov exponents becomes positive. ∗ Department of Physics, Saratov State University, 83 Astrakhanskaya Street, Saratov,410012, Russia † School of Fundamental Sciences, Massey University, Palmerston North, New ZealandKeywords: Spatiotemporal patterns, chimera, van der Pol oscillator, FitzHugh-nagumoneuron, spiral waves, spiral wave chimera, Nonlocal Interactions, Lyapunov exponentE-mail addresses: I.A. Shepelev ( [email protected] ), A.V. Bukh ( [email protected] ),S.S. Muni( [email protected] ), V.S. Anishchenko ( [email protected] ) ntroduction Investigation of auto-wave processes in nonlinear complex systems and theirmathematical models is one of the highly developed direction in nonlinearphysics and related fields. The auto-wave regime is the self-sustained waveregime in a non-equilibrium system which weakly depends on initial and bound-ary conditions. These systems can be described by a system of ordinary dif-ferential equations with diffusive coupling and an active nonlinearity [18, 53,13, 17, 50, 5, 4, 11]. Intensive studies of complex spatiotemporal behavior insuch models has begun from the 50s of XX century and continues to this day[21, 59, 11, 27, 22, 47, 33, 55, 25, 52, 49]. The interest to the study of au-towave structures in different nonlinear complex systems sharply increased inthe early 2000s. This has been mainly caused by the discovery of so-called”chimera states” which are hybrid states that emerge spontaneously, combiningboth coherent and incoherent parts. The first chimera state has been found ina ring of non-locally coupled identical phase oscillators by Kuramoto and Bat-togtokh in [20] in 2002. The term ”chimera state” has been proposed in [1] in2004 by Abrams and Strogatz. The non-locality of coupling means that eachindividual oscillator is symmetrically coupled with all adjacent oscillators fromthe neighborhood with a certain radius [46]. Later these states have been foundin diverse oscillatory models with nonlocal interaction such as the FitzHugh-Nagumo, the van der Pol, the Stuart-Landau, the Hindmarsh-Rose and others[48, 33, 28, 38, 41, 16, 31, 26, 57, 42]. However, the chimera structures havebeen discovered in systems with purely local coupling [23, 10, 47, 25, 44] andwith global one [56, 41]. The chimeras have been found in real experiments[14, 48, 51]. The potential relevance of chimera states also includes the phe-nomenon of unihemispheric sleep which is observed in birds and dolphins [36].They sleep with only one eye open, meaning that half of the brain is synchronouswith the other half being asynchronous. Furthermore, it is hypothesized thatchimera states are the route of both onset and termination of epileptic seizures[30, 39, 2]. Chimera can be realized even in the common brain dynamics as ithas been shown in the recent work [3].The investigation on the lattice dynamics of coupled oscillators has revealeda special type of spiral wave, namely spiral wave chimeras (SWC), first found nu-merically in two-dimensional nonlocally coupled systems [21, 45] and confirmedanalytically for the 2D system of nonlocally coupled phase oscillators [27, 24].The spiral wave chimera has been observed experimentally in [49] in catalyst-freeBelousovвҐҮZhabotinsky reaction. SWC has been discovered in different 2Dsystems [32, 47, 12, 40, 7, 43]. Usually the elongation of the coupling range leadsto the transformation of the common spiral wave to the spiral wave chimera, aswell as in the other models. However, despite the extensive numerical evidenceof SWC, their confirmation in many dynamical systems and the experimentalobservations, the processes that occur during the formation of SWC remainselusive. In our previous work [43], we had offered a hypothesis that the reasonbehind the formation of SWC in the lattice of bistable FHN oscillators is instrong increase of the sensitivity of a region around the wave center when the2oupling range r is elongated. We try to show that this phenomenon is commonin the oscillatory models, where the SWC are observed, namely lattices of thevan der Pol oscillators and of the FitzHugh-Nagumo oscillators. The interactionbetween oscillators has linear nonlocal character. Besides, we show the similarresults for a lattice of discrete-time elements (Nekorkin maps). In this section, we summarize two models considered in this article, namelyвҐҰ the van der Pol (vdP) and FitzHugh-Nagumo (FHN) oscillators. We alsointroduce the respective coupling schemes describing the two-dimensional layoutwith nonlocal interactions.
The dynamics of a single van der Pol oscillator is described by the followingsystem of ODEs: dxdt = y,dydt = µ (1 − x ) y − ω x, (1)where x and y are dynamical variables. The parameter µ determines the nonlin-earity degree, while the parameter ω is responsible for the oscillator frequency.The value of µ = 0 corresponds to the supercritical Andronov-Hopf bifurcationwhich is results in the birth of limit cycle for µ > . The Values of control pa-rameters are fixed as µ = 2 . and ω = 2 . in this study. These values correspondto the relaxation regime of self-oscillations.We consider the model of a spatially organized ensemble of oscillators whichis a 2D regular N × N lattice with an edge N = 100 and consists of nonlocallycoupled vdP oscillators (1). The interaction between elements is introducedlinearly into x -variables. This model is described by the following system ofnetwork equations: dx i,j dt = y i,j + σQ i,j P k,p ( x k,p − x i,j ) ,dy i,j dt = µ (1 − x i,j ) y i,j − ω x i,j ,i, j = 1 , ...N, (2)The double index of the dynamic variables x i,j and y i,j with i, j = 1 , ..., N determines the position of an element in the two-dimensional lattice. All theoscillators are identical in parameters and each of them is coupled with all thelattice elements from a square with side (1 + 2 P ) in the center of which thiselement is located. The integer P defines the nonlocality of coupling and iscalled the interaction interval. The case of P = 1 corresponds to the localcoupling, while P = N/ is the case of global coupling, when each element3nteracts with the whole system. The number of coupled units with the i, j thelement is determined by the parameter Q i,j , which is given by a number of k, p combinations with the following relations: ( max(1 , i − P ) k, p min( N, i + P ) , max(1 , j − P ) k, p min( N, j + P ) , (3)which represent zero flux boundary conditions (Neumann type) in the nonlocalcase [15]. We also use the notion of the coupling range r = P/N in analogywith the classical works on chimeras. The coupling strength is determined bythe value of σ , which is fixed in this study as σ = 0 . . The FHN oscillator is one of the simplest model which describes the neurondynamics. The behavior of a single oscillator is described by following systemof differential equations: ε dudt = u − αu − w,dwdt = γv − w + β, (4)where u and w denote a fast activator and a slow inhibitor variable, respectively.The parameter ε determines the time-scale separation and is fixed in this studyat ε = 0 . . The threshold parameters β and γ determines the oscillatory, ex-citable, or bistable behavior in the system, i.e., one (self-sustained or excitableregime) or three (bistable regime) times it intersecs the nullclines. We fix thevalue of β = 0 . throughout this study just to avoid the full symmetry of thesystem. Hence, when γ < . the regime is bistable with two stable foci anda saddle between them (bistable regime). The value of γ ≈ . corresponds tothe subcritical Andronov-Hopf bifurcation. When γ exceeds this value, a stablelimit cycle appears in the system (4) (self-sustained regime). We set the valueof γ according to the self-sustained regime as γ = 0 . .Similar to the case of the vdP lattice (2), we consider the FHN lattice dy-namics on a two-dimensional regular N × N lattice with N = 100 nodes andzero flux boundary conditions. The FHN coupled lattice dynamics is describedas follows: ε du i,j dt = u i,j − αu i,j − w i,j + εσQ P k,p ( u k,p − u i,j ) ,dw i,j dt = γu i,j − w i,j + β,i, j = 1 , ...N, (5)The coupling is linear and is introduced in the first dynamical variable u . Allthe oscillators are identical in parameters and each of them is coupled with thecoupling strength σ with all the lattice elements from a square with side (1+2 P )
4n the center of which this element is located, where P is the coupling interval.Thus, each element is coupled with Q neighbors, which are given by numbersof k, p combinations by formula (3) for the zero flux boundary conditions. Thecoupling strength σ is also fixed in this study and is equal to σ = 0 . .As initial conditions, we use the instantaneous state of the systems in theregime of a spiral wave for the case of local coupling, which has been obtained fora multitude of random initial values of the variables with a uniform distributionwithin x ∈ [ − , , y ∈ [ − , . When the coupling range r increases, we usethe instantaneous states of a corresponding model at the previous step by r asthe initial states for the following value of r . The system equations are integratedusing the Runge-Kutta 4th order method with time step dt = 0 . . All theregimes under study are obtained after the transient process of t trans = 10000 time units. One of the typical regimes realized in (2) for the zero flux BC is a spiral wave[29, 34, 58]. It’s feature is a rotation of the wave front in a spiral fashionaround a certain center in space. It is known that the spiral waves are typicalfor many systems of different nature. They play an important role in livingsystems [8, 9, 6, 35]. When the coupling is local ( P = 1 ) it is possible to observea regime with a different number of spiral waves. The wavelength of thesewaves is short enough. However, the number of waves can be equal to one too.This case is presented in fig.1(a). The wavefront propagates from the centerin a spiral fashion. Fig.1(b) demonstrates phase portrait projections for thewave center (WC) element and the other elements. It is clearly seen that theirdynamics are noticeably different. All the lattice elements except the elementof the wave center oscillate periodically, while the WC element oscillates quasi-harmonically, as indicated by the zero value of the maximal Lyapunov exponent( λ = 0 . ). We calculate the Rice frequency f Ri,j [37] for individual i, j -oscillators. It is calculated as following: f Ri,j = M i,j T , (6)The Rice frequency may be considered as the frequency ( f ) for harmonic os-cillations and the mean frequency ω for the chaotic or quasi-harmonic ones. Aspatial distribution of the mean frequency is represented in fig.1(c). All the os-cillators except the wave center element oscillate with the same frequency. TheSW oscillator is characterized by the frequency which is slightly higher than forthe rest part of the lattice.For the quantitative analysis of the dynamical regimes we plot a spatialdistribution of the indixes of local sensitivity (ILS) Λ i,j ( T ) , the method of cal-5 j i - - x i , j (a) j i . h f i i , j . (c) − . − . . − − y x (b) j i - - Λ i , j · [ − ] (d)Figure 1: (Color online) Spiral wave in (2) for the local coupling r = 0 . .(a)is a snapshot of the system state, (b) phase portrait projections for oscilla-tors with indexes i = 51 , j = 52 (wave center oscillator, green line) and i = 40 , j = 40 (synchronous region, black line), (c) spatial distribution of themean frequency f i,j , (d) spatial distribution of the indexes of local sensitivity(LIS) Λ i,j . Parameters: σ = 0 . , µ = 2 . , ω = 2 . , N = 100 culation of which has been described in [44]. The ILS evaluates the sensitivityof individual elements to weak perturbations. The ILS can be calculated foreach i th oscillator by the following formula: Λ i,j ( T ) := 1 T ln k ξ i,j ( T ) kk ξ , (0) k = 1 T ln N k ξ i,j ( T ) kk ξ (0) k , (7)where ξ (0) is an initial vector of the perturbation of the whole system, while ξ i,j ( T ) describes the local evolution of a perturbation of the i, j th oscillator( i, j th component of the perturbation). This characteristic enables us to eval-uate the sensitivity of each oscillator of the lattice and to highlight the mostunstable spatial regions. The spatial distribution of the ILS for the spiral waveunder study is presented in fig.1(d). This plot shows that there are no re-gions with noticeably high sensitivity. High-sensitive regions alternate withlow-sensitive ones. 6 .2 Van der Pol model: Spiral wave chimeras Now we consider the evolution of the spiral wave regime when the couplingrange r increases. At first, we study a wave regime when the value of r = 0 . (small non-locality). An example of the spiral wave for this value of r is shownin fig.2(a). The wavelength becomes noticeably longer than that of the previous j i - - x i , j (a) j i . h f i i , j . (c) − . − . . − − y x (b) j i . Λ i , j · [ − ] . (d)Figure 2: (Color online) Spiral wave in (2) for the nonlocal coupling r = 0 . .(a) is a snapshot of the system state, (b) phase portrait projections for oscillatorswith indexes i = 55 , j = 55 (wave center oscillator, green line) and i = 40 , j =40 (synchronous region, black line), (c) spatial distribution of the mean frequency f i,j , (d) spatial distribution of the LIS Λ i,j . Parameters: σ = 0 . , µ = 2 . , ω = 2 . , N = 100 case. In the previous case, the transformation to the spiral wave chimera stilldid not occur and an incoherent core is absent. However, the quantitativeanalysis of the regime shows that the dynamics has significantly changed. Atfirst, the value of the maximal Lyapunov exponent becomes positive ( λ =0 . ). It means that the chaotic oscillations appear in the system (2).Fig.2(b) shows the phase portrait projections of the wave center element andfor the elements outside the wave center. The first phase portrait projectioncorresponds to a chaotic attractor, and the second one demonstrates the limit7ycle of period-1. Thus, oscillations become chaotic only for the elements aroundthe wave center, while oscillations of the other elements remain periodic. As weshow below, this behavior is typical for the spiral wave chimera too. A spatialdistribution of the mean frequency presented in fig.2(c) illustrates that now agroup of oscillators with increased values of the frequency h f i forms aroundthe wave center. Moreover, this group already has the bell-like distribution of h f i (i.e. the maximal values of the frequency has the oscillator in the groupcenter) typical for the SWCs. The oscillators outside this group have the samefrequency.A spatial distribution of the ILS in the lattice (2) is shown in fig.2(d). Itcan be concluded from the plot that significant changes occur in the systemwith an increase in the coupling nonlocality. In the previous case of r = 0 . ,the oscillators in the wave centers are characterized by the similar values of Λ i,j , the ones outside the center, i.e. they have been low-sensitive and have notbeen distinguished from the other oscillators of the wavefront. When r = 0 . ,oscillators of the wave center become the most sensitive elements of the wholelattice and are characterized by the maximal values of the ILS. Consequently,this part of the system is most likely to develop instability and incoherence.We assume that hypersensitivity of the center of a spiral chimera is one ofthe main reason of the formation of the incoherence core around the center withincreasing in the number of the neighbors coupled with the element. To confirmor refute this hypothesis, we study the evolution of the system for the longercoupling range. At first, we study the case of r = 0 . . The stable wave regimefor this value of r is represented by a snapshot of the system state in fig.3(a).It is clearly seen that an incoherence core is formed around the wave center,i.e.formation of the spiral wave chimera takes place now. The features of thisregime is similar to that of the previous case with ( λ = 0 . ). Oscilla-tions corresponding to the incoherence core elements have a chaotic characterand correspond to a chaotic attractor, while oscillators outside this core demon-strate the regular dynamics and are characterized by the stable limit cycle.These differences are well visible in projections of the phase portrait of elementsof the clusters with the coherent and incoherent behavior, which are shown infig.3(b). Since the oscillator dynamics of the spiral core is chaotic, a value ofthe maximal Lyapunov exponent for the whole lattice for this regime is equal to λ = 0 . . It is known that the spiral wave chimera has similar features asin the phase chimera in the ensemble of Kuramoto oscillators, namely, it has thecharacteristic maximum of the mean frequency distribution in an center of theincoherence core [12, 49]. The similar distribution takes place in our case (seefig.3(c)). The frequency of the oscillators in the incoherence core smoothly de-creases as one moves away from the core center in any radial direction. Fig.3(d)shows a spatial distribution of the ILS Λ i,j . This confirms that the most sensi-tive region in the lattice (2) is the core of a spiral wave as well as in the caseabove. The incoherence core is formed exactly within this spatial region. Themost sensitive oscillators are located close to the core boundary, while oscillatorsinside the incoherence core are characterized by small values of the ILS. Indeed,the oscillations inside the core are noticeably less chaotic than ones in the core8 j i - - x i , j (a) j i . h f i i , j . (c) − . − . . − − y x (b) j i . Λ i , j · [ − ] . (d)Figure 3: (Color online) Spiral wave chimera in (2) for the nonlocal coupling r = 0 . . (a) is a snapshot of the system state, (b) phase portrait projections foroscillators with indexes i = 44 , j = 58 (oscillator of the incoherence core, greenline) and i = 40 , j = 40 (synchronous region, black line), (c) spatial distributionof the mean frequency f i,j , (d) spatial distribution of the LIS Λ i,j . Parameters: σ = 0 . , µ = 2 . , ω = 2 . , N = 100 boundary. These results enable us to assume that the reason behind the occur-rence of the incoherence core around the spiral wave center is associated with anincrease in the sensitivity of oscillators around this center with the elongationof the coupling range r . Hence, these elements become more and more unstableand this leads to chaotization of the oscillation process inside the high-sensitiveregion and as a result formation of the incoherence cluster takes place.Further elongation of the coupling range leads to an increase a number ofoscillators forming the incoherence core and accordingly to expansion of thiscore in space. An example of the spiral wave chimera for the long couplingrange r = 0 . is shown in a snapshot of the system state in fig.4(a). It shouldbe noted that the spatial configuration of the incoherence core has noticeablychanged in comparison with the case of the shorter r . Now, the core of theSW chimera has an interesting structure, namely alternating concentric layersof coherent and incoherent oscillators, namely one layer has the incoherent dis-9ribution of the instantaneous states, and the following layer is characterizedby the synchronous behavior of oscillators. A spatial distribution of the mean j i - - x i , j (a) j i - . - Λ i , j · [ − ] . (b)Figure 4: (Color online) Spiral wave chimera in (2) for the nonlocal coupling r = 0 . . (a) is a snapshot of the system state, (b) spatial distribution of the LIS Λ i,j . Parameters: σ = 0 . , µ = 2 . , ω = 2 . , N = 100 frequency is similar to the previous case and has typical shape for the spiralwave chimeras. Elements of the incoherence core remain the most sensitive inthe system and have the maximal values of the ILS, what is shown in fig.4(b).Moreover, a spatial structure of the ILS distribution for the incoherence core isalso has the ring-like shape, i.e. the ring layers with with high values of the ILS(incoherence layer) alternates with the low-sensitive layers (synchronous layer).If we continue to increase a value of the coupling range ( r > . ) thenthe model (2) switch to the regime of full synchronization when all the latticeelements oscillate periodically with the same instantaneous phases and ampli-tudes. The frequency of all the oscillators is also the same. Hence, a valueof the maximal Lyapunov exponent becomes equal to zero ( λ = 0 . for r = 0 . ). Spiral waves in the network of the FHN oscillators regime has been found in[40, 12, 54]. The dynamical regime of the individual oscillators is self-sustainedin the system under study (5). However, the interaction between the oscillatorsis introduced through a special rotational matrix in all of the above mentionedexamples. This type of coupling significantly changes the dynamics of the indi-vidual elements of the network. In the system under study, the coupling formbetween the elements is introduced in the first state variable ( u i,j variable, seethe eq. (5)). To the best of our knowledge, spiral wave chimeras has not beenfound in this type of coupling before. Our investigation confirms that formationof this type of chimera takes place in the model (5), when the coupling strength σ is sufficiently low. Note that the SWC’s are observed in the FHN lattice for10ignificantly lower coupling strength ( σ ≈ . for (5) while they are observedin the VdP lattice for comparatively large coupling strength of σ = 0 . for (2)).The system is characterized by the high-degree multistability when the cou-pling is local ( r = 0 . or P = 1 ), namely states with a different number ofspiral waves can be realized from various randomly distributed initial condi-tions. We chose the case when only one spiral wave is set in the lattice. Thisstate is illustrated by a snapshot of the system state in fig.5(a). Oscillations of j i - - x i , j (a) − − . . − − v u (b) j i - - Λ i , j · [ − ] (c)Figure 5: (Color online) Spiral wave in (5) for the local coupling r = 0 . .(a) is a snapshot of the system state, (b) phase portrait projections for oscil-lators with indexes i = 41 , j = 32 (wave center oscillator, green line) and i = 40 , j = 40 (synchronous region, black line), (c) spatial distribution of theLIS Λ i,j . Parameters: σ = 0 . , ε = 0 . , γ = 0 . , β = 0 . , N = 100 all the oscillators including the wave center oscillator have the regular charac-ter. A zero value of the maximal Lyapunov exponent ( λ = 0 . ) confirmsthis. The corresponding phase portrait projections are presented in fig.5(b). Atfirst sight, these attractors correspond to purely periodic oscillations and arethe same. However, an enlarged fragment of the trajectories shows that the os-cillations poorly express quasi-harmonic character. Furthermore, oscillations inthe wave center corresponds to the other attractor than in the other part of thelattice, but very similar. A spatial distribution of the ILS Λ i,j is demonstratedin fig.5(c). It shows that there is no spatial region with high sensitivity as wellas for the spiral chimera in the model (2). At that instant, the wave centeroscillator is the most sensitive element in the whole system. The frequency ofthis element is also slightly higher than for the other system elements. Let us now explore the evolution of the system with an increase in the couplingrange values. Growth of the coupling range lead to elongation of the wavelength.The spiral wave realized for the case of r = 0 . is illustrated in fig.6(a). Theslight incoherence already occurs around the wave center, but the incoherencecluster has not formed yet. Oscillations of the elements which are outside thewave center becomes fully periodic, while oscillations within this center appar-ently seem to become weakly chaotic, as evidenced by a low but positive value11f the maximal Lyapunov exponent λ = 0 . . Fig.6(b) illustrates the cor-responding phase portrait projections. A spatial distribution of Λ i,j presented j i - - u i , j (a) − − . . − − v u (b) j i - - . Λ i , j · [ − ] . (c)Figure 6: (Color online) Spiral wave in (5) for the nonlocal coupling r = 0 . .(a) is a snapshot of the system state, (b) phase portrait projections for oscil-lators with indexes i = 43 , j = 31 (wave center oscillator, green line) and i = 30 , j = 20 (synchronous region, black line), (c) spatial distribution of theLIS Λ i,j . Parameters: σ = 0 . , ε = 0 . , γ = 0 . , β = 0 . , N = 100 in fig.6(c) shows that now there is a spatial region around the wave center withthe highest values of the ILS. Oscillators close to the wave center becomes mostsensitive in the system (5), as well as for the lattice of vdP oscillators (2).Next we study the wave regime when the coupling range r is extended upto r = 0 . . The spiral wave completely transforms to the spiral wave chimerawith the presence of an incoherence core. This regime is shown in fig.7(a).Oscillations in the incoherence core remain weakly chaotic, which is seen fromthe phase portrait projection in fig.7(b). The chaos is so weak that the phaseportrait projection seems to correspond to a limit cycle. However, a value ofthe maximal exponent remains positive ( λ = 0 . ). Therefore, oscillationsin the lattice should be chaotic. Indeed, if we observe an enlarged fragmentof the phase portrait projection then it is possible to see that oscillations inthe incoherence core corresponds to a chaotic attractor. Oscillations in thecoherence region are also either quasi-periodic or slightly chaotic, but theircorresponding phase portrait projection is significantly more narrow. Fig.7(c)illustrates that the most sensitive oscillators of the system (5) are located inthe incoherence core as well as in the previous case. They are characterizedby the maximal values of the ILS ( Λ i,j ). It is means that the behavior in thisregion is less regular and more unstable. Moreover, the oscillators close to theincoherence core boundary are more sensitive than ones in the core center. Thesimilar behavior has been observed for the lattice of vdP oscillators (2) above.Further increase of the coupling range leads to an extension of the incoher-ence core and an elongation of the wavelength. An example of the spiral wavechimera in the case of a long coupling range r = 0 . is shown in snapshot ofthe system state in fig.8(a). Unlike the SWC in the lattice (2), the incoher-ence core in the present case has a more common spatial structure, namely theincoherence is observed within the whole incoherence cluster. A value of themaximal Lyapunov exponent becomes noticeably lower than for shorter r and is12 j i - - u i , j (a) − − . . − − v u (b) j i - Λ i , j · [ − ] (c)Figure 7: (Color online) Spiral wave chimera in (5) for the nonlocal coupling r = 0 . . (a) is a snapshot of the system state, (b) phase portrait projections foroscillators with indexes i = 54 , j = 54 (wave center oscillator, green line) and i = 77 , j = 8 (synchronous region, black line), (c) spatial distribution of the LIS Λ i,j . Parameters: σ = 0 . , ε = 0 . , γ = 0 . , β = 0 . , N = 100 j i - - u i , j (a) j i - - Λ i , j · [ − ] (b)Figure 8: (Color online) Spiral wave chimera in (5) for the nonlocal coupling r = 0 . . (a) is a snapshot of the system state, (b) spatial distribution of theLIS Λ i,j . Parameters: σ = 0 . , ε = 0 . , γ = 0 . , β = 0 . , N = 100 equal to λ = 0 . , hence the oscillations in the incoherence core are veryweakly chaotic. A spatial distribution of the indexes of local sensitivity Λ i,j isshown in fig.8(b). The elements with characterizing by the maximal values of Λ i,j are located in the incoherence cluster boundary, while oscillators inside thecluster are less sensitive. The same character of the ILS distribution is observedfor the shorter coupling ranges. A spatial distribution of the frequency in theincoherence core is also have a typical bell-like shape. Comparing the evolution of the spiral wave regime in the lattice of van derPol oscillators (2) and that of the FHN oscillators (5), we identify a similarscenario, which is observed for both systems. When the interaction between13lements has the local character, the wave center sensitivity does not differfrom the rest part of the system. All the oscillators demonstrate the regularbehavior. However, growth of the coupling range values leads to a sharp increasein the sensitivity of a spatial region around the wave center. Oscillations insidethe center become chaotic, while they remain regular outside the center core.Furthermore, this region extends in space. As a result, the incoherence appearsinside this region and incoherence cluster forms as the wave core, when thecoupling range r becomes sufficiently long. This core extends with growth of r up to sufficiently large values of r ≈ . . When the value of the couplingrange exceeds this threshold level, both models switch to the regime of completesynchronization, when all the elements oscillate periodically as a whole with thesame instantaneous amplitudes and phases.At that instant, the dynamics of the two models under study is not thesame. The lattice of vdP oscillators demonstrates more complex dynamics. Os-cillations in the incoherence cluster are characterized by more expressed chaoticbehavior than for the case of the FHN model, where oscillations is almost reg-ular. Apparently, this is the reason why the incoherence core in the model (2)has a complex spatial structure for the long coupling ranges. It is interesting to compare the evolution of the Lyapunov spectrum for bothsystems under study when the number of coupled neighbors increases. Are thesystems in a state of the hyperchaos (i.e. has more than one positive Lyapunovexponents (LE) λ k )? To answer these questions, we calculate the evolution ofthe first three Lyapunov exponents for both of the lattice (2) and (5), whenvalues of the coupling range r increase. At first we explore the dependence forthe vdP model (2) which is presented in fig.9(a). When the coupling is local,the system demonstrates the regular behavior with the zero maximal LE andnegative for others. A sharp switch of the maximal LE to positive value region isobserved already for r = 0 . , but for others LE’s remain negative. However, thesystem switches to the hyperchaotic regime beginning with a value of r ≥ . (the second LE becomes positive while the third LE remains negative). Themaximal LE reaches the maximum value when the coupling range becomesequal to r = 0 . and decreases for longer r . A sharp increase in the sensitivityof a region around the wave center is observed exactly for this value of couplingrange (the oscillator dynamics around the wave center becomes strongly chaotic,while the remaining part of the system remains to oscillate regularly). The thirdLE exceeds a zero value for r = 0 . . The special behavior of the model (2)is observed when the coupling range approaches to the value r = 0 . . Thevalues of all the LE’s decreases. At that instant, the size of the incoherencecluster shrinks, and a region with a smooth spatial profile occurs instead ofthe incoherence. At r = 0 . , all the three LE’s reach the minimal valueswithin r ∈ [0 .
1; 0 . , namely λ = 0 . and the other LE’s become negative.The spatiotemporal structure undergoes significant change for this value of r .14 -13-11-9-113 0.04 0.08 0.12 0.16 0.2 0.24 0.28 0.32 r λ P [ − ] λ λ λ (a) r λ P [ − ] λ λ λ (b)Figure 9: (Color inline) Dependences of the first three Lyapunov exponents λ , λ and λ on the coupling range r for the systems (2) (a) and (5) (b). Parametersof (2): σ = 0 . , µ = 2 . , ω = 2 . , N = 100 ; parameters of (5): σ = 0 . , ε = 0 . , γ = 0 . , β = 0 . , N = 100 The incoherence cluster completely disappears and a spatial region with thesynchronous dynamics begins to form instead of the incoherence core. Furtherincrease in the the coupling range leads to an occurrence of the incoherenceagain and growth of values of all the three first LE’s takes place. However,when values of r exceed . , the regime of spiral wave chimeras are destroyedand the system switches to the complete synchronization regime with purelyregular dynamics of the oscillators. At that we observe a sharp decline of the2nd and 3rd LE’s to negative values while the maximal LE λ becomes equalto zero within the calculation accuracy.The dynamics of the second model (5) is simpler in comparison with thelattice (2). The evolution of λ , , is exemplified in fig.9(b). When the couplingis local ( r = 0 . ), the maximal LE λ is equal to zero and the other LE’s are15egative. But λ becomes positive for r = 0 . and reaches the maximal value λ = 0 . , when the coupling range is equal to r = 0 . . Values of λ , aremaximal for this r . The spatiotemporal incoherence also appears in the FHNmodel within the high-sensitive area around the wave center, which forms whenthe coupling range is elongated. It should be noted that at the maximal valueof λ is significantly less than that of the van der Pol model. Moreover, thelattice (5) is never switched to the hyperchaotic regime, because values of λ , always remain negative. The second LE reaches a zero value at r = 0 . , butlatter does not lead to any qualitative changes of the SWC. The FHN modelswitches to the regime of complete synchronization (as well as the vdP model)when r exceeds the value r > . . The maximal LE becomes equal to 0 whilethe values of λ , sharply decreases. These results approve that the dynamics ofthe lattice (5) in the spiral wave chimera regime is characterized by a noticeableless chaotic dynamics than in the vdP model, which demonstrates the morecomplex behavior. Hence, the dependency of the maximal LE on the couplingrange has a similar character for both models. Hence, it enables us to concludethat formation mechanisms of SW chimera has the same nature.Thus, evolution of the maximal Lyapunov exponent and of the ILS distri-bution enables us to simply diagnose the transformation of the spiral wave tothe spiral wave chimera. Elongation of the coupling interval leads to a lossof the stability of a spiral wave core (center) (ILS values are maximal for thisspatial region). As a result, the type of oscillator dynamics within the wavecenter changes from regular to chaotic (or even hyperchaotic) one. At that in-stant, other oscillators of the lattice continue to oscillate almost periodically. Inconsequence of these processes, the incoherence cluster forms around the spiralwave core. Switching of the models to the synchronous regime is accompaniedby sharp decline of the second and third LE’s. Conclusions
We have studied the evolution of the auto-wave regimes by increasing the non-locality degree of two different networks connected in two-dimensional geometrywith the free flux boundary conditions. The first network consists of the van derPol oscillators and the second one is of the FitzHugh-Nagumo (FHN) oscillatorswith self-sustained dynamics. The main auto-wave regime for these systems isa spiral wave, which transforms into a spiral wave chimera with the growth ofthe nonlocality degree. The features of the spiral wave chimeras under studyare similar to the ones in the other models [7, 21, 47, 25, 45, 19]. For example,there is no characteristic bell-like spatial distribution of the mean oscillationfrequency. This enables us to assume that the formation of incoherence corearound the spiral wave center has the same features. Thus, the aim of thecurrent study is concentrated on search and explanation of some common quan-titative peculiarities of the formation of the spiral wave chimera from the spiralwave.The spiral waves appear in both models under study under the local coupling16nd also exists for low non-locality degree. However, in the van der Pol model,they are observed for a significantly higher coupling strength ( σ ≈ . ) thanthat of the FHN model ( σ ≈ . ). Our comparative study has demonstratedthat, eventhough the dynamics of a single oscillator in both of the models aredescribed by different dynamical equations, both systems demonstrate a similarbehavior when the non-locality degree increases. The dynamics of all the indi-vidual oscillators of both systems is regular (periodic or quasi-periodic), whichis confirmed by a zero value of the maximal Lyapunov exponent (LE). But evena small elongation of the coupling range (non-locality degree) leads to switchingof the oscillations of the spiral wave center from the regular to chaotic, while thedynamics of the remaining part of the system remains regular. Introducing non-local coupling induces a change in the lattice. We have calculated the indicesof local sensitivity (ILS) described in [44] for the quantitative analysis of thespatiotemporal auto-wave structures. The spatial distribution of ILS enables usto evaluate the most and least sensitive spatial regions in the structures. Whenthe coupling is local, the sensitivity is similar for the whole system. The in-troduction of the nonlocal coupling leads to the formation of a highly sensitivespatial region around the spiral wave center. Moreover, this region extends withthe growth of the nonlocal coupling. Values of the maximal LE also increaseup to a certain threshold value of the coupling range (sufficiently short), whenit reaches the maximum value. Chaotic oscillations are observed only close tothe wave center. This means that these oscillations are most chaotic for thethreshold level of the coupling range, but there is not yet any incoherence inthe lattices. Apparently, for this reason, oscillators stop to demonstrate thesynchronous behavior, when values of the coupling range exceed this level, andthe incoherence core begins to form around the wave center. The sensitivity ofthis core remains maximal in the lattice. The size of the core extends with theelongation of the coupling range. It should be noted that the spatiotemporaldynamics of the lattice of van der Pol oscillators are more complex than thatof the FHN model. The formation of the incoherence core is accompanied byswitching of the system to hyperchaotic regime, where several of the Lyapunovexponents are positive, while the regime with only one positive LE is observedin the second system under study. Indeed, the chaotic behavior of the oscillatorsin the incoherence core are weakly expressed by this model.Thus, we show that the features of formation of the incoherence core inthe lattice of oscillators under study are following: Introducing the non-localityinto the coupling leads to the formation of an instability region with increasingsensitivity around the spiral wave center. When the non-locality exceeds acertain threshold level, the elements near the centre of the wave become sounstable that they cease to oscillate synchronously with each other and formthe incoherence core. The size of the highly-sensitive spatial region with chaoticdynamics increases with the elongation of the coupling range. The incoherencecore accordingly expands too. A similar behavior has been discovered in thelattice of FHN oscillators in the bistable regime in [43] and for the lattice ofdiscrete-time oscillator in the appendix.17 Appendix
We also study the evolution of a spiral wave when the coupling range increasesfor the lattice of discrete-time oscillators (maps). A basic element of this latticeis the Nekorkin map described as following: x t +1 = x t + F ( x t ) − y t − βH ( x t − d ) ,y t +1 = y t + ε ( x t − J ) , (8)where x t is a variable that describes the dynamics of the membrane potentialof the nerve cell, y t is a variable that relates to the cumulative effect of allion currents across the membrane, functions F ( x t ) and H ( x t − d ) are given asfollows: F ( x t ) = x t ( x t − a )(1 − x t ) , < a < ,H ( x t ) = ( , x t > , , elsewhere . (9)The parameter ε > determines the characteristic time scale of y t , the parame-ter J controls the level of the membrane depolarization ( J < d ) , the parameters β > and d > determine the excitation threshold of bursting oscillations, t = 1 , , . . . represents the discrete time.A N × N
2D lattice of the nonlocally coupled Nekorkin maps is describedby the following system of equations: x t +1 i,j = x ti,j + F ( x ti,j ) − y ti,j − βH ( x ti,j − d )+ , + σ x B xi,j X m x ,n x (cid:2) f ( x tm x ,n x ) − f ( x ti,j ) (cid:3) ,y t +1 i,j = y ti,j + ε ( x ti,j − J ) , (10)where m x , n x ∈ N are indices of the nonlocal neighbours. The sum denotes thenonlocal coupling of range R x in a square domain. The parameter σ x denotesthe coupling strength between the elements in the x variable, B xi,j gives thenumber of nonlocally coupled neighbors of node ( i, j ).The numerical results show that when the nonlocal coupling strength σ x and the coupling range R x are varied, the model (10) can demonstrate all thetypical spiral wave patterns, including spiral wave chimeras, which were observedearlier. Examples of these states are presented from the left in fig.10. It is seenthat the evolution of a spiral wave with increasing values of the coupling rangehas the same character as that of the systems described above. However, theregion around the wave center has high sensitivity even when the coupling islocal. This happens as the wave center element oscillates chaotically. Even aslight elongation of the coupling range r leads to significant expansion of thehigh-sensitive region around the wave center in space. The formation of theincoherence core has already taken place for r = 0 . . The right column infig.10 illustrates the spatial distributions of the ILS with the growth of thecoupling strength. The evolution of the first three LEs for the system (10) is18emonstrated in fig.11. The behavior of all the LEs is simpler than that of themodels (2) and (5). The lattice is always in hyperchaotic regime (even for thevery short and very long coupling ranges) because all the three calculated LE’sare positive. The coupling range elongation leads to growth of the LE’s values.The value of λ ( r ) at first monotonically increases and after that almost doesnot change with increase in the coupling range. The value of λ ( r ) and λ ( r ) also demonstrate monotonous growth. When the model (10) switches to thesynchronous regime at r = 0 . values of all the three LEs sharply decrease, butremain positive. Acknowledgements
This work was funded by the Deutsche Forschungsgemeinschaft (DFG, GermanResearch Foundation) – Projektnummer. 163436311-SFB 910. I.A.S., A.V.B.and V.S.A. thank for the financial support provided by RFBR and DFG accord-ing to the research project
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Phys. Rev. E , 70:016212,Jul 2004. 23 j i - . - . x i , j . . (a) j i - . - . x i , j . . (c) j i . . x i , j . . (e) j i . . x i , j . . (g) j i . . Λ i , j · [ − ] . . (b) j i . . Λ i , j · [ − ] . . (d) j i . . Λ i , j · [ − ] . . (f) j i . . Λ i , j · [ − ] . . (h)Figure 10: (Color online) Spiral waves and spiral wave chimeras in (10) for σ x = 0 . . The left column illustrates snapshots of the system states and theright column shows spatial distributions of the ILS, when coupling radius equalto (a,b) r = 1 , (c,d) r = 5 , (e,f) r = 12 , (g,h) r = 20 . Parameters: J = 0 . , ε = 0 . , α = 0 . , β = 0 . , d = 0 . , N = 100 λ r [ − ] λ λ λ Figure 11: (Color inline) Dependences of the three first Lyapunov exponents λ , λ and λ on the coupling range r for the systems (10). Parameters: σ x = 0 . , J = 0 . , ε = 0 . , α = 0 . , β = 0 . , d = 0 . , N = 100= 100