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Spiral and target wave chimeras in a 2D network of nonlocally coupled van der Pol oscillators
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Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
Spiral and target wave chimeras in a 2D network of nonlocally coupled van der Pol oscillators. Bukh A.V.∗, Anishchenko V.S. Physics Department, Astrakhanskaya street, 83, Saratov, 410012 Russian Federation
a r t i c l e
i n f o
Article history: Received 8 October 2019 Accepted 12 October 2019 Available online xxx Keywords: Two-dimensional ensemble Nonlocal coupling Spiral wave chimera Target wave chimera,
a b s t r a c t We study numerically autowave structures in a two-dimensional lattice of van der Pol oscillators for the cases of local and nonlocal coupling between the individual oscillators. It is shown that classical spiral wave and target wave regimes are observed in the network. When the nonlocal coupling between the elements is introduced, complex spatio-temporal structures, including chimera states, can be realized. We reveal and describe single- and multi-core spiral wave chimeras (SWC), as well as a novel type of chimera state which is based on target waves and called a target wave chimera (TWC). We investigate and compare the features of SWC and TWC incoherent cores. Finally, we study how the system dynamics changes when the coupling parameters are varied and discuss the impact of external noise on the observed wave structures. © 2019 Elsevier Ltd. All rights reserved.
1. Introduction Exploring autowave structures in nonlinear media and their mathematical models is one of the most actual and highly developed research area in nonlinear physics and related fields. An autowave represens a self-sustained wave process in a nonequilibrium medium, which remains unchanged when both initial and boundary conditions slightly change. Diffusion equations with an active nonlinearity are often used as a mathematical tool to describe autowaves [1–11]. Investigations of autowave structures started in the second half of the XX century and have been actively continuing nowadays [1–41]. The interest to the study of autowave structures in different models of active media sharply increased in the early 20 0 0s. This was mainly caused by the discovery of socalled chimera structures in ensembles of nonlocally coupled identical oscillators. Autowave structures were analyzed using reactiondiffusion mathematical models for nonlinear media. This technique allows to model a medium as an ensemble of interacting active oscillators with local coupling. Each oscillator is connected only with the closest neighbours within the ensemble. The chimera states were first revealed when the ensemble elements are coupled nonlocally. The nonlocal coupling means that each individual oscillator is symmetrically coupled with a certain number of neighbouring oscillators, which is given by the coupling radius [42]. Using the nonlocal coupling enables one to find a bulk of more complex spatio-temporal structures which cannot be observed in the case ∗
of local coupling [10,31,42–46]. Thus, the structure, that was called later “a spiral wave chimera” (SWC), was first discovered numerically in a two-dimensional ensemble of nonlocally coupled phase oscillators and described in the pioneering works [23,25]. It is important to note that these results were obtained before the term “chimera structure” was introduced in the paper [44]. In the present work we study autowave structures which are based on spiral and target waves. These structures were first obtained in numerical simulation of the dynamics of a 2D lattice of nonlocally coupled van der Pol oscillators [47]. We consider and analyze in detail the network dynamics for local and nonlocal coupling between the oscillators. We show that in the presence of local coupling, only “classical” spiral and target waves can be realized when the control and coupling parameters are varied. More complex autowave structures, including chimera states, can be found in the considered 2D lattice when the nonlocal coupling is introduced and its range increases. We present numerical results for spiral wave chimeras (SWC) with single and several incoherent cores. We also describe a novel type of chimera state, which is based on target waves and called a target wave chimera (TWC). The latter is found when the nonlocal coupling becomes much stronger [47]. We analyze in detail the features of incoherent cores of the observed chimera states. We investigate the peculiarities of spiral and target wave regimes as the nonlocal coupling strength is varied and in the presence of external noise in the considered network.
Corresponding author. E-mail addresses: [email protected] (B. A.V.), [email protected] (A. V.S.).
https://doi.org/10.1016/j.chaos.2019.109492 0960-0779/© 2019 Elsevier Ltd. All rights reserved.
Please cite this article as: B. A.V. and A. V.S., Spiral and target wave chimeras in a 2D network of nonlocally coupled van der Pol oscillators., Chaos, Solitons and Fractals, https://doi.org/10.1016/j.chaos.2019.109492
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1 Fig. 1. Scheme of the coupling topology for the ensemble (1) with P = 2.
Fig. 2. Diagram of regimes in the network (1) with local coupling (P = 1) in the (ω, ε ) parameter plane for σ = 0.7. The area of incoherent oscillations is colored with grey (Incoherence); the area of synchronous oscillations is coloured with light-blue (Sync); areas of existence of spiral and target waves are shown with green (SW) and red (TW) colors, respectively.
2. Model under study We consider the dynamics of a two-dimensional lattice of coupled van der Pol oscillators with N × N elements (N = 100), which is described by the following equations:
x˙ i, j = yi, j + y˙ i, j =
σ
Bi, j
[xm,n − xi, j ],
m,n
ε 1 − x2i, j · yi, j − ω2 xi, j ,
(1)
where m, n ∈ {1, 2, . . . , N} are indices of oscillators which are coupled with the (i, j)th oscillator; ε and ω are the parameter of excitation and the natural frequency of the individual oscillator, respectively. In our calculations, the dynamics of the isolated van der Pol oscillator corresponds to relaxation oscillations. Double indices of the variables (xi,j , yi,j ) for i, j = 1, . . . , N correspond to the position of the oscillator on the two-dimensional plane. Parameter σ denotes the coupling strength between the oscillators in x variables, Bi,j gives the size of the coupling range for each (i, j) node of the lattice, i.e., the number of (m, n) combinations which satisfy the no-flux boundary conditions:
max(1, i − P ) m min(N, i + P ), max(1, j − P ) n min(N, j + P ), ( m = i ) ∪ ( n = j ).
(2)
Here P is the number of nearest neighbors which the (i, j)th oscillator is coupled with and denotes the coupling range. The scheme of the model (1) with nonlocal coupling topology is drawn in Fig. 1 for P = 2. 3. Numerical results for local coupling Our numerical simulations show that spiral wave and target wave structures can be realized in the lattice (1) when the values of control parameters ε and ω of the individual oscillators are varied for a fixed value of the coupling range P. We explore the system (1) for initial conditions randomly distributed in the interval x, y ∈ [−0.9 : 0.9] and start with the local coupling P = 1. The diagram of the regimes observed in the network (1) in this case is presented in Fig. 2 in the (ω, ε ) parameter plane. Four regions of dynamical regimes can be distinguished. It is seen that spiral waves and incoherent regimes dominate in the system dynamics and are realized for a wide range of parameters ω and ε variation. These regimes are observed for small values of ε , and only the incoherent behavior of the lattice elements is possible when the values of ε are large. Target wave regimes appear only for sufficiently large values of ω and ε (the red region in Fig. 2). Synchronous dynamics of all the network elements is observed within a rather narrow region (the light-blue region in the diagram) for moderate values of ε and large values of ω. Our detailed studies demonstrate that for certain values of ω and ε , a combined spatiotemporal structure is observed which consists of both spiral and target waves.
We now fix ω = 0.9 and consider the wave dynamics of the network (1) when the parameter ε is varied. The corresponding snapshots of amplitudes xij for spiral wave regimes are given in Fig. 3 for three different values of the excitation parameter ε . As follows from this figure, the wavelength of the spiral wave decreases with increasing ε . A further increase of the parameter ε ≥ 2.2 causes the spiral wave to destroy and the system dynamics to pass to the incoherent regime. The evolution of target wave structures is shown in Fig. 4 for three different values of ε and for fixed σ = 0.7, ω = 0.9, P = 1. As in the case of spiral wave regimes (Fig. 3), the wavelength of target waves also decreases when the value of ε increases (Fig. 4). In both regimes, the wavelength is inversely proportional to the parameter ε . The target wave mode is destroyed and replaced by the incoherent oscillations when ε ≥ 2.4 (Fig. 2). Our studies show that there is a multistability in the system (1). This means that various spatio-temporal structures can be realized at the same values of system parameters. This effect occurs when either initial conditions are varied or the parameters’ values change in the reverse direction. Comparing Figs. 3(c) and 4(c) one can see that depending on the initial conditions, both spiral wave and target wave regimes can be realized at the same parameter values. Moreover, the effect of multistability is observed within the finite range ε ∈ [1.65; 2.2]. 4. Spatio-temporal structures for nonlocal coupling. Birth of chimera structures Numerical investigations have shown that the nonlocal coupling in ensembles makes spatio-temporal structures more complicated that can lead to the appearance of spiral wave chimera states [23,25,28,29,35,41,47–51]. We now consider the 2D lattice of van der Pol oscillators (1) with nonlocal interaction. The diagram of the dynamical regimes in (1) is depicted in Fig. 5 in the (P, ε )) parameter plane for fixed ω = 0.9 and σ = 0.7. As can be seen from the diagram, for large values of ε and as the coupling range P is varied, regimes of spiral wave chimeras and target wave chimeras can appear in the system (1). The impact of nonlocal coupling on the spiral wave structure is illustrated in Fig. 6. Firstly, the wavelength of the spiral wave increases with increasing P from the local case P = 1 (Fig. 6(a) and (b)). Then, at P = 16 an incoherent core (spot) arises in the neighbourhood of the spiral wave center and this indicates the appearance of spiral wave chimera (Fig. 6(c)). The peculiarities of the lattice dynamics in the SWC regime are illustrated in Fig. 7 which includes the following numerical results: the snapshot of the system dynamics at P = 16, its 1D cut across the incoherent core (marked by the red vertical line in Fig. 7(a)),
Please cite this article as: B. A.V. and A. V.S., Spiral and target wave chimeras in a 2D network of nonlocally coupled van der Pol oscillators., Chaos, Solitons and Fractals, https://doi.org/10.1016/j.chaos.2019.109492
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Fig. 3. Snapshots of amplitudes xij of the lattice (1) for spiral wave regimes with ε = 0.5 (a), ε = 1.5 (b), and ε = 2.15 (c). Other parameters: σ = 0.7, ω = 0.9, P = 1.
Fig. 4. Snapshots of amplitudes xij of the network (1) for target wave regimes with ε = 1.65 (a), ε = 2.0 (b), and ε = 2.15 (c). Other parameters: σ = 0.7, ω = 0.9, P = 1.
Fig. 5. Diagram of regimes in the network (1) with nonlocal coupling in the (P, ε ) parameter plane at ω = 0.9, σ = 0.7. The blue region (Sync) corresponds to synchronous oscillations, the green (SW) and red (TW) regions – to spiral and target wave regimes, respectively, spiral and target wave chimera states are realized in the dark green (SWC) and dark red (TWC) regions, respectively.
the space-time diagram and the cross-correlation Ri,i0 dependence for this cut [52], phase portraits and time series for neighbouring oscillators from the coherent and incoherent clusters. As can be seen, the elements from the incoherent core oscillate chaotically
(Fig. 7(e) and (h)), while the oscillators from the coherent regions demonstrate regular behavior (Fig. 7(d) and (g)). This fact is confirmed by the distribution of the cross-correlation coefficient values (Fig. 7(f)), which are close to zero inside the incoherent core. The appearance of multi-core spiral wave chimera is shown in Fig. 8 when the coupling range P increases. Thus, one can conclude that when the nonlocal coupling is introduced between the oscillators in the lattice (1), the spiral wave structure becomes more complicated and this then results in the appearance of spiral wave chimeras when the coupling range P increases. We now consider the impact of the coupling range P on the target wave regime. As seen from the diagram in Fig. 5, when P > 1 and for large values of ε , target wave chimeras (TWC) appear in the network (1). Their appearance is illustrated in Fig. 9 which shows the evolution of the target wave regime (at P = 1, Fig. 9(a)) with increasing coupling range P. It is seen that a novel type of chimeras, that was called a target wave chimera [47], arises at P = 4 (Fig. 9(b)). Similarly to the SWC (Fig. 6(c)), the TWC structure is characterized by the formation of an incoherent core in the neighbourhood of the target wave source (Fig. 9(b) and (c). As in the case of SWC, we also investigate the features of the incoherent core dynamics
Fig. 6. Evolution of the spiral wave regime in the network (1) with increasing coupling range P. Snapshots of amplitudes xij for P = 1 (a), P = 4 (b), and P = 16 (c). Other parameters: σ = 0.7, ω = 0.9, ε = 1.
Please cite this article as: B. A.V. and A. V.S., Spiral and target wave chimeras in a 2D network of nonlocally coupled van der Pol oscillators., Chaos, Solitons and Fractals, https://doi.org/10.1016/j.chaos.2019.109492
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Fig. 7. Exploration of the SWC dynamics for P = 16 (Fig. 6(c)). (a) Snapshot of amplitudes xij , (b) its one-dimensional cut (marked by the vertical line in the snapshot (a)), (c) space-time diagram, phase portraits for oscillators from coherent (d) and incoherent (e) regions, (f) cross-correlation dependence, and time series for oscillators from coherent (g) and incoherent (h) regions. Other parameters: σ = 0.7, ω = 0.9, ε = 1.
Fig. 8. Transition from the spiral wave to the spiral wave chimera with increasing coupling range P. Snapshots of amplitudes xij for P = 1 (a), P = 5 (b), and P = 10 (c). Other parameters: σ = 0.7, ω = 0.9, ε = 0.8.
for the TWC. The corresponding numerical results are presented in Fig. 10 which shows the snapshot of the TWC for P = 8 (Fig. 10(c)), the cut of the snapshot of amplitudes across the incoherent core (marked by the red vertical line in Fig. 10(a)), the space-time diagram and cross-correlation dependence for this cut, phase portraits and time series for neighbouring oscillators from the coherent and incoherent clusters. It is seen that oscillations for the elements from both the incoherent core (Fig. 10(e) and (h) are regular, while the oscillators from the coherent region show quasiperiodic dynamics (Fig. 10(d) and (g). It is worth noting that in the case of incoherent core, the oscillations correspond to different regular attractors (Fig. 10(e)), i.e., there is a bistability. The values of the
cross-correlation coefficient are switched between different levels for neighbouring nodes (Fig. 10(f)). Such a feature of the oscillators from the TWC incoherent core is characteristic for a solitary state chimera, which was revealed for the first time in [53]. Thus, comparing Figs. 7 and 10 we can conclude that the dynamics of oscillators from the SWC and TWC incoherent cores are completely different. Our numerical studies indicate that the TWC can also be obtained from spiral wave structures. This process is illustrated in Fig. 11 when P increases. We note that in this case the incoherent core is topologically different from that for the TWC which appears from the target wave regime (Fig. 9).
Please cite this article as: B. A.V. and A. V.S., Spiral and target wave chimeras in a 2D network of nonlocally coupled van der Pol oscillators., Chaos, Solitons and Fractals, https://doi.org/10.1016/j.chaos.2019.109492
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Fig. 9. Evolution of the target wave with increasing coupling range P. Snapshots of amplitudes xij for P = 1 (a), P = 4 (b), and P = 8 (c). Other parameters: σ = 0.7, ω = 0.9, ε = 2.
Fig. 10. Peculiarities of the target wave chimera mode in the case of P = 8 (Fig. 9(c)). (a) Snapshot of amplitudes xij , (b) its one-dimensional cut (marked by the vertical line in the snapshot (a)), (c) space-time diagram, phase portraits for oscillators from coherent (d) and incoherent (e) regions, (f) cross-correlation dependence, and time series of oscillators from coherent (g) and incoherent (h) regions. Other parameters: σ = 0.7, ω = 0.9, ε = 2.
5. Effect of the coupling strength on wave structures We now explore how the coupling strength affects the lattice (1) dynamics. Using random initial conditions we first get the wave structure for σ = 0.7 and then save it as the initial conditions for further numerical simulation. Afterwards, we gradually increase and decrease the parameter σ from σ = 0.7. Our numerical results demonstrate that when σ is varied, the wavelength of both spiral and target waves changes. In particular, it is shown that the wavelength of both wave structures increases with increasing σ . It is
also established that spiral and target waves exist within different ranges in the coupling strength σ . In our calculations σ is varied within the interval [0,1]. Spiral waves are realized for σ ∈ [0.65; 1], while target waves are observed for σ ∈ [0.59; 0.83]. 6. Impact of external noise on spiral and target waves Finally, we analyze the effect of external noise with intensity D on the wave structures in the lattice (1). In this case all the network elements are affected by external noise simultaneously.
Please cite this article as: B. A.V. and A. V.S., Spiral and target wave chimeras in a 2D network of nonlocally coupled van der Pol oscillators., Chaos, Solitons and Fractals, https://doi.org/10.1016/j.chaos.2019.109492
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Fig. 11. Transition from the spiral wave to the target wave chimera with increasing coupling range P. Snapshots of amplitudes xij for P = 1 (a), P = 2 (b), and P = 8 (c). Other parameters: σ = 0.7, ω = 0.9, ε = 2.
Fig. 12. Impact of noise on the lattice elements in the regimes of spiral (a-c) and target (d-f) waves. Snapshots of amplitudes xij at D0 = 0.001 (a,d), D0 = 0.01 (b,e), and D0 = 0.1 (c,f). Other parameters: P = 1, ω = 0.9, ε = 2, σ = 0.7.
At first, the noise intensity is set to a certain value D = D0 and the system (1) is integrated for 104 discrete time steps. Then, the noise is switched off (D = 0), and the system is integrated once more for 104 discrete time steps. The resulting snapshot of the xij amplitudes after 2 × 104 discrete time steps is registered and analyzed. Numerical results for the influence of noise on spiral and target wave structures are presented in Fig. 12(a)–(c) and (d)–(f), respectively. As can be seen, both wave structures are robust towards the noise with intensity D0 < 0.01 (Fig. 12(a) and (d)). The spiral wave is destroyed and converted to the spiral wave with several wave origins when D0 ≥ 0.01 (Fig. 12(b) and (c)). The target wave behaves in a different way in the presence of noise. A noise-induced transition from the target wave to the spiral wave with several wave origins occurs for D0 ≥ 0.01 (Fig. 12(e) and (f)). 7. Conclusions In this paper we have presented the results of numerical simulation of the 2D network of nonlocally (in the general case) coupled van der Pol oscillators. The control parameters of the individual van der Pol oscillator correspond to the relaxation oscillatory regime. The dynamics of the 2D lattice has been studied for local and nonlocal coupling topology between the oscillators.
It has been shown that classical spiral and target waves can be realized in the network. When the nonlocal coupling is introduced in the lattice, the observed wave structures become more complicated. Increasing the coupling range results in the appearance of single- and multi-core spiral wave chimeras. A novel type of wave chimera, a target wave chimera, has been revealed in the case of nonlocal coupling and described in detail. The diagrams of dynamical regimes realized in the 2D lattice of coupled van der Pol oscillators have been constructed in the control parameters’ planes for the cases of local and nonlocal coupling. Thus, the relatively simple 2D lattice of nonlocally coupled van der Pol oscillators can demonstrate both known spiral and target wave regimes and the corresponding chimera structures. Taking into account the fact that spiral and target waves are observed when modeling and analyzing the dynamics of the heart muscle [7,17,19,21,24], we believe that the obtained results can be useful in cardiology.
Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Please cite this article as: B. A.V. and A. V.S., Spiral and target wave chimeras in a 2D network of nonlocally coupled van der Pol oscillators., Chaos, Solitons and Fractals, https://doi.org/10.1016/j.chaos.2019.109492
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Please cite this article as: B. A.V. and A. V.S., Spiral and target wave chimeras in a 2D network of nonlocally coupled van der Pol oscillators., Chaos, Solitons and Fractals, https://doi.org/10.1016/j.chaos.2019.109492
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Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
Spiral and target wave chimeras in a 2D network of nonlocally coupled van der Pol oscillators. Bukh A.V.∗, Anishchenko V.S. Physics Department, Astrakhanskaya street, 83, Saratov, 410012 Russian Federation
a r t i c l e
i n f o
Article history: Received 8 October 2019 Accepted 12 October 2019 Available online xxx Keywords: Two-dimensional ensemble Nonlocal coupling Spiral wave chimera Target wave chimera,
a b s t r a c t We study numerically autowave structures in a two-dimensional lattice of van der Pol oscillators for the cases of local and nonlocal coupling between the individual oscillators. It is shown that classical spiral wave and target wave regimes are observed in the network. When the nonlocal coupling between the elements is introduced, complex spatio-temporal structures, including chimera states, can be realized. We reveal and describe single- and multi-core spiral wave chimeras (SWC), as well as a novel type of chimera state which is based on target waves and called a target wave chimera (TWC). We investigate and compare the features of SWC and TWC incoherent cores. Finally, we study how the system dynamics changes when the coupling parameters are varied and discuss the impact of external noise on the observed wave structures. © 2019 Elsevier Ltd. All rights reserved.
1. Introduction Exploring autowave structures in nonlinear media and their mathematical models is one of the most actual and highly developed research area in nonlinear physics and related fields. An autowave represens a self-sustained wave process in a nonequilibrium medium, which remains unchanged when both initial and boundary conditions slightly change. Diffusion equations with an active nonlinearity are often used as a mathematical tool to describe autowaves [1–11]. Investigations of autowave structures started in the second half of the XX century and have been actively continuing nowadays [1–41]. The interest to the study of autowave structures in different models of active media sharply increased in the early 20 0 0s. This was mainly caused by the discovery of socalled chimera structures in ensembles of nonlocally coupled identical oscillators. Autowave structures were analyzed using reactiondiffusion mathematical models for nonlinear media. This technique allows to model a medium as an ensemble of interacting active oscillators with local coupling. Each oscillator is connected only with the closest neighbours within the ensemble. The chimera states were first revealed when the ensemble elements are coupled nonlocally. The nonlocal coupling means that each individual oscillator is symmetrically coupled with a certain number of neighbouring oscillators, which is given by the coupling radius [42]. Using the nonlocal coupling enables one to find a bulk of more complex spatio-temporal structures which cannot be observed in the case ∗
of local coupling [10,31,42–46]. Thus, the structure, that was called later “a spiral wave chimera” (SWC), was first discovered numerically in a two-dimensional ensemble of nonlocally coupled phase oscillators and described in the pioneering works [23,25]. It is important to note that these results were obtained before the term “chimera structure” was introduced in the paper [44]. In the present work we study autowave structures which are based on spiral and target waves. These structures were first obtained in numerical simulation of the dynamics of a 2D lattice of nonlocally coupled van der Pol oscillators [47]. We consider and analyze in detail the network dynamics for local and nonlocal coupling between the oscillators. We show that in the presence of local coupling, only “classical” spiral and target waves can be realized when the control and coupling parameters are varied. More complex autowave structures, including chimera states, can be found in the considered 2D lattice when the nonlocal coupling is introduced and its range increases. We present numerical results for spiral wave chimeras (SWC) with single and several incoherent cores. We also describe a novel type of chimera state, which is based on target waves and called a target wave chimera (TWC). The latter is found when the nonlocal coupling becomes much stronger [47]. We analyze in detail the features of incoherent cores of the observed chimera states. We investigate the peculiarities of spiral and target wave regimes as the nonlocal coupling strength is varied and in the presence of external noise in the considered network.
Corresponding author. E-mail addresses: [email protected] (B. A.V.), [email protected] (A. V.S.).
https://doi.org/10.1016/j.chaos.2019.109492 0960-0779/© 2019 Elsevier Ltd. All rights reserved.
Please cite this article as: B. A.V. and A. V.S., Spiral and target wave chimeras in a 2D network of nonlocally coupled van der Pol oscillators., Chaos, Solitons and Fractals, https://doi.org/10.1016/j.chaos.2019.109492
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1 Fig. 1. Scheme of the coupling topology for the ensemble (1) with P = 2.
Fig. 2. Diagram of regimes in the network (1) with local coupling (P = 1) in the (ω, ε ) parameter plane for σ = 0.7. The area of incoherent oscillations is colored with grey (Incoherence); the area of synchronous oscillations is coloured with light-blue (Sync); areas of existence of spiral and target waves are shown with green (SW) and red (TW) colors, respectively.
2. Model under study We consider the dynamics of a two-dimensional lattice of coupled van der Pol oscillators with N × N elements (N = 100), which is described by the following equations:
x˙ i, j = yi, j + y˙ i, j =
σ
Bi, j
[xm,n − xi, j ],
m,n
ε 1 − x2i, j · yi, j − ω2 xi, j ,
(1)
where m, n ∈ {1, 2, . . . , N} are indices of oscillators which are coupled with the (i, j)th oscillator; ε and ω are the parameter of excitation and the natural frequency of the individual oscillator, respectively. In our calculations, the dynamics of the isolated van der Pol oscillator corresponds to relaxation oscillations. Double indices of the variables (xi,j , yi,j ) for i, j = 1, . . . , N correspond to the position of the oscillator on the two-dimensional plane. Parameter σ denotes the coupling strength between the oscillators in x variables, Bi,j gives the size of the coupling range for each (i, j) node of the lattice, i.e., the number of (m, n) combinations which satisfy the no-flux boundary conditions:
max(1, i − P ) m min(N, i + P ), max(1, j − P ) n min(N, j + P ), ( m = i ) ∪ ( n = j ).
(2)
Here P is the number of nearest neighbors which the (i, j)th oscillator is coupled with and denotes the coupling range. The scheme of the model (1) with nonlocal coupling topology is drawn in Fig. 1 for P = 2. 3. Numerical results for local coupling Our numerical simulations show that spiral wave and target wave structures can be realized in the lattice (1) when the values of control parameters ε and ω of the individual oscillators are varied for a fixed value of the coupling range P. We explore the system (1) for initial conditions randomly distributed in the interval x, y ∈ [−0.9 : 0.9] and start with the local coupling P = 1. The diagram of the regimes observed in the network (1) in this case is presented in Fig. 2 in the (ω, ε ) parameter plane. Four regions of dynamical regimes can be distinguished. It is seen that spiral waves and incoherent regimes dominate in the system dynamics and are realized for a wide range of parameters ω and ε variation. These regimes are observed for small values of ε , and only the incoherent behavior of the lattice elements is possible when the values of ε are large. Target wave regimes appear only for sufficiently large values of ω and ε (the red region in Fig. 2). Synchronous dynamics of all the network elements is observed within a rather narrow region (the light-blue region in the diagram) for moderate values of ε and large values of ω. Our detailed studies demonstrate that for certain values of ω and ε , a combined spatiotemporal structure is observed which consists of both spiral and target waves.
We now fix ω = 0.9 and consider the wave dynamics of the network (1) when the parameter ε is varied. The corresponding snapshots of amplitudes xij for spiral wave regimes are given in Fig. 3 for three different values of the excitation parameter ε . As follows from this figure, the wavelength of the spiral wave decreases with increasing ε . A further increase of the parameter ε ≥ 2.2 causes the spiral wave to destroy and the system dynamics to pass to the incoherent regime. The evolution of target wave structures is shown in Fig. 4 for three different values of ε and for fixed σ = 0.7, ω = 0.9, P = 1. As in the case of spiral wave regimes (Fig. 3), the wavelength of target waves also decreases when the value of ε increases (Fig. 4). In both regimes, the wavelength is inversely proportional to the parameter ε . The target wave mode is destroyed and replaced by the incoherent oscillations when ε ≥ 2.4 (Fig. 2). Our studies show that there is a multistability in the system (1). This means that various spatio-temporal structures can be realized at the same values of system parameters. This effect occurs when either initial conditions are varied or the parameters’ values change in the reverse direction. Comparing Figs. 3(c) and 4(c) one can see that depending on the initial conditions, both spiral wave and target wave regimes can be realized at the same parameter values. Moreover, the effect of multistability is observed within the finite range ε ∈ [1.65; 2.2]. 4. Spatio-temporal structures for nonlocal coupling. Birth of chimera structures Numerical investigations have shown that the nonlocal coupling in ensembles makes spatio-temporal structures more complicated that can lead to the appearance of spiral wave chimera states [23,25,28,29,35,41,47–51]. We now consider the 2D lattice of van der Pol oscillators (1) with nonlocal interaction. The diagram of the dynamical regimes in (1) is depicted in Fig. 5 in the (P, ε )) parameter plane for fixed ω = 0.9 and σ = 0.7. As can be seen from the diagram, for large values of ε and as the coupling range P is varied, regimes of spiral wave chimeras and target wave chimeras can appear in the system (1). The impact of nonlocal coupling on the spiral wave structure is illustrated in Fig. 6. Firstly, the wavelength of the spiral wave increases with increasing P from the local case P = 1 (Fig. 6(a) and (b)). Then, at P = 16 an incoherent core (spot) arises in the neighbourhood of the spiral wave center and this indicates the appearance of spiral wave chimera (Fig. 6(c)). The peculiarities of the lattice dynamics in the SWC regime are illustrated in Fig. 7 which includes the following numerical results: the snapshot of the system dynamics at P = 16, its 1D cut across the incoherent core (marked by the red vertical line in Fig. 7(a)),
Please cite this article as: B. A.V. and A. V.S., Spiral and target wave chimeras in a 2D network of nonlocally coupled van der Pol oscillators., Chaos, Solitons and Fractals, https://doi.org/10.1016/j.chaos.2019.109492
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Fig. 3. Snapshots of amplitudes xij of the lattice (1) for spiral wave regimes with ε = 0.5 (a), ε = 1.5 (b), and ε = 2.15 (c). Other parameters: σ = 0.7, ω = 0.9, P = 1.
Fig. 4. Snapshots of amplitudes xij of the network (1) for target wave regimes with ε = 1.65 (a), ε = 2.0 (b), and ε = 2.15 (c). Other parameters: σ = 0.7, ω = 0.9, P = 1.
Fig. 5. Diagram of regimes in the network (1) with nonlocal coupling in the (P, ε ) parameter plane at ω = 0.9, σ = 0.7. The blue region (Sync) corresponds to synchronous oscillations, the green (SW) and red (TW) regions – to spiral and target wave regimes, respectively, spiral and target wave chimera states are realized in the dark green (SWC) and dark red (TWC) regions, respectively.
the space-time diagram and the cross-correlation Ri,i0 dependence for this cut [52], phase portraits and time series for neighbouring oscillators from the coherent and incoherent clusters. As can be seen, the elements from the incoherent core oscillate chaotically
(Fig. 7(e) and (h)), while the oscillators from the coherent regions demonstrate regular behavior (Fig. 7(d) and (g)). This fact is confirmed by the distribution of the cross-correlation coefficient values (Fig. 7(f)), which are close to zero inside the incoherent core. The appearance of multi-core spiral wave chimera is shown in Fig. 8 when the coupling range P increases. Thus, one can conclude that when the nonlocal coupling is introduced between the oscillators in the lattice (1), the spiral wave structure becomes more complicated and this then results in the appearance of spiral wave chimeras when the coupling range P increases. We now consider the impact of the coupling range P on the target wave regime. As seen from the diagram in Fig. 5, when P > 1 and for large values of ε , target wave chimeras (TWC) appear in the network (1). Their appearance is illustrated in Fig. 9 which shows the evolution of the target wave regime (at P = 1, Fig. 9(a)) with increasing coupling range P. It is seen that a novel type of chimeras, that was called a target wave chimera [47], arises at P = 4 (Fig. 9(b)). Similarly to the SWC (Fig. 6(c)), the TWC structure is characterized by the formation of an incoherent core in the neighbourhood of the target wave source (Fig. 9(b) and (c). As in the case of SWC, we also investigate the features of the incoherent core dynamics
Fig. 6. Evolution of the spiral wave regime in the network (1) with increasing coupling range P. Snapshots of amplitudes xij for P = 1 (a), P = 4 (b), and P = 16 (c). Other parameters: σ = 0.7, ω = 0.9, ε = 1.
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Fig. 7. Exploration of the SWC dynamics for P = 16 (Fig. 6(c)). (a) Snapshot of amplitudes xij , (b) its one-dimensional cut (marked by the vertical line in the snapshot (a)), (c) space-time diagram, phase portraits for oscillators from coherent (d) and incoherent (e) regions, (f) cross-correlation dependence, and time series for oscillators from coherent (g) and incoherent (h) regions. Other parameters: σ = 0.7, ω = 0.9, ε = 1.
Fig. 8. Transition from the spiral wave to the spiral wave chimera with increasing coupling range P. Snapshots of amplitudes xij for P = 1 (a), P = 5 (b), and P = 10 (c). Other parameters: σ = 0.7, ω = 0.9, ε = 0.8.
for the TWC. The corresponding numerical results are presented in Fig. 10 which shows the snapshot of the TWC for P = 8 (Fig. 10(c)), the cut of the snapshot of amplitudes across the incoherent core (marked by the red vertical line in Fig. 10(a)), the space-time diagram and cross-correlation dependence for this cut, phase portraits and time series for neighbouring oscillators from the coherent and incoherent clusters. It is seen that oscillations for the elements from both the incoherent core (Fig. 10(e) and (h) are regular, while the oscillators from the coherent region show quasiperiodic dynamics (Fig. 10(d) and (g). It is worth noting that in the case of incoherent core, the oscillations correspond to different regular attractors (Fig. 10(e)), i.e., there is a bistability. The values of the
cross-correlation coefficient are switched between different levels for neighbouring nodes (Fig. 10(f)). Such a feature of the oscillators from the TWC incoherent core is characteristic for a solitary state chimera, which was revealed for the first time in [53]. Thus, comparing Figs. 7 and 10 we can conclude that the dynamics of oscillators from the SWC and TWC incoherent cores are completely different. Our numerical studies indicate that the TWC can also be obtained from spiral wave structures. This process is illustrated in Fig. 11 when P increases. We note that in this case the incoherent core is topologically different from that for the TWC which appears from the target wave regime (Fig. 9).
Please cite this article as: B. A.V. and A. V.S., Spiral and target wave chimeras in a 2D network of nonlocally coupled van der Pol oscillators., Chaos, Solitons and Fractals, https://doi.org/10.1016/j.chaos.2019.109492
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Fig. 9. Evolution of the target wave with increasing coupling range P. Snapshots of amplitudes xij for P = 1 (a), P = 4 (b), and P = 8 (c). Other parameters: σ = 0.7, ω = 0.9, ε = 2.
Fig. 10. Peculiarities of the target wave chimera mode in the case of P = 8 (Fig. 9(c)). (a) Snapshot of amplitudes xij , (b) its one-dimensional cut (marked by the vertical line in the snapshot (a)), (c) space-time diagram, phase portraits for oscillators from coherent (d) and incoherent (e) regions, (f) cross-correlation dependence, and time series of oscillators from coherent (g) and incoherent (h) regions. Other parameters: σ = 0.7, ω = 0.9, ε = 2.
5. Effect of the coupling strength on wave structures We now explore how the coupling strength affects the lattice (1) dynamics. Using random initial conditions we first get the wave structure for σ = 0.7 and then save it as the initial conditions for further numerical simulation. Afterwards, we gradually increase and decrease the parameter σ from σ = 0.7. Our numerical results demonstrate that when σ is varied, the wavelength of both spiral and target waves changes. In particular, it is shown that the wavelength of both wave structures increases with increasing σ . It is
also established that spiral and target waves exist within different ranges in the coupling strength σ . In our calculations σ is varied within the interval [0,1]. Spiral waves are realized for σ ∈ [0.65; 1], while target waves are observed for σ ∈ [0.59; 0.83]. 6. Impact of external noise on spiral and target waves Finally, we analyze the effect of external noise with intensity D on the wave structures in the lattice (1). In this case all the network elements are affected by external noise simultaneously.
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Fig. 11. Transition from the spiral wave to the target wave chimera with increasing coupling range P. Snapshots of amplitudes xij for P = 1 (a), P = 2 (b), and P = 8 (c). Other parameters: σ = 0.7, ω = 0.9, ε = 2.
Fig. 12. Impact of noise on the lattice elements in the regimes of spiral (a-c) and target (d-f) waves. Snapshots of amplitudes xij at D0 = 0.001 (a,d), D0 = 0.01 (b,e), and D0 = 0.1 (c,f). Other parameters: P = 1, ω = 0.9, ε = 2, σ = 0.7.
At first, the noise intensity is set to a certain value D = D0 and the system (1) is integrated for 104 discrete time steps. Then, the noise is switched off (D = 0), and the system is integrated once more for 104 discrete time steps. The resulting snapshot of the xij amplitudes after 2 × 104 discrete time steps is registered and analyzed. Numerical results for the influence of noise on spiral and target wave structures are presented in Fig. 12(a)–(c) and (d)–(f), respectively. As can be seen, both wave structures are robust towards the noise with intensity D0 < 0.01 (Fig. 12(a) and (d)). The spiral wave is destroyed and converted to the spiral wave with several wave origins when D0 ≥ 0.01 (Fig. 12(b) and (c)). The target wave behaves in a different way in the presence of noise. A noise-induced transition from the target wave to the spiral wave with several wave origins occurs for D0 ≥ 0.01 (Fig. 12(e) and (f)). 7. Conclusions In this paper we have presented the results of numerical simulation of the 2D network of nonlocally (in the general case) coupled van der Pol oscillators. The control parameters of the individual van der Pol oscillator correspond to the relaxation oscillatory regime. The dynamics of the 2D lattice has been studied for local and nonlocal coupling topology between the oscillators.
It has been shown that classical spiral and target waves can be realized in the network. When the nonlocal coupling is introduced in the lattice, the observed wave structures become more complicated. Increasing the coupling range results in the appearance of single- and multi-core spiral wave chimeras. A novel type of wave chimera, a target wave chimera, has been revealed in the case of nonlocal coupling and described in detail. The diagrams of dynamical regimes realized in the 2D lattice of coupled van der Pol oscillators have been constructed in the control parameters’ planes for the cases of local and nonlocal coupling. Thus, the relatively simple 2D lattice of nonlocally coupled van der Pol oscillators can demonstrate both known spiral and target wave regimes and the corresponding chimera structures. Taking into account the fact that spiral and target waves are observed when modeling and analyzing the dynamics of the heart muscle [7,17,19,21,24], we believe that the obtained results can be useful in cardiology.
Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Please cite this article as: B. A.V. and A. V.S., Spiral and target wave chimeras in a 2D network of nonlocally coupled van der Pol oscillators., Chaos, Solitons and Fractals, https://doi.org/10.1016/j.chaos.2019.109492
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Please cite this article as: B. A.V. and A. V.S., Spiral and target wave chimeras in a 2D network of nonlocally coupled van der Pol oscillators., Chaos, Solitons and Fractals, https://doi.org/10.1016/j.chaos.2019.109492