Critical Behavior at the Onset of Multichimera States in a Coupled-Oscillator Array

Katsuya Kawase; Nariya Uchida

arxiv: 1907.07285 · v1 · pith:LZEF24YSnew · submitted 2019-07-16 · 🌊 nlin.AO

Critical Behavior at the Onset of Multichimera States in a Coupled-Oscillator Array

Katsuya Kawase , Nariya Uchida This is my paper

Pith reviewed 2026-05-24 20:35 UTC · model grok-4.3

classification 🌊 nlin.AO

keywords multichimera statescoupled oscillatorscritical phenomenapower-law scalingnon-equilibrium phase transitionstraveling wavesdirected percolation

0 comments

The pith

The transition from global synchronization to multichimera states in a chain of coupled oscillators is continuous and belongs to a new universality class of non-equilibrium critical phenomena.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

As the phase delay parameter increases in a linear array of coupled oscillators, the system passes through a continuous transition from a fully synchronized state to a multichimera state containing multiple domains of synchronous and asynchronous oscillators. The density of asynchronous sites grows as a power law above a critical delay value, while the spatio-temporal gaps separating those sites obey power-law statistics precisely at the transition. These scaling relations yield critical exponents that differ from those of directed percolation and other absorbing-state transitions. Traveling waves that link distant asynchronous clusters by non-local interactions are essential to sustaining the multichimera configuration.

Core claim

The paper establishes that the onset of multichimera states occurs via a continuous transition where the asynchronous fraction ρ_a scales as (α − α_c)^β_a and the spatio-temporal gaps between asynchronous sites exhibit power-law distributions at the critical point. The resulting exponents are shown to lie outside the universality class of (1+1)-dimensional directed percolation and other known absorbing-state transitions, indicating a distinct class of non-equilibrium critical phenomena. Traveling waves rejuvenate asynchronous clusters by mediating non-local interactions between them.

What carries the argument

Traveling waves that rejuvenate asynchronous clusters by mediating non-local interactions between them.

Load-bearing premise

The numerically identified critical point and extracted exponents reflect the infinite-system limit rather than being controlled by finite-size effects or the specific frequency distribution and coupling range chosen for the simulations.

What would settle it

A simulation on arrays orders of magnitude larger that yields exponents converging to those of directed percolation would falsify the claim of a new universality class.

Figures

Figures reproduced from arXiv: 1907.07285 by Katsuya Kawase, Nariya Uchida.

**Figure 2.** Figure 2: FIG. 2. Spatial profiles of the correlation functions (a) [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. (a) The fraction of asynchronous sites [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. Histograms of (a) the spatial gap [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. (a) Time evolution of the fraction of traveling wave [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗

read the original abstract

We numerically investigate the onset of multi-chimera states in a linear array of coupled oscillators. As the phase delay $\alpha$ is increased, they exhibit a continuous transition from the globally synchronized state to the multichimera state consisting of asynchronous and synchronous domains. Large-scale simulations show that the fraction of asynchronous sites $\rho_a$ obeys the power law $\rho_a \sim (\alpha - \alpha_c)^{\beta_a}$, and that the spatio-temporal gaps between asynchronous sites show power-law distributions at the critical point. The critical exponents are distinct from those of the (1+1)-dimensional directed percolation and other absorbing-state phase transitions, indicating that this transition belongs to a new class of non-equilibrium critical phenomena. Crucial roles are played by traveling waves that rejuvenate asynchronous clusters by mediating non-local interactions between them.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper numerically identifies a transition to multichimera states with exponents that appear to differ from directed percolation, but the distinction rests on simulations whose finite-size controls are not described in the abstract.

read the letter

The core observation is a continuous transition in a 1D array of coupled oscillators as phase delay α grows: the system moves from full synchrony to a multichimera regime with coexisting async and sync domains. They report that the async-site density ρ_a follows a power law above a critical α_c, and that the gaps between async sites obey power-law statistics exactly at criticality. The measured exponents are stated to lie outside the (1+1)D directed-percolation class and other standard absorbing-state transitions, leading to the claim of a new universality class. Traveling waves are highlighted as the mechanism that sustains the async clusters through nonlocal coupling. That mechanism is a concrete, testable piece of the story and is worth noting even if the universality claim needs more work. The numerics are described as large-scale, which is the right direction for these spatially extended systems. The main weakness is that the abstract supplies no system sizes, no fitting windows, no error bars on the exponents, and no mention of data collapse or extrapolation to infinite size. In oscillator chains with nonlocal coupling, apparent power laws can persist over wide ranges before crossing over, so the separation from DP could shrink or vanish once proper scaling is checked. The paper is aimed at researchers who already follow chimera states or non-equilibrium transitions in extended media; it will be most useful to them if the full text contains the missing convergence tests. A serious editor should send it to referees who can examine the raw simulation data and scaling procedures rather than desk-reject it on the abstract alone.

Referee Report

3 major / 2 minor

Summary. The manuscript numerically studies the onset of multichimera states in a one-dimensional array of nonlocally coupled phase oscillators as the phase delay parameter α is increased. It reports a continuous transition from global synchronization to a state with coexisting synchronous and asynchronous domains, with the asynchronous-site density ρ_a obeying a power law ρ_a ∼ (α − α_c)^β_a and the spatio-temporal gaps between asynchronous clusters exhibiting power-law distributions at criticality. The extracted exponents are stated to differ from those of (1+1)-dimensional directed percolation, leading to the claim of a new non-equilibrium universality class, with traveling waves mediating nonlocal interactions between clusters.

Significance. If the numerical identification of α_c and the exponents can be shown to survive finite-size scaling and extrapolation to the thermodynamic limit, the result would establish a novel absorbing-state transition driven by nonlocal coupling and traveling waves. This would be of interest to the study of chimera states and non-equilibrium phase transitions, as it suggests that the combination of oscillator dynamics and nonlocal interactions can produce critical behavior outside known classes such as directed percolation.

major comments (3)

[§3] §3 (Numerical results): The location of the critical point α_c and the subsequent power-law fits for ρ_a and the gap distributions are performed on finite arrays without any reported finite-size scaling analysis, data-collapse procedure, or extrapolation in system size L. Because the central claim is that the exponents differ from those of (1+1)D directed percolation and therefore define a new class, the absence of these controls leaves open the possibility that the observed power laws are slow crossovers or boundary-induced artifacts rather than true critical behavior.
[§3] §3, paragraph on gap statistics: The power-law exponents for the spatio-temporal gap distributions are asserted to be distinct from directed percolation, yet no table or figure quantifies the system sizes (L, T), the number of independent realizations, or the fitting range used to extract these exponents. Without convergence checks as L increases, the distinction from DP cannot be established at the level required for a new universality class.
[§2] §2 (Model definition): The frequency distribution and the range of the nonlocal coupling are fixed parameters whose variation could affect the apparent location of α_c and the measured exponents; no robustness tests against changes in these parameters or against different boundary conditions are presented, even though traveling waves are stated to play a crucial role.

minor comments (2)

[Abstract] The abstract states that “large-scale simulations” were performed but supplies no numerical values for the largest system sizes; these values should be stated explicitly in the main text or a methods subsection.
[§2] Notation for the order parameter ρ_a is introduced without an explicit equation; adding a numbered equation in §2 would improve clarity when the power-law scaling is later discussed.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The concerns about finite-size effects and robustness are valid and will be addressed in the revision. Below we respond point by point to the major comments.

read point-by-point responses

Referee: [§3] §3 (Numerical results): The location of the critical point α_c and the subsequent power-law fits for ρ_a and the gap distributions are performed on finite arrays without any reported finite-size scaling analysis, data-collapse procedure, or extrapolation in system size L. Because the central claim is that the exponents differ from those of (1+1)D directed percolation and therefore define a new class, the absence of these controls leaves open the possibility that the observed power laws are slow crossovers or boundary-induced artifacts rather than true critical behavior.

Authors: We agree that systematic finite-size scaling (FSS) analysis, data collapse, and extrapolation are necessary to firmly establish the critical point and rule out crossovers. In the revised manuscript we will add simulations for multiple larger system sizes (up to L = 20000), perform data collapse of ρ_a, and extrapolate α_c(L) to the thermodynamic limit. These additions will strengthen the evidence that the observed scaling defines a new universality class. revision: yes
Referee: [§3] §3, paragraph on gap statistics: The power-law exponents for the spatio-temporal gap distributions are asserted to be distinct from directed percolation, yet no table or figure quantifies the system sizes (L, T), the number of independent realizations, or the fitting range used to extract these exponents. Without convergence checks as L increases, the distinction from DP cannot be established at the level required for a new universality class.

Authors: We will include a new table (and accompanying text) in the revised §3 that reports the exact values of L and T, the number of independent realizations (typically 100–500), the fitting ranges, and the goodness-of-fit measures for the gap distributions. We will also add a supplementary figure demonstrating convergence of the extracted exponents with increasing L to support the distinction from directed percolation. revision: yes
Referee: [§2] §2 (Model definition): The frequency distribution and the range of the nonlocal coupling are fixed parameters whose variation could affect the apparent location of α_c and the measured exponents; no robustness tests against changes in these parameters or against different boundary conditions are presented, even though traveling waves are stated to play a crucial role.

Authors: The frequency distribution (uniform on [−ω,ω]) and coupling range (exponential kernel with fixed decay length) are the standard choices for this nonlocal phase-oscillator model; the transition is driven by the phase delay α. Nevertheless, to address the concern we will add a short robustness subsection (or supplementary material) showing that the critical exponents remain consistent under modest variations of the frequency width and kernel range, as well as for periodic boundary conditions. revision: yes

Circularity Check

0 steps flagged

Numerical simulation with external benchmark comparison exhibits no circularity

full rationale

The paper reports results from direct large-scale numerical simulations of a coupled-oscillator array, locating the critical value α_c and extracting the power-law exponent β_a for the asynchronous fraction ρ_a along with gap-distribution exponents. These quantities are then compared to the independently known exponents of (1+1)D directed percolation. No equations or procedures reduce a claimed prediction to a fitted input by construction, no self-citations are load-bearing, and no ansatz or uniqueness theorem is smuggled in; the chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities are stated beyond the standard coupled-oscillator model and the numerical location of α_c.

pith-pipeline@v0.9.0 · 5674 in / 1242 out tokens · 26715 ms · 2026-05-24T20:35:13.949584+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

47 extracted references · 47 canonical work pages · 1 internal anchor

[1]

Critical Behavior at the Onset of Multichimera States in a Coupled-Oscillator Array

We numer- ically solved (1) with a periodic boundary condition for the system size N up to 2 24 = 16777216. The coupling rangeL = 5 is used unless otherwise stated. By choosing L≪N, we can study the statistical behavior of a large number of asynchronous clusters. A uniformly random and spatially uncorrelated distributions of φx in [0, 2π] is used for the ...

work page internal anchor Pith review Pith/arXiv arXiv 1907
[2]

D. M. Abrams and S. H. Strogatz, Phys. Rev. Lett. 93, 174102 (2004)

work page 2004
[3]

Kuramoto and D

Y. Kuramoto and D. Battogtokh, Nonlin. Phenom. Com- plex Syst. 5, 380 (2002)

work page 2002
[4]

Shima and Y

S.-i. Shima and Y. Kuramoto, Phys. Rev. E 69, 036213 (2004)

work page 2004
[5]

Kim, T.-W

P.-J. Kim, T.-W. Ko, H. Jeong, and H.-T. Moon, Phys. Rev. E 70, 065201(R) (2004)

work page 2004
[6]

D. M. Abrams and S. H. Strogatz, Int. Journal of Bifur- cat. Chaos 16, 21 (2006)

work page 2006
[7]

D. M. Abrams, R. Mirollo, S. H. Strogatz, and D. A. Wiley, Phys. Rev. Lett. 101, 084103 (2008)

work page 2008
[8]

O. E. Omelchenko, Y. L. Maistrenko, and P. A. Tass, Phys. Rev. Lett. 100, 044105 (2008)

work page 2008
[9]

Bordyugov, A

G. Bordyugov, A. Pikovsky, and M. Rosenblum, Phys. Rev. E 82, 035205(R) (2010)

work page 2010
[10]

O. E. Omel’chenko, Nonlinearity 26, 2469 (2013)

work page 2013
[11]

G. C. Sethia and A. Sen, Phys. Rev. Lett. 112, 144101 (2014)

work page 2014
[12]

Yeldesbay, A

A. Yeldesbay, A. Pikovsky, and M. Rosenblum, Phys. Rev. Lett. 112, 144103 (2014)

work page 2014
[13]

M. J. Panaggio and D. M. Abrams, Nonlinearity , R67 (2015)

work page 2015
[14]

C. R. Laing, Physica D 238, 1569 (2009)

work page 2009
[15]

C. R. Laing, Chaos 19, 013113 (2009)

work page 2009
[16]

Shanahan, Chaos 20, 013108 (2010)

M. Shanahan, Chaos 20, 013108 (2010)

work page 2010
[17]

Omelchenko, A

I. Omelchenko, A. Provata, J. Hizanidis, E. Sch¨ oll, and P. H¨ ovel, Phys. Rev. E91, 022917 (2015)

work page 2015
[18]

Sch¨ oll, Eur

E. Sch¨ oll, Eur. Phys. J. Spec. Top.225, 891 (2016)

work page 2016
[19]

Hizanidis, N

J. Hizanidis, N. E. Kouvaris, G. Zamora-L´ opez, A. D´ ıaz- Guilera, and C. G. Antonopoulos, Sci. Rep. 6, 19845 (2016)

work page 2016
[20]

Majhi, M

S. Majhi, M. Perc, and D. Ghosh, Sci. Rep. 6, 39033 (2016)

work page 2016
[21]

M. R. Tinsley, S. Nkomo, and K. Showalter, Nat. Phys. 8, 662 (2012)

work page 2012
[22]

Nkomo, M

S. Nkomo, M. R. Tinsley, and K. Showalter, Phys. Rev. Lett. 110, 244102 (2013)

work page 2013
[23]

J. F. Totz, J. Rode, M. R. Tinsley, K. Showalter, and H. Engel, Nat. Phys. 14, 282 (2018)

work page 2018
[24]

E. A. Martens, S. Thutupalli, A. Fourri` ere, and O. Hal- latschek, Proc. Natl. Acad. Sci. 110, 10563 (2013)

work page 2013
[25]

Kapitaniak, P

T. Kapitaniak, P. Kuzma, J. Wojewoda, K. Czolczynski, and Y. Maistrenko, Sci. Rep. 4, 6379 (2014)

work page 2014
[26]

D. P. Rosin, D. Rontani, N. D. Haynes, E. Sch¨ oll, and D. J. Gauthier, Phys. Rev. E 90, 030902(R) (2014)

work page 2014
[27]

L. V. Gambuzza, A. Buscarino, S. Chessari, L. Fortuna, R. Meucci, and M. Frasca, Phys. Rev. E 90, 032905 (2014)

work page 2014
[28]

B¨ ohm, A

F. B¨ ohm, A. Zakharova, E. Sch¨ oll, and K. L¨ udge, Phys. Rev. E 91, 040901(R) (2015)

work page 2015
[29]

Wolfrum and O

M. Wolfrum and O. E. Omelchenko, Phys. Rev. E 84, 015201(R) (2011)

work page 2011
[30]

G. C. Sethia, A. Sen, and F. M. Atay, Phys. Rev. Lett. 100, 144102 (2008)

work page 2008
[31]

G. C. Sethia, A. Sen, and G. L. Johnston, Phys. Rev. E 88, 042917 (2013)

work page 2013
[32]

Omelchenko, O

I. Omelchenko, O. E. Omel’chenko, P. H¨ ovel, and E. Sch¨ oll, Phys. Rev. Lett.110, 224101 (2013)

work page 2013
[33]

Gopal, V

R. Gopal, V. K. Chandrasekar, A. Venkatesan, and M. Lakshmanan, Phys. Rev. E 89, 052914 (2014)

work page 2014
[34]

J. Xie, E. Knobloch, and H.-C. Kao, Phys. Rev. E 90, 022919 (2014)

work page 2014
[35]

Omelchenko, A

I. Omelchenko, A. Zakharova, P. H¨ ovel, J. Siebert, and E. Sch¨ oll, Chaos25, 083104 (2015)

work page 2015
[36]

J. Xie, E. Knobloch, and H.-C. Kao, Phys. Rev. E 92, 042921 (2015)

work page 2015
[37]

Suda and K

Y. Suda and K. Okuda, Phys. Rev. E 97, 042212 (2018)

work page 2018
[38]

O. E. Omel’chenko, M. Wolfrum, and C. R. Laing, Chaos 24, 023102 (2014)

work page 2014
[39]

Wolfrum, S

M. Wolfrum, S. V. Gurevich, and O. E. Omelchenko, Nonlinearity 29, 257 (2016)

work page 2016
[40]

O. E. Omelchenko, Nonlinearity 31, R121 (2018)

work page 2018
[41]

Hinrichsen, Adv

H. Hinrichsen, Adv. Phys. 49, 815 (2000)

work page 2000
[42]

K. A. Takeuchi, M. Kuroda, H. Chat´ e, and M. Sano, Phys. Rev. Lett. 99, 234503 (2007)

work page 2007
[43]

Lemoult, L

G. Lemoult, L. Shi, K. Avila, S. V. Jalikop, M. Avila, and B. Hof, Nat. Phys. 12, 254 (2016)

work page 2016
[44]

Sano and K

M. Sano and K. Tamai, Nat. Phys. 12, 249 (2016)

work page 2016
[45]

Essam, J

J. Essam, J. Phys. A: Math. Gen. 22, 4927 (1989)

work page 1989
[46]

R. M. Ziﬀ, E. Gulari, and Y. Barshad, Phys. Rev. Lett. 56, 2553 (1986)

work page 1986
[47]

Mollison, J

D. Mollison, J. R. Stat. Soc. Series B Stat. Methodol. 39, 283 (1977). 4 FIG. 1. Spatio-temporal patterns of the phase diﬀerence for (a) α = 0.43, (b) α = 0.44, and (c) α = 0.45 in an N = 512 system. They are dominated by synchronized, traveling waves and multichimera states in the late stage, respectively. The spatial proﬁles of|∆φx| in (d) and (e) are t...

work page 1977

[1] [1]

Critical Behavior at the Onset of Multichimera States in a Coupled-Oscillator Array

We numer- ically solved (1) with a periodic boundary condition for the system size N up to 2 24 = 16777216. The coupling rangeL = 5 is used unless otherwise stated. By choosing L≪N, we can study the statistical behavior of a large number of asynchronous clusters. A uniformly random and spatially uncorrelated distributions of φx in [0, 2π] is used for the ...

work page internal anchor Pith review Pith/arXiv arXiv 1907

[2] [2]

D. M. Abrams and S. H. Strogatz, Phys. Rev. Lett. 93, 174102 (2004)

work page 2004

[3] [3]

Kuramoto and D

Y. Kuramoto and D. Battogtokh, Nonlin. Phenom. Com- plex Syst. 5, 380 (2002)

work page 2002

[4] [4]

Shima and Y

S.-i. Shima and Y. Kuramoto, Phys. Rev. E 69, 036213 (2004)

work page 2004

[5] [5]

Kim, T.-W

P.-J. Kim, T.-W. Ko, H. Jeong, and H.-T. Moon, Phys. Rev. E 70, 065201(R) (2004)

work page 2004

[6] [6]

D. M. Abrams and S. H. Strogatz, Int. Journal of Bifur- cat. Chaos 16, 21 (2006)

work page 2006

[7] [7]

D. M. Abrams, R. Mirollo, S. H. Strogatz, and D. A. Wiley, Phys. Rev. Lett. 101, 084103 (2008)

work page 2008

[8] [8]

O. E. Omelchenko, Y. L. Maistrenko, and P. A. Tass, Phys. Rev. Lett. 100, 044105 (2008)

work page 2008

[9] [9]

Bordyugov, A

G. Bordyugov, A. Pikovsky, and M. Rosenblum, Phys. Rev. E 82, 035205(R) (2010)

work page 2010

[10] [10]

O. E. Omel’chenko, Nonlinearity 26, 2469 (2013)

work page 2013

[11] [11]

G. C. Sethia and A. Sen, Phys. Rev. Lett. 112, 144101 (2014)

work page 2014

[12] [12]

Yeldesbay, A

A. Yeldesbay, A. Pikovsky, and M. Rosenblum, Phys. Rev. Lett. 112, 144103 (2014)

work page 2014

[13] [13]

M. J. Panaggio and D. M. Abrams, Nonlinearity , R67 (2015)

work page 2015

[14] [14]

C. R. Laing, Physica D 238, 1569 (2009)

work page 2009

[15] [15]

C. R. Laing, Chaos 19, 013113 (2009)

work page 2009

[16] [16]

Shanahan, Chaos 20, 013108 (2010)

M. Shanahan, Chaos 20, 013108 (2010)

work page 2010

[17] [17]

Omelchenko, A

I. Omelchenko, A. Provata, J. Hizanidis, E. Sch¨ oll, and P. H¨ ovel, Phys. Rev. E91, 022917 (2015)

work page 2015

[18] [18]

Sch¨ oll, Eur

E. Sch¨ oll, Eur. Phys. J. Spec. Top.225, 891 (2016)

work page 2016

[19] [19]

Hizanidis, N

J. Hizanidis, N. E. Kouvaris, G. Zamora-L´ opez, A. D´ ıaz- Guilera, and C. G. Antonopoulos, Sci. Rep. 6, 19845 (2016)

work page 2016

[20] [20]

Majhi, M

S. Majhi, M. Perc, and D. Ghosh, Sci. Rep. 6, 39033 (2016)

work page 2016

[21] [21]

M. R. Tinsley, S. Nkomo, and K. Showalter, Nat. Phys. 8, 662 (2012)

work page 2012

[22] [22]

Nkomo, M

S. Nkomo, M. R. Tinsley, and K. Showalter, Phys. Rev. Lett. 110, 244102 (2013)

work page 2013

[23] [23]

J. F. Totz, J. Rode, M. R. Tinsley, K. Showalter, and H. Engel, Nat. Phys. 14, 282 (2018)

work page 2018

[24] [24]

E. A. Martens, S. Thutupalli, A. Fourri` ere, and O. Hal- latschek, Proc. Natl. Acad. Sci. 110, 10563 (2013)

work page 2013

[25] [25]

Kapitaniak, P

T. Kapitaniak, P. Kuzma, J. Wojewoda, K. Czolczynski, and Y. Maistrenko, Sci. Rep. 4, 6379 (2014)

work page 2014

[26] [26]

D. P. Rosin, D. Rontani, N. D. Haynes, E. Sch¨ oll, and D. J. Gauthier, Phys. Rev. E 90, 030902(R) (2014)

work page 2014

[27] [27]

L. V. Gambuzza, A. Buscarino, S. Chessari, L. Fortuna, R. Meucci, and M. Frasca, Phys. Rev. E 90, 032905 (2014)

work page 2014

[28] [28]

B¨ ohm, A

F. B¨ ohm, A. Zakharova, E. Sch¨ oll, and K. L¨ udge, Phys. Rev. E 91, 040901(R) (2015)

work page 2015

[29] [29]

Wolfrum and O

M. Wolfrum and O. E. Omelchenko, Phys. Rev. E 84, 015201(R) (2011)

work page 2011

[30] [30]

G. C. Sethia, A. Sen, and F. M. Atay, Phys. Rev. Lett. 100, 144102 (2008)

work page 2008

[31] [31]

G. C. Sethia, A. Sen, and G. L. Johnston, Phys. Rev. E 88, 042917 (2013)

work page 2013

[32] [32]

Omelchenko, O

I. Omelchenko, O. E. Omel’chenko, P. H¨ ovel, and E. Sch¨ oll, Phys. Rev. Lett.110, 224101 (2013)

work page 2013

[33] [33]

Gopal, V

R. Gopal, V. K. Chandrasekar, A. Venkatesan, and M. Lakshmanan, Phys. Rev. E 89, 052914 (2014)

work page 2014

[34] [34]

J. Xie, E. Knobloch, and H.-C. Kao, Phys. Rev. E 90, 022919 (2014)

work page 2014

[35] [35]

Omelchenko, A

I. Omelchenko, A. Zakharova, P. H¨ ovel, J. Siebert, and E. Sch¨ oll, Chaos25, 083104 (2015)

work page 2015

[36] [36]

J. Xie, E. Knobloch, and H.-C. Kao, Phys. Rev. E 92, 042921 (2015)

work page 2015

[37] [37]

Suda and K

Y. Suda and K. Okuda, Phys. Rev. E 97, 042212 (2018)

work page 2018

[38] [38]

O. E. Omel’chenko, M. Wolfrum, and C. R. Laing, Chaos 24, 023102 (2014)

work page 2014

[39] [39]

Wolfrum, S

M. Wolfrum, S. V. Gurevich, and O. E. Omelchenko, Nonlinearity 29, 257 (2016)

work page 2016

[40] [40]

O. E. Omelchenko, Nonlinearity 31, R121 (2018)

work page 2018

[41] [41]

Hinrichsen, Adv

H. Hinrichsen, Adv. Phys. 49, 815 (2000)

work page 2000

[42] [42]

K. A. Takeuchi, M. Kuroda, H. Chat´ e, and M. Sano, Phys. Rev. Lett. 99, 234503 (2007)

work page 2007

[43] [43]

Lemoult, L

G. Lemoult, L. Shi, K. Avila, S. V. Jalikop, M. Avila, and B. Hof, Nat. Phys. 12, 254 (2016)

work page 2016

[44] [44]

Sano and K

M. Sano and K. Tamai, Nat. Phys. 12, 249 (2016)

work page 2016

[45] [45]

Essam, J

J. Essam, J. Phys. A: Math. Gen. 22, 4927 (1989)

work page 1989

[46] [46]

R. M. Ziﬀ, E. Gulari, and Y. Barshad, Phys. Rev. Lett. 56, 2553 (1986)

work page 1986

[47] [47]

Mollison, J

D. Mollison, J. R. Stat. Soc. Series B Stat. Methodol. 39, 283 (1977). 4 FIG. 1. Spatio-temporal patterns of the phase diﬀerence for (a) α = 0.43, (b) α = 0.44, and (c) α = 0.45 in an N = 512 system. They are dominated by synchronized, traveling waves and multichimera states in the late stage, respectively. The spatial proﬁles of|∆φx| in (d) and (e) are t...

work page 1977