We demonstrate that chimera behavior can be observed in ensembles of phase oscillators with unidirectional coupling. For a small network consisting of only three identical oscillators (cyclic triple), tiny chimera islands arise in the parameter space. They are surrounded by developed chaotic switching behavior caused by a collision of rotating waves propagating in opposite directions. For larger networks, as we show for a hundred oscillators (cyclic century), the islands merge into a single chimera continent, which incorporates the world of chimeras of different configurations. The phenomenon inherits from networks with intermediate ranges of the unidirectional coupling and it diminishes as the coupling range decreases.

1.
Y.
Kuramoto
and
D.
Battogtokh
, “
Coexistence of coherence and incoherence in nonlocally coupled phase oscillators
,”
Nonlinear Phenom. Complex Syst.
5
,
380
(
2002
).
2.
D. M.
Abrams
and
S. H.
Strogatz
, “
Chimera states for coupled oscillators
,”
Phys. Rev. Lett.
93
,
174102
(
2004
).
3.
M. J.
Panaggio
and
D. M.
Abrams
, “
Chimera states: Coexistence of coherence and incoherence in networks of coupled oscillators
,”
Nonlinearity
28
,
R67
(
2015
).
4.
E.
Schöll
, “
Synchronization patterns and chimera states in complex networks: Interplay of topology and dynamics
,”
Eur. Phys. J. Spec. Top.
225
,
891
919
(
2016
).
5.
O.
Omel’chenko
and
E.
Knobloch
, “
Chimerapedia: Coherence–incoherence patterns in one, two and three dimensions
,”
New J. Phys.
21
,
093034
(
2019
).
6.
A.
Zakharova
,
Chimera Patterns in Networks: Interplay between Dynamics, Structure, Noise, and Delay
(
Springer Nature
,
2020
).
7.
F.
Parastesh
,
S.
Jafari
,
H.
Azarnoush
,
Z.
Shahriari
,
Z.
Wang
,
S.
Boccaletti
, and
M.
Perc
, “
Chimeras
,”
Phys. Rep.
898
,
1
(
2021
).
8.
J.
Hizanidis
,
N. E.
Kouvaris
,
G.
Zamora-Lopez
,
A.
Diaz-Guilera
, and
C.
Antopoulos
, “
Chimera-like states in modular neural networks
,”
Sci. Rep.
6
,
19845
(
2016
).
9.
N. C.
Rattenborg
,
C. J.
Amlaner
, and
S. L.
Lima
, “
Behavioral, neurophysiological and evolutionary perspectives on unihemispheric sleep
,”
Neurosci. Biobehav. Rev.
24
,
817
(
2000
).
10.
A.
Rothkegel
and
K.
Lehnertz
, “
Irregular macroscopic dynamics due to chimera states in small-world networks of pulse-coupled oscillators
,”
New J. Phys.
16
,
055006
(
2014
).
11.
Z.
Wang
and
Z.
Liu
, “
A brief review of chimera state in empirical brain networks
,”
Front. Physiol.
11
,
724
(
2020
).
12.
A. E.
Motter
,
S. A.
Myers
,
M.
Angel
, and
T.
Nishikawa
, “
Spontaneous synchrony in power-grid networks
,”
Nat. Phys.
9
,
191
(
2013
).
13.
L. M.
Pecora
,
F.
Sorrentino
,
A.
Hagerstrom
,
T.
Murphy
, and
R.
Roy
, “
Cluster synchronization and isolated desynchronization in complex networks with symmetries
,”
Nat. Commun.
5
,
4079
(
2014
).
14.
N.
Lazarides
and
G.
Tsironis
, “
Superconducting metamaterials
,”
Phys. Rep.
752
,
1
67
(
2018
).
15.
I.
Belykh
,
R.
Jeter
, and
V.
Belykh
, “
Foot force models of crowd dynamics on a wobbly bridge
,”
Sci. Adv.
3
,
e1701512
(
2017
).
16.
J.
Hart
,
L.
Larger
,
T.
Murphy
, and
R.
Roy
, “
Delayed dynamical systems: Networks, chimeras and reservoir computing
,”
Phil. Trans. R. Soc. A
377
,
20180123
(
2019
).
17.
F.
Hellmann
,
P.
Schultz
,
P.
Jaros
,
R.
Levchenko
,
T.
Kapitaniak
, and
J.
Kurths
, “
Network-induced multistability through lossy coupling and exotic solitary states
,”
Nat. Commun.
11
,
592
(
2020
).
18.
H.
Taher
,
S.
Olmi
, and
E.
Schöll
, “
Enhancing power grid synchronization and stability through time-delayed feedback control
,”
Phys. Rev. E
100
,
062306
(
2019
).
19.
J. C.
Gonzales-Avella
,
M. G.
Cosenza
, and
M. S.
Miguel
, “
Localized coherence in two interacting populations of social agents
,”
Physica A
399
,
24
(
2014
).
20.
A.
Pikovsky
, “
Chimeras on a social-type network
,”
Math. Model. Nat. Phenom.
16
,
15
(
2021
).
21.
O.
Omelchenko
,
M.
Wolfrum
, and
Yu.
Maistrenko
, “
Chimera states as chaotic spatiotemporal patterns
,”
Phys. Rev. E
81
,
065201(R)
(
2010
).
22.
V.
Dziubak
,
V.
Maistrenko
,
Yu.
Maistrenko
, and
M.
Timme
(unpublished).
23.
J.
Xie
,
E.
Knobloch
, and
H. C.
Kao
, “
Multicluster and traveling chimera states in nonlocal phase-coupled oscillators
,”
Phys. Rev. E
90
,
022919
(
2014
).
24.
C.
Bick
and
E. A.
Martens
, “
Controlling chimeras
,”
New J. Phys.
17
,
033030
(
2015
).
25.
T.
Kapitaniak
,
P.
Kuzma
,
J.
Wojewoda
,
K.
Czolczynski
, and
Yu.
Maistrenko
, “
Imperfect chimera states for coupled pendula
,”
Sci. Rep.
4
,
6379
(
2014
).
26.
P.
Jaros
,
Y.
Maistrenko
, and
T.
Kapitaniak
, “
Chimera states on the route from coherence to rotating waves
,”
Phys. Rev. E
91
,
022907
(
2015
).
27.
Y.
Maistrenko
,
S.
Brezetsky
,
P.
Jaros
,
R.
Levchenko
, and
T.
Kapitaniak
, “
Smallest chimera states
,”
Phys. Rev. E
95
,
010203(R)
(
2017
).
28.
P.
Jaros
,
S.
Brezetsky
,
R.
Levchenko
,
D.
Dudkowski
,
T.
Kapitaniak
, and
Yu.
Maistrenko
, “
Solitary states for coupled oscillators with inertia
,”
Chaos
28
,
011103
(
2018
).
29.
S.
Brezetsky
,
P.
Jaros
,
R.
Levchenko
,
T.
Kapitaniak
, and
Yu.
Maistrenko
, “
Chimera complexity
,”
Phys. Rev. E
103
,
L050204
(
2021
).
30.
I.
Belykh
,
B.
Brister
, and
V.
Belykh
, “
Bistability of patterns of synchrony in Kuramoto oscillators with inertia
,”
Chaos
26
,
094822
(
2016
);
[PubMed]
B. N.
Brister
,
V. N.
Belykh
,
I. V.
Belykh
, “
When three is a crowd: Chaos from clusters of Kuramoto oscillators with inertia
,”
Phys. Rev. E
101
,
062206
(
2020
).
[PubMed]
31.
P.
Ashwin
and
O.
Burylko
, “
Weak chimeras in minimal networks of coupled phase oscillators
,”
Chaos
25
,
013106
(
2015
).
32.
R.
Berner
,
A.
Polanska
,
E.
Schöll
, and
S.
Yanchuk
, “
Solitary states in adaptive nonlocal oscillator networks
,”
Eur. Phys. J. Spec. Top.
229
,
2183
(
2020
).
33.
L.
Schülen
,
D.
Janzen
,
E.
Medeiros
, and
A.
Zakharova
, “
Solitary states in multiplex neural networks: Onset and vulnerability
,”
Chaos Soliton. Fract.
145
,
110670
(
2021
).
34.
V.
Maistrenko
,
O.
Sudakov
, and
Yu.
Maistrenko
, “
Spiral wave chimeras for coupled oscillators with inertia
,”
Eur. Phys. J. Spec. Top.
229
,
2327
(
2020
).
35.
V.
Maistrenko
,
O.
Sudakov
, and
O.
Osiv
, “
Chimera and solitary states in 3D oscillator networks with inertia
,”
Chaos
30
,
063113
(
2020
).
36.
N.
Kruk
,
Yu.
Maistrenko
, and
H.
Koeppl
, “
Solitary states in the mean-field limit
,”
Chaos
30
,
111104
(
2020
).
37.
S.
Olmi
,
A.
Navas
,
S.
Boccaletti
, and
A.
Torcini
, “
Hysteretic transitions in the Kuramoto model with inertia
,”
Phys. Rev. E
90
,
042905
(
2014
);
S.
Olmi
,
E. A.
Martens
,
S.
Thutupalli
, and
A.
Torcini
, “
Intermittent chaotic chimeras for coupled rotators
,”
ibid.
92
,
030901(R)
(
2015
);
S.
Olmi
, “
Chimera states in coupled Kuramoto oscillators with inertia
,”
Chaos
25
,
123125
(
2015
).
[PubMed]
38.
Y.
Zhang
,
Z. G.
Nicolaou
,
J. D.
Hart
,
R.
Roy
, and
A. E.
Motter
, “
Critical switching in globally attractive chimeras
,”
Phys. Rev. X
10
,
011044
(
2020
).
39.
This kind of chaotic switching dynamics may be associated with intermittent chimeras identified in Ref. 37 for two-population Kuramoto model with inertia. Developed switching behavior between chaotic chimera states was reported recently in Refs. 29 and 38.
40.
D.
Oliveira
,
M.
Robnik
, and
E.
Leonel
, “
Shrimp-shape domains in a dissipative kicked rotator
,”
Chaos
21
,
043122
(
2011
).
41.
V.
Klinshov
,
D.
Shchapin
,
S.
Yanchuk
,
M.
Wolfrum
,
O.
D’Huys
, and
V.
Nekorkin
, “
Embedding the dynamics of a single delay system into a feed-forward ring
,”
Phys. Rev. E
96
,
042217
(
2017
).
42.
J. D.
Hart
,
L.
Larger
,
T. E.
Murphy
, and
R.
Roy
, “
Delayed dynamical systems: Networks, chimeras and reservoir computing
,”
Phil. Trans. R. Soc. A
377
,
20180123
(
2019
).
43.
V.
Semenov
,
A.
Zakharova
,
Y.
Maistrenko
, and
E.
Schöll
, “
Delayed-feedback chimera states: Forced multiclusters and stochastic resonance
,”
Europhys. Lett.
115
,
10005
(
2016
).
44.
Y.-S.
Park
et al.
Linear analysis of feedforward ring oscillators
,”
IEICE Trans. Electron E93-C
9
,
1467
1470
(
2010
).
45.
D.
Brunner
,
B.
Penkovsky
,
R.
Levchenko
,
E.
Schöll
,
L.
Larger
, and
Y.
Maistrenko
, “
Two-dimensional spatiotemporal complexity in dual-delayed nonlinear feedback systems: Chimeras and dissipative solitons
,”
Chaos
28
,
103106
(
2018
).
46.
V.
Semenov
and
Yu.
Maistrenko
, “
Dissipative solitons for bistable delayed-feedback systems
,”
Chaos
28
,
101103
(
2018
).
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