We disclose a new class of patterns, called patched patterns, in arrays of non-locally coupled excitable units with attractive and repulsive interactions. The self-organization process involves the formation of two types of patches, majority and minority ones, characterized by uniform average spiking frequencies. Patched patterns may be temporally periodic, quasiperiodic, or chaotic, whereby chaotic patterns may further develop interfaces comprised of units with average frequencies in between those of majority and minority patches. Using chaos and bifurcation theory, we demonstrate that chaos typically emerges via a torus breakup and identify the secondary bifurcation that gives rise to chaotic interfaces. It is shown that the maximal Lyapunov exponent of chaotic patched patterns does not decay, but rather converges to a finite value with system size. Patched patterns with a smaller wavenumber may exhibit diffusive motion of chaotic interfaces, similar to that of the incoherent part of chimeras.
REFERENCES
1.
Y.
Kuramoto
and D.
Battogtokh
, Nonlinear Phenom. Complex Syst.
5
, 380
(2002
), available at http://www.j-npcs.org/abstracts/vol2002/v5no4/v5no4p380.html.2.
D. M.
Abrams
and S. H.
Strogatz
, Phys. Rev. Lett.
93
, 174102
(2004
). 3.
D. M.
Abrams
, R.
Mirollo
, S. H.
Strogatz
, and D. A.
Wiley
, Phys. Rev. Lett.
101
, 084103
(2008
). 4.
A.
Zakharova
, Chimera Patterns in Networks: Interplay Between Dynamics, Structure, Noise, and Delay—Understanding Complex Systems
(Springer Nature
, Switzerland
, 2020
).5.
F.
Parastesh
, S.
Jafari
, H.
Azarnoush
, Z.
Shahriari
, Z.
Wang
, S.
Boccaletti
, and M.
Perc
, Phys. Rep.
898
, 1
(2021
). 6.
M. J.
Panaggio
and D. M.
Abrams
, Nonlinearity
28
, R67
(2015
). 7.
S. W.
Haugland
, J. Phys.: Complexity
2
, 032001
(2021
). 8.
O. E.
Omel’chenko
and E.
Knobloch
, New J. Phys.
21
, 093034
(2019
). 9.
O. E.
Omel’chenko
, Nonlinearity
31
, R121
(2018
). 10.
C. R.
Laing
, Physica D
240
, 1960
(2011
). 11.
O. E.
Omel’chenko
, Nonlinearity
26
, 2469
(2013
). 12.
M.
Wolfrum
, O. E.
Omel’chenko
, S.
Yanchuk
, and Y. L.
Maistrenko
, Chaos
21
, 013112
(2011
). 13.
M.
Wolfrum
and O. E.
Omel’chenko
, Phys. Rev. E
84
, 015201(R)
(2011
). 14.
O. E.
Omel’chenko
, M.
Wolfrum
, and Y. L.
Maistrenko
, Phys. Rev. E
81
, 065201(R)
(2010
). 15.
E. M.
Izhikevich
, Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting
(MIT Press
, Cambridge, MA
, 2007
).16.
B.
Lindner
, J.
García-Ojalvo
, A.
Neiman
, and L.
Schimansky-Geier
, Phys. Rep.
392
, 321
(2004
). 17.
I.
Franović
, S.
Yanchuk
, S.
Eydam
, I.
Bačić
, and M.
Wolfrum
, Chaos
30
, 083109
(2020
). 18.
I.
Franović
, K.
Todorović
, M.
Perc
, N.
Vasović
, and N.
Burić
, Phys. Rev. E
92
, 062911
(2015
). 19.
I.
Franović
, K.
Todorović
, N.
Vasović
, and N.
Burić
, Phys. Rev. Lett.
108
, 094101
(2012
). 20.
S.
Alonso
and M.
Bär
, Phys. Rev. Lett.
110
, 158101
(2013
). 21.
S.
Scialla
, A.
Loppini
, M.
Patriarca
, and E.
Heinsalu
, Phys. Rev. E
103
, 052211
(2021
). 22.
J. P.
Keener
and J.
Sneyd
, Mathematical Physiology
(Springer
, New York
, 2009
).23.
Z.
Wang
, Z.
Rostami
, S.
Jafari
, F. E.
Alsaadi
, M.
Slavinec
, and M.
Perc
, Chaos, Solitons Fractals
128
, 229
(2019
). 24.
A. S.
Mikhailov
and K.
Showalter
, Phys. Rep.
425
, 79
(2006
). 25.
D. V.
Alexandrov
, I. A.
Bashkirtseva
, M.
Crucifix
, and L. B.
Ryashko
, Phys. Rep.
902
, 1
(2021
). 26.
S.
Terrien
, V. A.
Pammi
, B.
Krauskopf
, N. G. R.
Broderick
, and S.
Barbay
, Phys. Rev. E
103
, 012210
(2021
). 27.
A.
Ceni
, P.
Ashwin
, and L.
Livi
, Cogn. Comput.
12
, 330
(2019
). 28.
R.
Ronge
and M. A.
Zaks
, Phys. Rev. E
103
, 012206
(2021
). 29.
R.
Ronge
and M. A.
Zaks
, Eur. Phys. J. Spec. Top.
230
, 2717
(2021
). 30.
P.
Ashwin
and O.
Burylko
, Chaos
25
, 013106
(2015
). 31.
M.
Owen
, C.
Laing
, and S.
Coombes
, New J. Phys.
9
, 378
(2007
). 32.
C. R.
Laing
, Front. Comput. Neurosci.
10
, 53
(2016
). 33.
C. R.
Laing
and O. E.
Omel’chenko
, Chaos
30
, 043117
(2020
). 34.
P. C.
Bressloff
, J. Phys. A Math. Theor.
45
, 033001
(2012
). 35.
K.
Zhang
, J. Neurosci.
16
, 2112
(1996
). 36.
I.
Franović
, O. E.
Omel’chenko
, and M.
Wolfrum
, Phys. Rev. E
104
, L052201
(2021
). 37.
I.
Franović
, S. R.
Eydam
, N.
Semenova
, and A.
Zakharova
, Chaos
32
, 011104
(2022
). 38.
N.
Semenova
, A.
Zakharova
, V.
Anishchenko
, and E.
Schöll
, Phys. Rev. Lett.
117
, 014102
(2016
). 39.
A.
Zakharova
, N.
Semenova
, V.
Anishchenko
, and E.
Schöll
, Chaos
27
, 114320
(2017
). 40.
I.
Omelchenko
, O. E.
Omel’chenko
, P.
Hövel
, and E.
Schöll
, Phys. Rev. Lett.
110
, 224101
(2013
). 41.
S. M.
Baer
and T.
Erneux
, SIAM J. Appl. Math.
46
, 721
(1986
). 42.
I.
Tsuda
, Chaos
19
, 015113
(2009
). 43.
K.
Kaneko
and I.
Tsuda
, Chaos
13
, 926
(2003
). 44.
I.
Tsuda
and T.
Umemura
, Chaos
13
, 937
(2003
). 45.
I.
Franović
, O. E.
Omel’chenko
, and M.
Wolfrum
, Chaos
28
, 071105
(2018
). 46.
S. R.
Eydam
, I.
Franović
, and M.
Wolfrum
, Phys. Rev. E
99
, 042207
(2019
). 47.
M.
Wolfrum
, O. E.
Omel’chenko
, and J.
Sieber
, Chaos
25
, 053113
(2015
). 48.
T.
Nomura
and L.
Glass
, Phys. Rev. E
53
, 6353
(1996
). 49.
A.
Pikovsky
and A.
Politi
, Lyapunov Exponents: A Tool to Explore Complex Dynamics
(Cambridge University Press
, Cambridge
, 2016
).50.
G.
Benettin
, L.
Galgani
, and J.-M.
Strelcyn
, Phys. Rev. A
14
, 2338
(1976
). 51.
I.
Omelchenko
, A.
Zakharova
, P.
Hövel
, J.
Siebert
, and E.
Schöll
, Chaos
25
, 083104
(2015
). 52.
Y. L.
Maistrenko
, A.
Vasylenko
, O.
Sudakov
, R.
Levchenko
, and V. L.
Maistrenko
, Int. J. Bifurcat. Chaos
24
, 1440014
(2014
). 53.
Y.
Kawamura
, Phys. Rev. E
75
, 056204
(2007
). 54.
H.
Schmidt
and D.
Avitabile
, Chaos
30
, 033133
(2020
). 55.
D. V.
Kasatkin
, V. V.
Klinshov
, and V. I.
Nekorkin
, Phys. Rev. E
99
, 022203
(2019
). 56.
O.
Omel’chenko
and C. R.
Laing
, Proc. R. Soc. A
478
, 20210817
(2022
). 57.
I.
Omelchenko
, Y.
Maistrenko
, P.
Hövel
, and E.
Schöll
, Phys. Rev. Lett.
106
, 234102
(2011
). © 2022 Author(s). Published under an exclusive license by AIP Publishing.
2022
Author(s)
You do not currently have access to this content.
