Nonlinear systems possessing nonattracting chaotic sets, such as chaotic saddles, embedded in their state space may oscillate chaotically for a transient time before eventually transitioning into some stable attractor. We show that these systems, when networked with nonlocal coupling in a ring, are capable of forming chimera states, in which one subset of the units oscillates periodically in a synchronized state forming the coherent domain, while the complementary subset oscillates chaotically in the neighborhood of the chaotic saddle constituting the incoherent domain. We find two distinct transient chimera states distinguished by their abrupt or gradual termination. We analyze the lifetime of both chimera states, unraveling their dependence on coupling range and size. We find an optimal value for the coupling range yielding the longest lifetime for the chimera states. Moreover, we implement transversal stability analysis to demonstrate that the synchronized state is asymptotically stable for network configurations studied here.
REFERENCES
1.
Y.
Kuramoto
and D.
Battogtokh
, “Coexistence of coherence and incoherence in nonlocally coupled phase oscillators
,” Nonlinear Phenom. Complex Syst.
5
, 380
–385
(2002
).2.
D.
Abrams
and S.
Strogatz
, “Chimera states for coupled oscillators
,” Phys. Rev. Lett.
93
, 174102
(2004
). 3.
C. R.
Laing
and C. C.
Chow
, “Stationary bumps in networks of spiking neurons
,” Neural Comput.
13
, 1473
–1494
(2001
). 4.
C. R.
Laing
, “Exact neural fields incorporating gap junctions
,” SIAM J. Appl. Dyn. Syst.
14
, 1899
–1929
(2015
). 5.
H.
Schmidt
and D.
Avitabile
, “Bumps and oscillons in networks of spiking neurons
,” Chaos
30
, 033133
(2020
). 6.
I.
Franović
, O. E.
Omel’chenko
, and M.
Wolfrum
, “Bumps, chimera states, and Turing patterns in systems of coupled active rotators
,” Phys. Rev. E
104
, L052201
(2021
). 7.
M. J.
Panaggio
and D. M.
Abrams
, “Chimera states: Coexistence of coherence and incoherence in networks of coupled oscillators
,” Nonlinearity
28
, R67
–R87
(2015
). 8.
E.
Schöll
, “Synchronization patterns and chimera states in complex networks: Interplay of topology and dynamics
,” Eur. Phys. J. Spec. Top.
225
, 891
–919
(2016
). 9.
F. P.
Kemeth
, S. W.
Haugland
, L.
Schmidt
, I. G.
Kevrekidis
, and K.
Krischer
, “A classification scheme for chimera states
,” Chaos
26
, 094814
(2016
). 10.
O. E.
Omel’chenko
, “The mathematics behind chimera states
,” Nonlinearity
31
, R121
–R164
(2018
). 11.
F.
Parastesh
, S.
Jafari
, H.
Azarnoush
, Z.
Shahriari
, Z.
Wang
, S.
Boccaletti
, and M.
Perc
, “Chimeras
,” Phys. Rep.
898
, 1
–114
(2021
). 12.
M.
Wolfrum
, O. E.
Omel’chenko
, S.
Yanchuk
, and Y. L.
Maistrenko
, “Spectral properties of chimera states
,” Chaos
21
, 013112
(2011
). 13.
M.
Wolfrum
and O. E.
Omel’chenko
, “Chimera states are chaotic transients
,” Phys. Rev. E
84
, 015201
(2011
). 14.
O. E.
Omel’chenko
, M.
Wolfrum
, and Y. L.
Maistrenko
, “Chimera states as chaotic spatiotemporal patterns
,” Phys. Rev. E
81
, 065201
(2010
). 15.
I.
Omelchenko
, Y.
Maistrenko
, P.
Hövel
, and E.
Schöll
, “Loss of coherence in dynamical networks: Spatial chaos and chimera states
,” Phys. Rev. Lett.
106
, 234102
(2011
). 16.
C.
Gu
, G.
St-Yves
, and J.
Davidsen
, “Spiral wave chimeras in complex oscillatory and chaotic systems
,” Phys. Rev. Lett.
111
, 134101
(2013
). 17.
M. R.
Tinsley
, S.
Nkomo
, and K.
Showalter
, “Chimera and phase-cluster states in populations of coupled chemical oscillators
,” Nat. Phys.
8
, 662
–665
(2012
). 18.
S.
Nkomo
, M. R.
Tinsley
, and K.
Showalter
, “Chimera states in populations of nonlocally coupled chemical oscillators
,” Phys. Rev. Lett.
110
, 244102
(2013
). 19.
J. F.
Totz
, J.
Rode
, M. R.
Tinsley
, K.
Showalter
, and H.
Engel
, “Spiral wave chimera states in large populations of coupled chemical oscillators
,” Nat. Phys.
14
, 282
–285
(2018
). 20.
M.
Wickramasinghe
and I. Z.
Kiss
, “Spatially organized dynamical states in chemical oscillator networks: Synchronization, dynamical differentiation, and chimera patterns
,” PLoS One
8
, e80586
(2013
). 21.
L.
Schmidt
, K.
Schönleber
, K.
Krischer
, and V.
García-Morales
, “Coexistence of synchrony and incoherence in oscillatory media under nonlinear global coupling
,” Chaos
24
, 013102
(2014
). 22.
E. A.
Martens
, S.
Thutupalli
, A.
Fourriere
, and O.
Hallatschek
, “Chimera states in mechanical oscillator networks
,” Proc. Natl. Acad. Sci. U.S.A.
110
, 10563
–10567
(2013
). 23.
T.
Kapitaniak
, P.
Kuzma
, J.
Wojewoda
, K.
Czolczynski
, and Y.
Maistrenko
, “Imperfect chimera states for coupled pendula
,” Sci. Rep.
4
, 6379
(2014
). 24.
A. M.
Hagerstrom
, T. E.
Murphy
, R.
Roy
, P.
Hövel
, I.
Omelchenko
, and E.
Schöll
, “Experimental observation of chimeras in coupled-map lattices
,” Nat. Phys.
8
, 658
–661
(2012
). 25.
D.
Dudkowski
, Y.
Maistrenko
, and T.
Kapitaniak
, “Different types of chimera states: An interplay between spatial and dynamical chaos
,” Phys. Rev. E
90
, 032920
(2014
). 26.
D.
Dudkowski
, Y.
Maistrenko
, and T.
Kapitaniak
, “Occurrence and stability of chimera states in coupled externally excited oscillators
,” Chaos
26
, 116306
(2016
). 27.
A.
Mishra
, C.
Hens
, M.
Bose
, P. K.
Roy
, and S. K.
Dana
, “Chimeralike states in a network of oscillators under attractive and repulsive global coupling
,” Phys. Rev. E
92
, 062920
(2015
). 28.
M. G.
Clerc
, S. C. M.
A. Ferré
, M. A.
García-Ñustes
, and R. G.
Rojas
, “Chimera-type states induced by local coupling
,” Phys. Rev. E
93
, 052204
(2016
). 29.
J. D.
Skufca
, J. A.
Yorke
, and B.
Eckhardt
, “Edge of chaos in a parallel shear flow
,” Phys. Rev. Lett.
96
, 174101
(2006
). 30.
B.
Eckhardt
, T. M.
Schneider
, B.
Hof
, and J.
Westerweel
, “Turbulence transition in pipe flow
,” Annu. Rev. Fluid Mech.
39
, 447
–468
(2007
). 31.
E. L.
Rempel
and A. C.-L.
Chian
, “Origin of transient and intermittent dynamics in spatiotemporal chaotic systems
,” Phys. Rev. Lett.
98
, 014101
(2007
). 32.
M.
Joglekar
, U.
Feudel
, and J. A.
Yorke
, “Geometry of the edge of chaos in a low-dimensional turbulent shear flow model
,” Phys. Rev. E
91
, 052903
(2015
). 33.
G.
Ansmann
, K.
Lehnertz
, and U.
Feudel
, “Self-induced switchings between multiple space-time patterns on complex networks of excitable units
,” Phys. Rev. X
6
, 011030
(2016
). 34.
T.
Lilienkamp
, J.
Christoph
, and U.
Parlitz
, “Features of chaotic transients in excitable media governed by spiral and scroll waves
,” Phys. Rev. Lett.
119
, 054101
(2017
). 35.
V.
Lucarini
and T.
Bódai
, “Transitions across melancholia states in a climate model: Reconciling the deterministic and stochastic points of view
,” Phys. Rev. Lett.
122
, 158701
(2019
). 36.
Y.-C.
Lai
and T.
Tél
, Transient Chaos: Complex Dynamics on Finite Time Scales
(Springer Science & Business Media
, 2011
), Vol. 173.37.
J. P.
Crutchfield
and K.
Kaneko
, “Are attractors relevant to turbulence?
,” Phys. Rev. Lett.
60
, 2715
–2718
(1988
). 38.
Y.-C.
Lai
and R. L.
Winslow
, “Geometric properties of the chaotic saddle responsible for supertransients in spatiotemporal chaotic systems
,” Phys. Rev. Lett.
74
, 5208
–5211
(1995
). 39.
E. S.
Medeiros
, R. O.
Medrano-T
, I. L.
Caldas
, and U.
Feudel
, “Boundaries of synchronization in oscillator networks
,” Phys. Rev. E
98
, 030201
(2018
). 40.
E. S.
Medeiros
, R. O.
Medrano-T
, I. L.
Caldas
, T.
Tél
, and U.
Feudel
, “State-dependent vulnerability of synchronization
,” Phys. Rev. E
100
, 052201
(2019
). 41.
E. S.
Medeiros
, R. O.
Medrano-T
, I. L.
Caldas
, and U.
Feudel
, “The impact of chaotic saddles on the synchronization of complex networks of discrete-time units
,” J. Phys.: Complex.
2
, 035002
(2021
). 42.
L.
Pecora
, T.
Carroll
, G.
Johnson
, D.
Mar
, and K. S.
Fink
, “Synchronization stability in coupled oscillator arrays: Solution for arbitrary configurations
,” Int. J. Bifurcat. Chaos
10
, 273
–290
(2000
). 43.
C.
Mitra
, A.
Choudhary
, S.
Sinha
, J.
Kurths
, and R. V.
Donner
, “Multiple-node basin stability in complex dynamical networks
,” Phys. Rev. E
95
, 032317
(2017
). 44.
A.
Wolf
, J. B.
Swift
, H. L.
Swinney
, and J. A.
Vastano
, “Determining Lyapunov exponents from a time series
,” Physica D
16
, 285
–317
(1985
). 45.
M.
Wolfrum
and O. E.
Omel’chenko
, “Chimera states are chaotic transients
,” Phys. Rev. E
84
, 015201
(2011
). 46.
T.
Lilienkamp
and U.
Parlitz
, “Susceptibility of transient chimera states
,” Phys. Rev. E
102
, 032219
(2020
). © 2023 Author(s). Published under an exclusive license by AIP Publishing.
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