We investigate the basin of attraction properties and its boundaries for chimera states in a circulant network of Hénon maps. It is known that coexisting basins of attraction lead to a hysteretic behaviour in the diagrams of the density of states as a function of a varying parameter. Chimera states, for which coherent and incoherent domains occur simultaneously, emerge as a consequence of the coexistence of basin of attractions for each state. Consequently, the distribution of chimera states can remain invariant by a parameter change, and it can also suffer subtle changes when one of the basins ceases to exist. A similar phenomenon is observed when perturbations are applied in the initial conditions. By means of the uncertainty exponent, we characterise the basin boundaries between the coherent and chimera states, and between the incoherent and chimera states. This way, we show that the density of chimera states can be not only moderately sensitive but also highly sensitive to initial conditions. This chimera’s dilemma is a consequence of the fractal and riddled nature of the basin boundaries.
References
1.
Y.
Kuramoto
and D.
Battogtokh
, “Coexistence of coherence and incoherence in nonlocally coupled phase oscillators
,” Nonlinear Phenom. Complex Syst.
5
, 380
(2002
).2.
K.
Umberger
, C.
Grebogi
, E.
Ott
, and B.
Afeyan
, “Spatio-temporal dynamics in a dispersively coupled chain of nonlinear oscillators
,” Phys. Rev. A
39
, 4835
(1989
).3.
D. M.
Abrams
and S. H.
Strogatz
, “Chimera states for coupled oscillators
,” Phys. Rev. Lett.
93
, 174102
(2004
).4.
D. M.
Abrams
, R.
Mirollo
, S. H.
Strogatz
, and D. A.
Wiley
, “Solvable model for chimera states of coupled oscillators
,” Phys. Rev. Lett.
101
, 084103
(2008
).5.
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
).6.
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
).7.
O. E.
Omel’chenko
, Y. L.
Maistrenko
, and P. A.
Tass
, “Chimera states: The natural link between coherence and incoherence
,” Phys. Rev. Lett.
100
, 044105
(2008
).8.
R. G.
Andrzejak
, G.
Ruzzene
, and I.
Malvestio
, “Generalized synchronization between chimera states
,” Chaos
27
, 053114
(2017
).9.
M. S.
Santos
, J. D.
Szezech Jr
, A. M.
Batista
, I. L.
Caldas
, R. L.
Viana
, and S. R.
Lopes
, “Recurrence quantification analysis of chimera states
,” Phys. Lett. A
379
, 2188
(2015
).10.
N.
Yao
, Z.-G.
Huang
, C.
Grebogi
, and Y.-C.
Lai
, “Emergence of multicluster chimera states
,” Sci. Rep.
5
, 12988
(2015
).11.
M. S.
Santos
, J. D.
Szezech
, F. S.
Borges
, K. C.
Iarosz
, I. L.
Caldas
, A. M.
Batista
, R. L.
Viana
, and J.
Kurths
, “Chimera-like states in a neuronal network model of the cat brain
,” Chaos Solitons Fractals
101
, 86
(2017
).12.
D.
Dudkowski
, Y.
Maistrenko
, and T.
Kapitaniak
, “Occurrence and stability of chimera states in coupled externally excited oscillators
,” Chaos
26
, 116306
(2016
).13.
M. R.
Tinsley
, S.
Nkomo
, and K.
Showalter
, “Chimera and phase-cluster states in populations of coupled chemical oscillators
,” Nat. Phys.
8
, 662
(2012
).14.
E. A.
Martens
, S.
Thutupalli
, A.
Fourrière
, and O.
Hallatschek
, “Chimera states in mechanical oscillator networks
,” Proc. Natl. Acad. Sci.
110
, 10563
(2013
).15.
T.
Kapitaniak
, P.
Kuzma
, J.
Wojewoda
, K.
Czolczynski
, and Y.
Maistrenko
, “Imperfect chimera states for coupled pendula
,” Sci. Rep.
4
, 6379
(2014
).16.
L. B.
Gambuzza
, A.
Buscarino
, S.
Chessari
, L.
Fortuna
, R.
Meucci
, and M.
Frasca
, “Experimental investigation of chimera states with quiescent and synchronous domains in coupled electronic oscillators
,” Phys. Rev. E
90
, 032905
(2014
).17.
E. A.
Martens
, M. J.
Panaggio
, and D. M.
Abrams
, “Basins of attraction for chimera states
,” New J. Phys.
18
, 022002
(2016
).18.
S.
Rakshit
, B. K.
Bera
, M.
Perc
, and D.
Ghosh
, “Basin stability for chimera states
,” Sci. Rep.
7
, 2412
(2017
).19.
M.
Hénon
, “Numerical study of quadratic area-preserving mappings
,” Quart. Appl. Math.
27
, 291
(1969
).20.
A.
Politi
and A.
Torcini
, “Periodic orbits in coupled Hénon maps: Lyapunov and multifractal analysis
,” Chaos
2
, 293
(1992
).21.
V.
Astakhov
, A.
Shabunin
, W.
Uhm
, S.
Kim
, “Multistability formation, synchronization loss in coupled Hénon maps: Two sides of the single bifurcational mechanism
,” Phys. Rev. E
63
, 056212
(2001
).22.
V.
Santos
, J. D.
Szezech Jr
, M. S.
Baptista
, A. M.
Batista
, and I. L.
Caldas
, “Unstable dimension variability structure in the parameter space of coupled Hénon maps
,” Appl. Math. Comput.
286
, 23
(2016
).23.
N. I.
Semenova
, G. I.
Strelkova
, V. S.
Anishchenko
, and A.
Zakharova
, “Temporal intermittency and the lifetime of chimera states in ensembles of nonlocally coupled chaotic oscillators
,” Chaos
27
, 061102
(2017
).24.
J. C.
Alexander
, J. A.
Yorke
, Z.
You
, and I.
Kan
, “Riddled basins
,” Int. J. Bifurcat. Chaos
2
, 795
(1992
).25.
E.
Ott
, J. C.
Sommerer
, J. C.
Alexander
, I.
Kan
, and J. A.
Yorke
, “Scaling behavior of chaotic systems with riddled basins
,” Phys. Rev. Lett.
71
, 4134
(1993
).26.
P.
Ashwin
, J.
Buescu
, and I.
Stewart
, “Bubbling of attractors and synchronisation of chaotic oscillators
,” Phys. Lett. A
193
, 126
(1994
).27.
J. F.
Heagy
, T. L.
Carrol
, and L. M.
Pecora
, “Experimental and numerical evidence for riddled basins in coupled chaotic systems
,” Phys. Rev. Lett.
73
, 3528
(1994
).28.
M.
Woltering
and M.
Markus
, “Riddled basins in a model for the Belousov-Zhabotinsky reaction
,” Chem. Phys. Lett.
321
, 473
(2000
).29.
K.
Kaneko
, “Overview of coupled map lattices
,” Chaos
2
, 279
(1992
).30.
N.
Semenova
, A.
Zakharova
, E.
Schöll
, and V.
Anishchenko
, “Does hyperbolicity impede emergence of chimera states in networks of nonlocally coupled chaotic oscillators?
,” Europhys. Lett.
112
, 40002
(2015
).31.
I.
Omelchenko
, B.
Riemenschneider
, P.
Hövel
, Y.
Maistrenko
, and E.
Schöll
, “Transition from spatial coherence to incoherence in coupled chaotic systems
,” Phys. Rev. E
85
, 026212
(2012
).32.
R.
Gopal
, V. K.
Chandrasekar
, A.
Venkatesan
, and M.
Lakshman
, “Observation and characterization of chimera states in coupled dynamical systems with nonlocal coupling
,” Phys. Rev. E
89
, 052914
(2014
).33.
P. J.
Menck
, J.
Heitzig
, N.
Marwan
, and J.
Kurths
, “How basin stability complements the linear-stability paradigm
,” Nat. Phys.
9
, 89
(2013
).34.
P. J.
Menck
, J.
Heitzig
, J.
Kurths
, and H. J.
Schellnhuber
, “How dead ends undermine power grid stability
,” Nat. Commun.
5
, 3969
(2014
).35.
P.
Schultz
, P. J.
Menck
, J.
Heitzig
, and J.
Kurths
, “Potentials and limits to basin stability estimation
,” New J. Phys.
19
, 023005
(2017
).36.
S. W.
McDonald
, C.
Grebogi
, E.
Ott
, and J. A.
Yorke
, “Fractal basin boundaries
,” Physica D
17
, 125
(1985
).37.
C.
Grebogi
, E.
Kostelich
, E.
Ott
, J. A.
Yorke
, “Multi-dimensioned intertwined basin boundaries: Basin structure of the kicked double rotor
,” Physica D
25
, 347
(1987
).38.
J.
Aguirre
, R. L.
Viana
, and M. A. F.
Sanjuán
, “Fractal structures in nonlinear dynamics
,” Rev. Mod. Phys.
81
, 333
(2009
).39.
E. S.
Medeiros
, I. L.
Caldas
, M. S.
Baptista
, and U.
Feudel
, “Trapping phenomenon attenuates the consequence of tipping points for limit cycles
,” Sci. Rep.
7
, 42351
(2017
).© 2018 Author(s).
2018
Author(s)
You do not currently have access to this content.
