Labyrinth chaos was discovered by Otto Rössler and René Thomas in their endeavor to identify the necessary mathematical conditions for the appearance of chaotic and hyperchaotic motion in continuous flows. Here, we celebrate their discovery by considering a single labyrinth walk system and an array of coupled labyrinth chaos systems that exhibit complex, chaotic behavior, reminiscent of chimera-like states, a peculiar synchronization phenomenon. We discuss the properties of the single labyrinth walk system and review the ability of coupled labyrinth chaos systems to exhibit chimera-like states due to the unique properties of their space-filling, chaotic trajectories, which amounts to elegant, hyperchaotic walks. Finally, we discuss further implications in relation to the labyrinth walk system by showing that even though it is volume-preserving, it is not force-conservative.
REFERENCES
1.
R.
Abraham
and J. E.
Marsden
, Foundations of Mechanics
, 2nd ed. (Addison-Wesley
, 1987
).2.
D. M.
Abrams
and S. H.
Strogatz
, “Chimera states for coupled oscillators
,” Phys. Rev. Lett.
93
(17
), 174102
(2004
). 3.
A. V.
Andreev
, A. G.
Balanov
, T. M.
Fromhold
, M. T.
Greenaway
, A. E.
Hramov
, W.
Li
, V. V.
Makarov
, and A. M.
Zagoskin
, “Chaos and hyperchaos in driven interacting quantum systems,” arXiv:quant-ph:1907.03602 (2020).4.
C. G.
Antonopoulos
, V.
Basios
, J.
Demongeot
, P.
Nardone
, and R.
Thomas
, “Linear and nonlinear arabesques: A study of closed chains of negative 2-element circuits
,” Int. J. Bifurcat. Chaos
23
, 1330033
(2013
). 5.
V. I.
Arnold
, V.
Kozlov
, A. I.
Neishtadt
, and E.
Khukhro
, Mathematical Aspects of Classical and Celestial Mechanics, 3rd ed., Encyclopaedia of Mathematical Sciences (Springer, 2010).6.
V.
Basios
and C. G.
Antonopoulos
, “Hyperchaos and labyrinth chaos: Revisiting Thomas–Rössler systems
,” J. Theor. Biol.
460
, 153
–159
(2019
). 7.
K. E.
Chlouverakis
and J. C.
Sprott
, “Hyperlabyrinth chaos: From chaotic walks to spatiotemporal chaos
,” Chaos
17
, 023110
(2007
). 8.
L.
Chunguang
and C.
Guanrong
, “Chaos and hyperchaos in the fractional-order Rössler equations
,” Physica A
341
, 55
–61
(2004
). 9.
T.
Dombre
, U.
Frisch
, J. M.
Greene
, M.
Hénon
, A.
Mehr
, and A. M.
Soward
, “Chaotic streamlines in the ABC flows
,” J. Fluid Mech.
167
, 353
–391
(1986
). 10.
S.
Friedlander
, A.
Gilbert
, and M.
Vishik
, “Hydrodynamic instability for certain ABC flows
,” Geophys. Astrophys. Fluid Dyn.
73
, 97
–107
(1993
). 11.
P.
Gaspard
and G.
Nicolis
, “What can we learn from homoclinic orbits in chaotic dynamics?
,” J. Stat. Phys.
31
, 499
–518
(1983
). 12.
13.
R.
Gilmore
, C.
Letellier
, and M.
Lefranc
, “Chaos topology
,” Scholarpedia
3
(7
), 4592
(2008
). 14.
15.
B. J.
Hiley
, “Active information and teleportation,” in Epistemological and Experimental Perspectives on Quantum Physics, edited by D. Greenberger et al. (Kluwer, the Netherlands, 1999), pp. 113–126.16.
B. J.
Hiley
, “From the Heisenberg picture to Bohm: A new perspective on active information and its relation to Shannon information,” in Quantum Theory: Reconsidering of Foundations, edited by A. Khrennikov (Växjö University Press, 2002), pp. 141–162.17.
E. M. S.
Hitzer
, “Imaginary eigenvalues and complex eigenvectors explained by real geometry,” in Applications of Geometric Algebra in Computer Science and Engineering, edited by L. Dorst, C. Doran, and J. Lasenby (Birkhäuser, Boston, MA, 2002).18.
J.
Hizanidis
, N.
Kouvaris
, G.
Zamora-López
, A. D.
Guilera
, and C. G.
Antonopoulos
, “Chimera-like states in modular neural networks
,” Sci. Rep.
6
, 19845
(2016
). 19.
J.
Hofbauer
and K.
Sigmund
, Evolutionary Games and Population Dynamics
(Cambridge University Press
, 1998
).20.
K.
Ikeda
and K.
Matsumoto
, “High-dimensional chaotic behavior in systems with time-delayed feedback
,” Physica D
29
(1–2
), 223
–235
(1987
). 21.
K.
Kaneko
, “From globally coupled maps to complex-systems biology
,” Chaos
25
, 097608
(2015
). 22.
M.
Kaufman
, C.
Soulé
, and R.
Thomas
, “A new necessary condition on interaction graphs for multistationarity
,” J. Theor. Biol.
248
, 675
–685
(2007
). 23.
M.
Kaufman
and R.
Thomas
, “Emergence of complex behavior from simple circuit structures
,” C. R. Acad. Sci. Paris Biol.
326
, 205
–214
(2003
).24.
Y.
Kuramoto
and D.
Battogtokh
, “Coexistence of coherence and incoherence in nonlocally coupled phase oscillators
,” Nonlinear Phenom. Complex Syst.
5
(4
), 380
–385
(2002
).25.
C.
Letellier
and O.
Rössler
, “Rössler attractor
,” Scholarpedia
1
(10
), 1721
(2006
). 26.
C.
Letellier
and O.
Rössler
, “Hyperchaos
,” Scholarpedia
2
(8
), 1936
(2007
). 27.
T.
Manos
, C.
Skokos
, and C. G.
Antonopoulos
, “Probing the local dynamics of periodic orbits by the generalized alignment index (GALI) method
,” Int. J. Bifurcat. Chaos
22
(09
), 1250218
(2012
). 28.
G. J.
Martyna
, M. L.
Klein
, and M.
Tuckerman
, “Nosé–Hoover chains: The canonical ensemble via continuous dynamics
,” J. Chem. Phys.
97
(4
), 2635
–2643
(1992
). 29.
G.
Nicolis
and V.
Basios
, Chaos, Information Processing and Paradoxical Games
(World Scientific
, 2015
).30.
G.
Nicolis
and C.
Nicolis
, Foundations of Complex Systems—Nonlinear Dynamics, Statistical Physics, Information and Prediction
(World Scientific
, 2010
).31.
D. D.
Nolte
, Introduction to Modern Dynamics: Chaos, Networks, Space and Time
(Oxford University Press
, 2015
).32.
M. J.
Panaggio
and D. M.
Abrams
, “Chimera states: Coexistence of coherence and incoherence in networks of coupled oscillators
,” Nonlinearity
28
(3
), R67
–R87
(2015
). 33.
F.
Parastesh
, S.
Jafari
, H.
Azarnoush
, B.
Hatef
, and A.
Bountis
, “Imperfect chimeras in a ring of four-dimensional simplified Lorenz systems
,” Chaos Solitons Fractals
110
, 203
–208
(2018
). 34.
O.
Rössler
, “An equation for hyperchaos
,” Phys. Lett. A
71
, 155
–157
(1979
). 35.
O.
Rössler
and C.
Letellier
, Chaos: The World of Nonperiodic Oscillations, Understanding Complex Systems Series (Springer, 2020).36.
A.
Schmidt
, T.
Kasimatis
, J.
Hizanidis
, A.
Provata
, and P.
Hövel
, “Chimera patterns in two-dimensional networks of coupled neurons
,” Phys. Rev. E
95
, 032224
(2017
). 37.
E.
Sigalas
, A. B. J.
Li
, and C. G.
Antonopoulos
, “Emergence of chimera-like states in prefrontal-cortex macaque intracranial recordings,” in International Workshop on Pattern Recognition in Neuroimaging (PRNI), Singapore (IEEE, 2018).38.
C.
Skokos
, C.
Antonopoulos
, T. C.
Bountis
, and M. N.
Vrahatis
, “Detecting order and chaos in Hamiltonian systems by the SALI method
,” J. Phys. A: Math. Gen.
36
, 6269
–6284
(2004
). 39.
C.
Skokos
, T. C.
Bountis
, and C.
Antonopoulos
, “Geometrical properties of local dynamics in Hamiltonian systems: The generalized alignment index (GALI) method
,” Physica D
231
, 30
–54
(2007
). 40.
C.
Soulé
, “Graphic requirements for multistationarity
,” ComPlexUs
1
, 123
–133
(2003
). 41.
J. C.
Sprott
, Elegant Chaos: Algebraically Simple Chaotic Flows
(World Scientific
, 2010
).42.
J. C.
Sprott
and K. E.
Chlouverakis
, “Labyrinth chaos
,” Int. J. Bifurcat. Chaos
17
(6
), 2097
–2108
(2007
). 43.
A. F.
Stevenson
, “Note on the existence and determination of a vector potential
,” Q. Appl. Math.
12
(2
), 194
–198
(1954
). 44.
G.
Teschl
, Ordinary Differential Equations and Dynamical Systems
(American Mathematical Society
, 2012
).45.
D.
Thieffry
and M.
Kaufman
, “Prologue to the special issue of JTB dedicated to the memory of René Thomas (1928–2017): A journey through biological circuits, logical puzzles and complex dynamics
,” J. Theor. Biol.
474
, 42
–47
(2019
). 46.
R.
Thomas
, “Deterministic chaos seen in terms of feedback circuits: Analysis, synthesis, ‘labyrinth chaos’
,” Int. J. Bifurcat. Chaos
9
(10
), 1889
–1905
(1999
). 47.
R.
Thomas
, V.
Basios
, M.
Eiswirth
, T.
Kruel
, and O.
Rössler
, “Hyperchaos of arbitrary order generated by a single feedback circuit, and the emergence of chaotic walks
,” Chaos
14
(3
), 669
–674
(2004
). 48.
R.
Thomas
and P.
Nardone
, “A further understanding of phase space partition diagrams
,” Int. J. Bifurcat. Chaos
19
(03
), 785
–804
(2009
). 49.
50.
M. E.
Tuckerman
, Y.
Liu
, G.
Ciccotti
, and G. J.
Martyna
, “Non-Hamiltonian molecular dynamics: Generalizing Hamiltonian phase space principles to non-Hamiltonian systems
,” J. Chem. Phys.
115
, 1678
–1702
(2001
). 51.
M. E.
Tuckerman
, C. J.
Mundy
, and G. J.
Martyna
, “On the classical statistical mechanics of non-Hamiltonian systems
,” Europhys. Lett.
45
, 149
–155
(1999
). 52.
L.
Zarfaty
, A.
Peletskyi
, E.
Barkai
, and S.
Denisov
, “Infinite horizon billiards: Transport at the border between Gauss and Lévy universality classes
,” Phys. Rev. E
100
, 042140
(2019
). © 2020 Author(s).
2020
Author(s)
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