Extreme multistability (EM) is characterized by the emergence of infinitely many coexisting attractors or continuous families of stable states in dynamical systems. EM implies complex and hardly predictable asymptotic dynamical behavior. We analyze a model for pendulum clocks coupled by springs and suspended on an oscillating base and show how EM can be induced in this system by specifically designed coupling. First, we uncover that symmetric coupling can increase the dynamical complexity. In particular, the coexistence of multiple isolated attractors and continuous families of stable periodic states is generated in a symmetric cross-coupling scheme of four pendulums. These coexisting infinitely many states are characterized by different levels of phase synchronization between the pendulums, including anti-phase and in-phase states. Some of the states are characterized by splitting of the pendulums into groups with silent sub-threshold and oscillating behavior, respectively. The analysis of the basins of attraction further reveals the complex dependence of EM on initial conditions.
REFERENCES
1.
2.
S.
Yanchuk
, A. C.
Roque
, E. E. N.
Macau
, and J.
Kurths
, Eur. Phys. J. Spec. Top.
230
, 2711
(2021
). 3.
I. I.
Blekhman
, Synchronization in Science and Technology
(ASME Press
, New York
, 1988
).4.
A.
Arenas
, A.
Díaz-Guilera
, J.
Kurths
, Y.
Moreno
, and C.
Zhou
, Phys. Rep.
469
, 93
(2008
). 5.
M. E. J.
Newman
, SIAM Rev.
45
, 167
(2003
).6.
T.
Gross
and B.
Blasius
, J. R. Soc. Interface
5
, 259
(2008
).7.
S.
Yanchuk
, V.
Jirsa
, J.
Cabral
, O.
Popovych
, and A.
Torcini
, “From structure to function in neuronal networks: Effects of adaptation, time-delays, and noise,” in Frontiers Research Topics (Frontiers Media SA, Lausanne, 2022).8.
A.
Pikovsky
, M.
Rosenblum
, and J.
Kurths
, Synchronization: A Universal Concept in Nonlinear Sciences, Cambridge Nonlinear Science Series (Cambridge University Press, 2001).9.
G. V.
Osipov
, J.
Kurths
, and C.
Zhou
, Synchronization in Oscillatory Networks
(Springer Science & Business Media
, 2007
).10.
S.
Boda
, S.
Ujvári
, A.
Tunyagi
, and Z.
Néda
, Eur. J. Phys.
34
, 1451
(2013
). 11.
S.
Chhabria
, K. A.
Blaha
, F.
Della Rossa
, and F.
Sorrentino
, Chaos
28
, 111102
(2018
). 12.
T.
Ikeguchi
and Y.
Shimada
, in Proceedings of the 5th International Conference on Applications in Nonlinear Dynamics, Understanding Complex Systems, edited by V. In, P. Longhini, and A. Palacios (Springer International Publishing, Cham, 2019), pp. 141–152.13.
L. M.
Pecora
and T. L.
Carroll
, Phys. Rev. Lett.
80
, 2109
(1998
). 14.
T.
Dahms
, J.
Lehnert
, and E.
Schöll
, Phys. Rev. E
86
, 016202
(2012
). 15.
L.
Lücken
and S.
Yanchuk
, Phys. D: Nonlinear Phenom.
241
, 350
(2012
).16.
D. M.
Abrams
and S. H.
Strogatz
, Phys. Rev. Lett.
93
, 174102
(2004
). 17.
I.
Omelchenko
, Y.
Maistrenko
, P.
Hövel
, and E.
Schöll
, Phys. Rev. Lett.
106
, 234102
(2011
). 18.
D. V.
Kasatkin
, S.
Yanchuk
, E.
Schöll
, and V. I.
Nekorkin
, Phys. Rev. E
96
, 062211
(2017
). 19.
M. C.
Soriano
, J.
García-Ojalvo
, C. R.
Mirasso
, and I.
Fischer
, Rev. Mod. Phys.
85
, 421
(2013
). 20.
M. R.
Tinsley
, S.
Nkomo
, and K.
Showalter
, Nat. Phys.
8
, 662
(2012
). 21.
C.
Hammond
, H.
Bergman
, and P.
Brown
, Trends Neurosci.
30
, 357
(2007
). 22.
B.
Blasius
, A.
Huppert
, and L.
Stone
, Nature
399
, 354
(1999
). 23.
A. A.
Tsonis
and P. J.
Roebber
, Phys. A
333
, 497
(2004
). 24.
U.
Feudel
, Int. J. Bifurc. Chaos
18
, 1607
(2008
). 25.
A. N.
Pisarchik
and U.
Feudel
, Phys. Rep.
540
, 167
(2014
). 26.
D.
Dudkowski
, K.
Czołczyński
, and T.
Kapitaniak
, Mech. Syst. Signal Process.
166
, 108446
(2022
). 27.
F. T.
Arecchi
, Chaos
1
, 357
(1991
). 28.
J.
Foss
, A.
Longtin
, B.
Mensour
, and J.
Milton
, Phys. Rev. Lett.
76
, 708
(1996
). 29.
S. B.
Power
and R.
Kleeman
, J. Phys. Oceanogr.
23
, 1670
(1993
). 30.
C.
Masoller
, Phys. Rev. Lett.
88
, 034102
(2002
). 31.
J.
Borresen
and S.
Lynch
, Int. J. Bifurc. Chaos
12
, 129
(2002
). 32.
U.
Feudel
, C.
Grebogi
, B. R.
Hunt
, and J. A.
Yorke
, Phys. Rev. E
54
, 71
(1996
). 33.
S.
Yanchuk
and M.
Wolfrum
, SIAM J. Appl. Dyn. Syst.
9
, 519
(2010
). 34.
A. G.
Balanov
, N. B.
Janson
, and E.
Schöll
, Phys. Rev. E
71
, 016222
(2005
). 35.
U.
Feudel
, C.
Grebogi
, L.
Poon
, and J. A.
Yorke
, Chaos, Solitons Fractals
9
, 171
(1998
). 36.
H.
Sun
, S. K.
Scott
, and K.
Showalter
, Phys. Rev. E
60
, 3876
(1999
). 37.
C. N.
Ngonghala
, U.
Feudel
, and K.
Showalter
, Phys. Rev. E
83
, 056206
(2011
). 38.
C.
Hens
, S. K.
Dana
, and U.
Feudel
, Chaos
25
, 053112
(2015
). 39.
C. R.
Hens
, R.
Banerjee
, U.
Feudel
, and S. K.
Dana
, Phys. Rev. E
85
, 035202
(2012
). 40.
S.
Pal
, B.
Sahoo
, and S.
Poria
, Phys. Scr.
89
, 045202
(2014
). 41.
A.
Politi
, G. L.
Oppo
, and R.
Badii
, Phys. Rev. A
33
, 4055
(1986
). 42.
A.
Pikovsky
and P.
Rosenau
, Phys. D: Nonlinear Phenom.
218
, 56
(2006
). 43.
J. S.
Lamb
and J. A.
Roberts
, Phys. D: Nonlinear Phenom.
112
, 1
(1998
). 44.
P.
Ashwin
, C.
Bick
, and O.
Burylko
, Front. Appl. Math. Stat.
2
, 7
(2016
). 45.
O.
Burylko
, A.
Mielke
, M.
Wolfrum
, and S.
Yanchuk
, SIAM J. Appl. Dyn. Syst.
17
, 2076
(2018
). 46.
S. H.
Strogatz
, D. M.
Abrams
, A.
McRobie
, B.
Eckhardt
, and E.
Ott
, Nature
438
, 43
(2005
). 47.
D.
Dudkowski
, J.
Wojewoda
, K.
Czołczyński
, and T.
Kapitaniak
, Chaos
30
, 011102
(2020
). 48.
D.
Dudkowski
, P.
Jaros
, K.
Czołczyński
, and T.
Kapitaniak
, Nonlinear Dyn.
102
, 1541
(2020
). 49.
K.
Czołczyński
, P.
Perlikowski
, A.
Stefański
, and T.
Kapitaniak
, Chaos
21
, 023129
(2011
). 50.
M.
Kapitaniak
, K.
Czolczynski
, P.
Perlikowski
, A.
Stefanski
, and T.
Kapitaniak
, Phys. Rep.
517
, 1
(2012
). 51.
R.
Berner
, J.
Fialkowski
, D. V.
Kasatkin
, V. I.
Nekorkin
, S.
Yanchuk
, and E.
Schöll
, Chaos
29
, 103134
(2019
). 52.
P.
Ebrahimzadeh
, M.
Schiek
, and Y.
Maistrenko
, Chaos
32
, 103118
(2022
).53.
S.
Brezetsky
, P.
Jaros
, R.
Levchenko
, T.
Kapitaniak
, and Y.
Maistrenko
, Phys. Rev. E
103
, L050204
(2021
). © 2023 Author(s). Published under an exclusive license by AIP Publishing.
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