In this paper, we experimentally verify the phenomenon of chaotic synchronization in coupled forced oscillators. The study is focused on the model of three double pendula locally connected via springs. Each of the individual oscillators can behave both periodically and chaotically, which depends on the parameters of the external excitation (the shaker). We investigate the relation between the strength of coupling between the upper pendulum bobs and the precision of their synchronization, showing that the system can achieve practical synchronization, within which the nodes preserve their chaotic character. We determine the influence of the pendula parameters and the strength of coupling on the synchronization precision, measuring the differences between the nodes’ motion. The results obtained experimentally are confirmed by numerical simulations. We indicate a possible mechanism causing the desynchronization of the system’s smaller elements (lower pendula bobs), which involves their motion around the unstable stationary position and possible transient dynamics. The results presented in this paper may be generalized into typical models of pendula and pendula-like coupled systems, exhibiting chaotic dynamics.

1.
E.
Ott
,
Chaos in Dynamical Systems
(
Cambridge University Press
,
2002
).
2.
S.
Wiggins
,
Introduction to Applied Nonlinear Dynamical Systems and Chaos
(
Springer
,
2003
).
3.
M. W.
Hirsch
,
S.
Smale
, and
R. L.
Devaney
,
Differential Equations, Dynamical Systems, and an Introduction to Chaos
(
Academic Press
,
2012
).
4.
J.
Awrejcewicz
and
C.-H.
Lamarque
,
Bifurcation and Chaos in Nonsmooth Mechanical Systems
(
World Scientific
,
2003
).
5.
R. J.
Field
and
L.
Gyorgyi
,
Chaos in Chemistry and Biochemistry
(
World Scientific
,
1993
).
6.
M.
Akhmet
and
M. O.
Fen
,
Replication of Chaos in Neural Networks, Economics and Physics
(
Springer
,
2015
).
7.
Y. L.
Maistrenko
,
E.
Mosekilde
, and
D.
Postnov
,
Chaotic Synchronization: Applications to Living Systems
(
World Scientific Publishing Company
,
2002
).
8.
L. M.
Pecora
and
T. L.
Carroll
, “
Synchronization in chaotic systems
,”
Phys. Rev. Lett.
64
,
821
(
1990
).
9.
A. E.
Hramov
and
A. A.
Koronovskii
, “
An approach to chaotic synchronization
,”
Chaos
14
(
3
),
603
(
2004
).
10.
G. M.
Mahmoud
,
M. A.
Al-Kashif
, and
S. A.
Aly
, “
Basic properties and chaotic synchronization of complex Lorenz system
,”
Int. J. Mod. Phys. C
18
(
02
),
253
(
2007
).
11.
R.
Mainieri
and
J.
Rehacek
, “
Projective synchronization in three-dimensional chaotic systems
,”
Phys. Rev. Lett.
82
,
3042
(
1999
).
12.
H. U.
Voss
, “
Anticipating chaotic synchronization
,”
Phys. Rev. E
61
,
5115
(
2000
).
13.
J.
Lu
,
T.
Zhou
, and
S.
Zhang
, “
Chaos synchronization between linearly coupled chaotic systems
,”
Chaos, Solitons Fractals
14
(
4
),
529
(
2002
).
14.
H.
Yu
and
Y.
Liu
, “
Chaotic synchronization based on stability criterion of linear systems
,”
Phys. Lett. A
314
(
4
),
292
(
2003
).
15.
Y.
Zhang
and
J.
Sun
, “
Chaotic synchronization and anti-synchronization based on suitable separation
,”
Phys. Lett. A
330
(
6
),
442
(
2004
).
16.
D.
Chen
,
W.
Zhao
,
J. C.
Sprott
, and
X.
Ma
, “
Application of Takagi-Sugeno fuzzy model to a class of chaotic synchronization and anti-synchronization
,”
Nonlinear Dyn.
73
(
3
),
1495
(
2013
).
17.
U.
Parlitz
,
L. O.
Chua
,
L.
Kocarev
,
K. S.
Halle
, and
A.
Shang
, “
Transmission of digital signals by chaotic synchronization
,”
Int. J. Bifurcation Chaos
02
(
04
),
973
(
1992
).
18.
T. L.
Carroll
,
J. F.
Heagy
, and
L. M.
Pecora
, “
Transforming signals with chaotic synchronization
,”
Phys. Rev. E
54
,
4676
(
1996
).
19.
G.
Kolumban
,
M. P.
Kennedy
, and
L. O.
Chua
, “
The role of synchronization in digital communications using chaos. II. Chaotic modulation and chaotic synchronization
,”
IEEE Trans. Circuits Syst. I: Fundam. Theory Appl.
45
(
11
),
1129
(
1998
).
20.
L.
Kocarev
and
U.
Parlitz
, “
General approach for chaotic synchronization with applications to communication
,”
Phys. Rev. Lett.
74
,
5028
(
1995
).
21.
U.
Parlitz
,
L.
Kocarev
,
T.
Stojanovski
, and
H.
Preckel
, “
Encoding messages using chaotic synchronization
,”
Phys. Rev. E
53
,
4351
(
1996
).
22.
R.
He
and
P. G.
Vaidya
, “
Implementation of chaotic cryptography with chaotic synchronization
,”
Phys. Rev. E
57
,
1532
(
1998
).
23.
C.-C.
Wang
and
J.-P.
Su
, “
A new adaptive variable structure control for chaotic synchronization and secure communication
,”
Chaos, Solitons Fractals
20
(
5
),
967
(
2004
).
24.
D.
Chen
,
R.
Zhang
,
X.
Ma
, and
S.
Liu
, “
Chaotic synchronization and anti-synchronization for a novel class of multiple chaotic systems via a sliding mode control scheme
,”
Nonlinear Dyn.
69
(
1-2
),
35
(
2012
).
25.
L. O.
Chua
,
L.
Kocarev
,
K.
Eckert
, and
M.
Itoh
, “
Experimental chaos synchronization in Chua’s circuit
,”
Int. J. Bifurcation Chaos
02
(
03
),
705
(
1992
).
26.
A.
Kittel
,
J.
Parisi
, and
K.
Pyragas
, “
Generalized synchronization of chaos in electronic circuit experiments
,”
Physica D
112
(
3-4
),
459
(
1998
).
27.
A.
Buscarino
,
L.
Fortuna
, and
M.
Frasca
, “
Experimental robust synchronization of hyperchaotic circuits
,”
Physica D
238
(
18
),
1917
(
2009
).
28.
V.
Makarenko
and
R.
Llinás
, “
Experimentally determined chaotic phase synchronization in a neuronal system
,”
Proc. Natl. Acad. Sci. U.S.A.
95
(
26
),
15747
(
1998
).
29.
W.
Wang
,
I. Z.
Kiss
, and
J. L.
Hudson
, “
Experiments on arrays of globally coupled chaotic electrochemical oscillators: Synchronization and clustering
,”
Chaos
10
(
1
),
248
(
2000
).
30.
J. M.
Cruz
,
M.
Rivera
, and
P.
Parmananda
, “
Experimental observation of different types of chaotic synchronization in an electrochemical cell
,”
Phys. Rev. E
75
,
035201
(
2007
).
31.
R.
Roy
and
K. S.
Thornburg
, “
Experimental synchronization of chaotic lasers
,”
Phys. Rev. Lett.
72
,
2009
(
1994
).
32.
S.
Sivaprakasam
,
E. M.
Shahverdiev
,
P. S.
Spencer
, and
K. A.
Shore
, “
Experimental demonstration of anticipating synchronization in chaotic semiconductor lasers with optical feedback
,”
Phys. Rev. Lett.
87
,
154101
(
2001
).
33.
Y.
Liu
,
Y.
Takiguchi
,
P.
Davis
,
T.
Aida
,
S.
Saito
, and
J. M.
Liu
, “
Experimental observation of complete chaos synchronization in semiconductor lasers
,”
Appl. Phys. Lett.
80
(
23
),
4306
(
2002
).
34.
L.
Kocarev
,
K. S.
Halle
,
K.
Eckert
,
L. O.
Chua
, and
U.
Parlitz
, “
Experimental demonstration of secure communications via chaotic synchronization
,”
Int. J. Bifurcation Chaos
02
(
03
),
709
(
1992
).
35.
B.
Nana
,
P.
Woafo
, and
S.
Domngang
, “
Chaotic synchronization with experimental application to secure communications
,”
Commun. Nonlinear Sci.
14
(
5
),
2266
(
2009
).
36.
E.
Sánchez
,
M. A.
Matías
, and
V.
Pérez-Muñuzuri
, “
Analysis of synchronization of chaotic systems by noise: An experimental study
,”
Phys. Rev. E
56
,
4068
(
1997
).
37.
J. C.
Quinn
,
P. H.
Bryant
,
D. R.
Creveling
,
S. R.
Klein
, and
H. D. I.
Abarbanel
, “
Parameter and state estimation of experimental chaotic systems using synchronization
,”
Phys. Rev. E
80
,
016201
(
2009
).
38.
Y.
Yu
and
S.
Zhang
, “
Global synchronization of three coupled chaotic systems with ring connection
,”
Chaos, Solitons Fractals
24
(
5
),
1233
(
2005
).
39.
W.
Lim
and
S.-Y.
Kim
, “
Mechanism for the partial synchronization in three coupled chaotic systems
,”
Phys. Rev. E
71
,
036221
(
2005
).
40.
N.
Tsukamoto
,
S.
Miyazaki
, and
H.
Fujisaka
, “
Synchronization and intermittency in three-coupled chaotic oscillators
,”
Phys. Rev. E
67
,
016212
(
2003
).
41.
A. P.
Kuznetsov
,
N. A.
Migunova
,
I. R.
Sataev
,
Y. V.
Sedova
, and
L. V.
Turukina
, “
From chaos to quasi-periodicity
,”
Regul. Chaotic Dyn.
20
,
189
(
2015
).
42.
J.
Zhang
,
L.
Zhang
,
X.
An
,
H.
Luo
, and
K. E.
Yao
, “
Adaptive coupled synchronization among three coupled chaos systems and its application to secure communications
,”
J. Wirel. Commun. Netw.
2016
(
1
),
134
.
43.
D.
Dudkowski
,
J.
Wojewoda
,
K.
Czolczynski
, and
T.
Kapitaniak
, “
Transient chimera-like states for forced oscillators
,”
Chaos
30
(
1
),
011102
(
2020
).
44.
D.
Dudkowski
,
J.
Wojewoda
,
K.
Czolczynski
, and
T.
Kapitaniak
, “
Is it really chaos? The complexity of transient dynamics of double pendula
,”
Nonlinear Dyn.
102
(
2
),
759
(
2020
).
45.
M. A.
Kiseleva
,
E. V.
Kudryashova
,
N. V.
Kuznetsov
,
O. A.
Kuznetsova
,
G. A.
Leonov
,
M. V.
Yuldashev
, and
R. V.
Yuldashev
, “
Hidden and self-excited attractors in Chua circuit: Synchronization and SPICE simulation
,”
Int. J. Parallel Emergent Distrib. Syst.
33
(
5
),
513
(
2018
).
46.
N. V.
Kuznetsov
,
G. A.
Leonov
,
M. V.
Yuldashev
, and
R. V.
Yuldashev
, “
Hidden attractors in dynamical models of phase-locked loop circuits: Limitations of simulation in MATLAB and SPICE
,”
Commun. Nonlinear Sci. Numer. Simul.
51
,
39
(
2017
).
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