Despite the prominent importance of the Lyapunov exponents for characterizing chaos, it still remains a challenge to measure them for large experimental systems, mainly because of the lack of recurrences in time series analysis. Here, we develop a method to overcome this difficulty, valid for highly symmetric systems such as systems with global coupling, for which the dimensionality of recurrence analysis can be reduced drastically. We test our method numerically with two globally coupled systems, namely, logistic maps and limit-cycle oscillators with global coupling. The evaluated exponent values are successfully compared with the true ones obtained by the standard numerical method. We also describe a few techniques to improve the accuracy of the proposed method.

1.
U.
Frisch
,
Turbulence: The Legacy of A. N. Kolmogorov
(
Cambridge University Press
,
1995
).
2.
W.
Wang
,
I. Z.
Kiss
, and
J.
Hudson
,
Chaos
10
,
248
(
2000
).
3.
E. M.
Cherry
, and
F. H.
Fenton
,
New J. Phys.
10
,
125016
(
2008
).
4.
E.
Ott
,
Chaos in Dynamical Systems
, 2nd ed. (
Cambridge University Press
,
Cambridge
,
2002
).
5.
J.-P.
Eckmann
, and
D.
Ruelle
,
Rev. Mod. Phys.
57
,
617
(
1985
).
6.
D.
Ruelle
,
Commun. Math. Phys.
87
,
287
(
1982
).
7.
B.
Andrievskii
, and
A.
Fradkov
,
Autom. Remote Control
65
,
505
(
2004
).
8.
N.
Balci
,
A. L.
Mazzucato
,
J. M.
Restrepo
, and
G. R.
Sell
,
Mon. Weather Rev.
140
,
2308
(
2012
).
9.
A.
Pikovsky
, and
A.
Politi
,
Lyapunov Exponents: A Tool to Explore Complex Dynamics
(
Cambridge University Press
,
2016
).
10.
I.
Shimada
, and
T.
Nagashima
,
Prog. Theor. Phys.
61
,
1605
(
1979
).
11.
G.
Benettin
,
L.
Galgani
,
A.
Giorgilli
, and
J.-M.
Strelcyn
,
Meccanica
15
,
9
(
1980
).
12.
H.
Kantz
, and
T.
Schreiber
,
Nonlinear Time Series Analysis
(
Cambridge University Press
,
2004
), Vol. 7.
13.
J.
Pathak
,
Z.
Lu
,
B. R.
Hunt
,
M.
Girvan
, and
E.
Ott
,
Chaos
27
,
121102
(
2017
).
14.
M.
Sano
, and
Y.
Sawada
,
Phys. Rev. Lett.
55
,
1082
(
1985
).
15.
J.-P.
Eckmann
,
S. O.
Kamphorst
,
D.
Ruelle
, and
S.
Ciliberto
,
Phys. Rev. A
34
,
4971
(
1986
).
16.
K.
Kaneko
,
Physica D
41
,
137
(
1990
).
17.
N.
Nakagawa
, and
Y.
Kuramoto
,
Physica D
75
,
74
(
1994
).
18.
N.
Nakagawa
, and
Y.
Kuramoto
,
Physica D
80
,
307
(
1995
).
19.
S.
De Monte
,
F.
d’Ovidio
,
S.
Danø
, and
P. G.
Sørensen
,
Proc. Natl. Acad. Sci. U.S.A.
104
,
18377
(
2007
).
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