In 1665, Huygens observed that two pendulum clocks hanging from the same board became synchronized in antiphase after hundreds of swings. On the other hand, modern experiments with metronomes placed on a movable platform show that they often tend to synchronize in phase, not antiphase. Here, we study both in-phase and antiphase synchronization in a model of pendulum clocks and metronomes and analyze their long-term dynamics with the tools of perturbation theory. Specifically, we exploit the separation of timescales between the fast oscillations of the individual pendulums and the much slower adjustments of their amplitudes and phases. By scaling the equations appropriately and applying the method of multiple timescales, we derive explicit formulas for the regimes in the parameter space where either antiphase or in-phase synchronization is stable or where both are stable. Although this sort of perturbative analysis is standard in other parts of nonlinear science, surprisingly it has rarely been applied in the context of Huygens’s clocks. Unusual features of our approach include its treatment of the escapement mechanism, a small-angle approximation up to cubic order, and both a two- and three-timescale asymptotic analysis.

1.
A. T.
Winfree
,
The Geometry of Biological Time
(
Springer-Verlag
,
1980
).
2.
A.
Pikovsky
,
M.
Rosenblum
, and
J.
Kurths
,
Synchronization: A Universal Concept in Nonlinear Sciences
(
Cambridge University Press
,
2001
).
3.
S. H.
Strogatz
,
Sync
(
Hyperion
,
2003
).
4.
I. I.
Blekhman
,
Synchronization in Science and Technology
(
American Society of Mechanical Engineers Press
,
1988
).
5.
C.
Huygens
, Oeuvres complètes de Christiaan Huygens, edited by M. Nijhoff (Societé Hollandaise des Sciences, 1893), Vol. 5, pp. 241–262.
6.
J. G.
Yoder
,
Unrolling Time: Christiaan Huygens and the Mathematization of Nature
(
Cambridge University Press
,
2004
).
7.
J. P.
Ramirez
and
H.
Nijmeijer
, “
The secret of the synchronized pendulums
,”
Phys. World
33
,
36
(
2020
).
8.
J.
Ellicott
, “
An account of the influence which two pendulum clocks were observed to have upon each other
,”
Philos. Trans. R. Soc.
41
,
126
128
(
1740
).
9.
J.
Ellicott
, “
Further observations and experiments concerning the two clocks above mentioned
,”
Philos. Trans. R. Soc.
41
,
128
135
(
1740
).
10.
W.
Ellis
, “
On sympathetic influence between clocks
,”
Mon. Not. R. Astron. Soc.
33
,
480
(
1873
).
11.
D.
Korteweg
, “
Les horloges sympathiques de Huygens
,”
Arch. Neerlandaises Serie II
11
,
273
295
(
1906
).
12.
M.
Bennett
,
M. F.
Schatz
,
H.
Rockwood
, and
K.
Wiesenfeld
, “
Huygens’s clocks
,”
Proc. R. Soc. A
458
,
563
579
(
2002
).
13.
W.
Oud
,
H.
Nijmeijer
, and
A. Y.
Pogromsky
, “A study of Huijgens’ synchronization: Experimental results,” in Group Coordination and Cooperative Control (Springer, 2006), pp. 191–203.
14.
M.
Senator
, “
Synchronization of two coupled escapement-driven pendulum clocks
,”
J. Sound Vib.
291
,
566
603
(
2006
).
15.
R.
Dilão
, “
Antiphase and in-phase synchronization of nonlinear oscillators: The Huygens’s clocks system
,”
Chaos
19
,
023118
(
2009
).
16.
K.
Czolczynski
,
P.
Perlikowski
,
A.
Stefanski
, and
T.
Kapitaniak
, “
Clustering of Huygens’ clocks
,”
Prog. Theor. Phys.
122
,
1027
1033
(
2009
).
17.
K.
Czołczyński
,
P.
Perlikowski
,
A.
Stefański
, and
T.
Kapitaniak
, “
Why two clocks synchronize: Energy balance of the synchronized clocks
,”
Chaos
21
,
023129
(
2011
).
18.
K.
Czolczynski
,
P.
Perlikowski
,
A.
Stefanski
, and
T.
Kapitaniak
, “
Huygens’ odd sympathy experiment revisited
,”
Int. J. Bifurcat. Chaos
21
,
2047
2056
(
2011
).
19.
K.
Czolczynski
,
P.
Perlikowski
,
A.
Stefanski
, and
T.
Kapitaniak
, “
Synchronization of the self-excited pendula suspended on the vertically displacing beam
,”
Commun. Nonlinear Sci. Numer. Simul.
18
,
386
400
(
2013
).
20.
M.
Kapitaniak
,
K.
Czolczynski
,
P.
Perlikowski
,
A.
Stefanski
, and
T.
Kapitaniak
, “
Synchronization of clocks
,”
Phys. Rep.
517
,
1
69
(
2012
).
21.
V.
Jovanovic
and
S.
Koshkin
, “
Synchronization of Huygens’ clocks and the Poincaré method
,”
J. Sound Vib.
331
,
2887
2900
(
2012
).
22.
J. P.
Ramirez
,
R. H.
Fey
, and
H.
Nijmeijer
, “
Synchronization of weakly nonlinear oscillators with Huygens’ coupling
,”
Chaos
23
,
033118
(
2013
).
23.
J. P.
Ramirez
,
R.
Fey
,
K.
Aihara
, and
H.
Nijmeijer
, “
An improved model for the classical Huygens experiment on synchronization of pendulum clocks
,”
J. Sound Vib.
333
,
7248
7266
(
2014
).
24.
J. P.
Ramirez
,
K.
Aihara
,
R.
Fey
, and
H.
Nijmeijer
, “
Further understanding of Huygens’ coupled clocks: The effect of stiffness
,”
Physica D
270
,
11
19
(
2014
).
25.
J. P.
Ramirez
,
L. A.
Olvera
,
H.
Nijmeijer
, and
J.
Alvarez
, “
The sympathy of two pendulum clocks: Beyond Huygens’ observations
,”
Sci. Rep.
6
,
23580
(
2016
).
26.
J. P.
Ramirez
and
H.
Nijmeijer
, “
The Poincaré method: A powerful tool for analyzing synchronization of coupled oscillators
,”
Indag. Math.
27
,
1127
1146
(
2016
).
27.
H. M.
Oliveira
and
L. V.
Melo
, “
Huygens synchronization of two clocks
,”
Sci. Rep.
5
,
11548
(
2015
).
28.
A. R.
Willms
,
P. M.
Kitanov
, and
W. F.
Langford
, “
Huygens’ clocks revisited
,”
R. Soc. Open Sci.
4
,
170777
(
2017
).
29.
K.
Wiesenfeld
, “
Huygens’s odd sympathy recreated
,”
Soc. Politica
11
,
15
22
(
2017
), available at https://socpol.uvvg.ro/huygenss-odd-sympathy-recreated/.
30.
J.
Pantaleone
, “
Synchronization of metronomes
,”
Am. J. Phys.
70
,
992
1000
(
2002
).
31.
N. V.
Kuznetsov
,
G. A.
Leonov
,
H.
Nijmeijer
, and
A. Y.
Pogromsky
, “Synchronization of two metronomes,” in Proceedings of the 3rd IFAC Workshop (PSYCO’07), 29–31 August 2007, Saint Petersburg, Russia (Springer, 2007), pp. 49–52.
32.
H.
Ulrichs
,
A.
Mann
, and
U.
Parlitz
, “
Synchronization and chaotic dynamics of coupled mechanical metronomes
,”
Chaos
19
,
043120
(
2009
).
33.
Y.
Wu
,
N.
Wang
,
L.
Li
, and
J.
Xiao
, “
Anti-phase synchronization of two coupled mechanical metronomes
,”
Chaos
22
,
023146
(
2012
).
34.
A.
Bahraminasab
, see https://www.youtube.com/watch?v=W1TMZASCR-I for “Synchronisation” (2007).
35.
Ikeguchi-Laboratory
, see https://www.youtube.com/watch?v=JWToUATLGzs for “Synchronization of Thirty Two Metronomes” (2012).
36.
MythBusters
, see https://www.youtube.com/watch?v=e-c6S6SdkPo for “N-Sync” (2014).
37.
A. M.
Lepschy
,
G.
Mian
, and
U.
Viaro
, “
Feedback control in ancient water and mechanical clocks
,”
IEEE Trans. Educ.
35
,
3
10
(
1992
).
38.
F. C.
Moon
and
P. D.
Stiefel
, “
Coexisting chaotic and periodic dynamics in clock escapements
,”
Philos. Trans. R. Soc. A
364
,
2539
2564
(
2006
).
39.
A. V.
Roup
,
D. S.
Bernstein
,
S. G.
Nersesov
,
W. M.
Haddad
, and
V.
Chellaboina
, “
Limit cycle analysis of the verge and foliot clock escapement using impulsive differential equations and Poincaré maps
,”
Int. J. Control
76
,
1685
1698
(
2003
).
40.
A.
Rowlings
,
The Science of Clocks and Watches
(
Caldwell Industries
,
Luling, TX
,
1944
).
41.
J.
Guckenheimer
and
P.
Holmes
,
Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields
(
Springer
,
1983
).
42.
S. H.
Strogatz
,
Nonlinear Dynamics and Chaos
(
Addison-Wesley
,
1994
).
43.
C. M.
Bender
and
S. A.
Orszag
,
Advanced Mathematical Methods for Scientists and Engineers
(
Springer
,
1999
).
44.
M. H.
Holmes
,
Introduction to Perturbation Methods
(
Springer
,
1995
).
45.
A.
Dhooge
,
W.
Govaerts
,
Y. A.
Kuznetsov
,
H. G. E.
Meijer
, and
B.
Sautois
, “
New features of the software matcont for bifurcation analysis of dynamical systems
,”
Math. Comput. Model. Dyn. Syst.
14
,
147
175
(
2008
).
46.
A. L.
Fradkov
and
B.
Andrievsky
, “
Synchronization and phase relations in the motion of two-pendulum system
,”
Int. J. Non Linear Mech.
42
,
895
901
(
2007
).
47.
M.
Kumon
,
R.
Washizaki
,
J.
Sato
,
R.
Mizumoto
, and
Z.
Iwai
, “Controlled synchronization of two 1-DOF coupled oscillators,” in Proceedings of the 15th IFAC World Congress, Barcelona (Elsevier, 2002), pp. 3–10.
You do not currently have access to this content.