The number N of stable fixed points of locally coupled Kuramoto models depends on the topology of the network on which the model is defined. It has been shown that cycles in meshed networks play a crucial role in determining N because any two different stable fixed points differ by a collection of loop flows on those cycles. Since the number of different loop flows increases with the length of the cycle that carries them, one expects N to be larger in meshed networks with longer cycles. Simultaneously, the existence of more cycles in a network means more freedom to choose the location of loop flows differentiating between two stable fixed points. Therefore, N should also be larger in networks with more cycles. We derive an algebraic upper bound for the number of stable fixed points of the Kuramoto model with identical frequencies, under the assumption that angle differences between connected nodes do not exceed π/2. We obtain Nk=1c[2Int(nk/4)+1], which depends both on the number c of cycles and on the spectrum of their lengths {nk}. We further identify network topologies carrying stable fixed points with angle differences larger than π/2, which leads us to conjecture an upper bound for the number of stable fixed points for Kuramoto models on any planar network. Compared to earlier approaches that give exponential upper bounds in the total number of vertices, our bounds are much lower and therefore much closer to the true number of stable fixed points.

1.
Y.
Kuramoto
, “
Self-entrainment of a population of coupled non-linear oscillators
,”
International Symposium on Mathe397 matical Problems in Theoretical Physics
, Lecture Notes in Physics Vol. 39, edited by
H.
Araki
(
Springer
,
Berlin, Heidelberg
,
1975
), pp.
420
422
.
2.
Y.
Kuramoto
,
Prog. Theor. Phys. Suppl.
79
,
223
(
1984
).
4.
J. A.
Acebrón
,
L. L.
Bonilla
,
C. J.
Pérez Vicente
,
F.
Ritort
, and
R.
Spigler
,
Rev. Mod. Phys.
77
,
137
(
2005
).
5.
F.
Dörfler
and
F.
Bullo
,
Automatica
50
,
1539
(
2014
).
6.
G. B.
Ermentrout
,
J. Math. Biol.
22
,
1
(
1985
).
7.
J. L.
van Hemmen
and
W. F.
Wreskinski
,
J. Stat. Phys.
72
,
145
(
1993
).
8.
D.
Aeyels
and
J. A.
Rogge
,
Prog. Theor. Phys.
112
,
921
(
2004
).
9.
R. E.
Mirollo
and
S. H.
Strogatz
,
Physica D
205
,
249
(
2005
).
10.
D.
Mehta
,
N. S.
Daleo
,
F.
Dörfler
, and
J. D.
Hauenstein
,
Chaos
25
,
053103
(
2015
).
11.
T.
Chen
,
D.
Mehta
, and
M.
Niemerg
, e-print arXiv:1603.05905 (
2016
).
12.
J. A.
Rogge
and
D.
Aeyels
,
J. Phys. A
37
,
11135
(
2004
).
13.
D. A.
Wiley
,
S. H.
Strogatz
, and
M.
Girvan
,
Chaos
16
,
015103
(
2006
).
14.
P. J.
Menck
,
J.
Heitzig
,
J.
Kurths
, and
H. J.
Schellnhuber
,
Nat. Commun.
5
,
3969
(
2014
).
15.
F.
Dörfler
,
M.
Chertkov
, and
F.
Bullo
,
Proc. Natl. Acad. Sci. U. S. A.
110
,
2005
(
2013
).
16.
R.
Delabays
,
T.
Coletta
, and
P.
Jacquod
,
J. Math. Phys.
57
,
032701
(
2016
).
17.
A. J.
Korsak
,
IEEE Trans. Power Appar. Syst.
PAS-91
,
1093
(
1972
).
18.
C. J.
Tavora
and
O. J. M.
Smith
,
IEEE Trans. Power Appar. Syst.
PAS-91
,
1138
(
1972
).
19.
A. R.
Bergen
and
V.
Vittal
,
Power Systems Analysis
(
Prentice Hall
,
2000
).
20.
S. J.
Skar
, “
Stability of power systems and other systems of second order differential equations
,” Ph.D. thesis,
Iowa State University
,
1980
.
21.
A.
Araposthatis
,
S.
Sastry
, and
P.
Varayia
,
Int. J. Electr. Power Energy Syst.
3
,
115
(
1981
).
22.
J.
Bailleul
and
C. I.
Byrnes
, in
Proceedings of the 21st IEEE Conference on Decision and Control
(
IEEE
,
1982
), Vol. 2, p.
919
.
23.
Y.
Tamura
,
H.
Mori
, and
S.
Iwamoto
,
IEEE Trans. Power Appar. Syst.
PAS-102
,
1115
(
1983
).
24.
A.
Klos
and
J.
Wojcicka
,
Int. J. Electr. Power Energy Syst.
13
,
268
(
1991
).
25.
H. D.
Nguyen
and
K. S.
Turitsyn
, in
2014 IEEE PES General Meeting - Conference Exposition
(
IEEE
,
2014
).
26.
N.
Janssens
and
A.
Kamagate
,
Int. J. Electr. Power Energy Syst.
25
,
591
(
2003
).
27.
T.
Coletta
,
R.
Delabays
,
I.
Adagideli
, and
P.
Jacquod
,
New J. Phys.
18
,
103042
(
2016
).
28.
J.
Ochab
and
P. F.
Góra
,
Acta Phys. Pol. B Proc. Suppl.
3
,
453
(
2010
).
29.
P. F. C.
Tilles
,
F. F.
Ferreira
, and
H. A.
Cerdeira
,
Phys. Rev. E
83
,
066206
(
2011
).
30.
T. K.
Roy
and
A.
Lahiri
,
Chaos, Solitons Fractals
45
,
888
(
2012
).
32.
A.-L.
Do
,
S.
Boccaletti
, and
T.
Gross
,
Phys. Rev. Lett.
108
,
194102
(
2012
).
33.
N.
Biggs
,
Algebraic Graph Theory
, 2nd ed. (
Cambridge University Press
,
1993
).
34.
R. A.
Horn
and
C. R.
Johnson
,
Matrix Analysis
(
Cambridge University Press
,
New York
,
1986
).
35.
D.
Manik
,
M.
Timme
, and
D.
Witthaut
, e-print arXiv:1611.09825 (
2016
).
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