The phenomenon of frequency and phase synchronization in stochastic systems requires a revision of concepts originally phrased in the context of purely deterministic systems. Various definitions of an instantaneous phase are presented and compared with each other with special attention paid to their robustness with respect to noise. We review the results of an analytic approach describing noise-induced phase synchronization in a thermal two-state system. In this context exact expressions for the mean frequency and the phase diffusivity are obtained that together determine the average length of locking episodes. A recently proposed method to quantify frequency synchronization in noisy potential systems is presented and exemplified by applying it to the periodically driven noisy harmonic oscillator. Since this method is based on a threshold crossing rate pioneered by Rice the related phase velocity is termed the Rice frequency. Finally, we discuss the relation between the phenomenon of stochastic resonance and noise-enhanced phase coherence by applying the developed concepts to the periodically driven bistable Kramers oscillator.

1.
V. I.
Arnold
,
Trans. Am. Math. Soc.
42
,
213
(
1965
);
E. Ott, Chaos in Dynamical Systems (Cambridge University Press, Cambridge, 1993).
2.
R. L. Stratonovich, Topics in the Theory of Random Noise (Gordon and Breach, New York, 1967).
3.
H.
Fujisaka
and
T.
Yamada
,
Prog. Theor. Phys.
69
,
32
(
1983
);
A. S.
Pikovsky
,
Z. Phys. B: Condens. Matter
55
,
149
(
1984
);
L. M.
Pecora
and
T. L.
Carroll
,
Phys. Rev. Lett.
64
,
821
(
1990
).
4.
N. F.
Rulkov
,
M. M.
Sushchik
,
L. S.
Tsimring
, and
H. D. I.
Abarbanel
,
Phys. Rev. E
51
,
980
(
1995
);
L.
Kocarev
and
U.
Parlitz
,
Phys. Rev. Lett.
76
,
1816
(
1996
).
5.
M. G.
Rosenblum
,
A. S.
Pikovsky
, and
J.
Kurths
,
Phys. Rev. Lett.
78
,
4193
(
1997
);
S.
Taherion
and
Y. C.
Lai
,
Phys. Rev. E
59
,
R6247
(
1999
).
6.
M. G.
Rosenblum
,
A. S.
Pikovsky
, and
J.
Kurths
,
Phys. Rev. Lett.
76
,
1804
(
1996
).
7.
S. K.
Han
,
T. G.
Yim
,
D. E.
Postnov
, and
O. V.
Sosnovtseva
,
Phys. Rev. Lett.
83
,
1771
(
1999
).
8.
E. M.
Izhikevich
,
Int. J. Bifur. Chaos
10
,
1171
(
2000
);
B.
Hu
and
C.
Zhou
,
Phys. Rev. E
63
,
026201
(
2001
).
9.
V. Anishchenko, A. Neiman, A. Astakhov, T. Vadiavasova, and L. Schimansky-Geier, Chaotic and Stochastic Processes in Dynamic Systems (Springer, Berlin, 2002).
10.
L. Schimansky-Geier, V. Anishchenko, and A. Neiman, in Neuro-informatics, edited by S. Gielen and F. Moss, Vol. 4, in Handbook of Biological Physics, series editor A. J. Hoff (Elsevier Science, New York, 2001).
11.
A.
Neiman
,
X.
Pei
,
D.
Russell
,
W.
Wojtenek
,
L.
Wilkens
,
F.
Moss
,
H. A.
Braun
,
M. T.
Huber
, and
K.
Voigt
,
Phys. Rev. Lett.
82
,
660
(
1999
);
S.
Coombes
and
P. C.
Bressloff
,
Phys. Rev. E
60
,
2086
(
1999
);
S.
Coombes
and
P. C.
Bressloff
,
Phys. Rev. E
63
,
059901
(
2001
);
W.
Singer
,
Neuron
24
,
49
(
1999
);
R. C.
Elson
,
A. I.
Selverston
,
R.
Huerta
,
N. F.
Rulkov
,
M. I.
Rabinovich
, and
H. D. I.
Abarbanel
,
Phys. Rev. Lett.
81
,
5692
(
1998
);
P.
Tass
,
M. G.
Rosenblum
,
J.
Weule
,
J.
Kurths
,
A.
Pikovsky
,
J.
Volkmann
,
A.
Schnitzler
, and
H.-J.
Freund
,
Phys. Rev. Lett.
81
,
3291
(
1998
);
R.
Ritz
and
T. J.
Sejnowski
,
Curr. Opin. Neurobiol.
7
,
536
(
1997
).
12.
B.
Schäfer
,
M. G.
Rosenblum
, and
J.
Kurths
,
Nature (London)
392
,
239
(
1998
).
13.
B.
Blasius
,
A.
Huppert
, and
L.
Stone
,
Nature (London)
399
,
354
(
1999
).
14.
J. P. M.
Heald
and
J.
Stark
,
Phys. Rev. Lett.
84
,
2366
(
2000
).
15.
J. A.
Freund
,
A. B.
Neiman
, and
L.
Schimansky-Geier
,
Europhys. Lett.
50
,
8
(
2000
).
16.
L.
Callenbach
,
P.
Hänggi
,
S. J.
Linz
,
J. A.
Freund
, and
L.
Schimansky-Geier
,
Phys. Rev. E
65
,
051110
(
2002
).
17.
S. O.
Rice
,
Bell Syst. Tech. J.
23/24
,
1
(
1944
/1945);
note Sec. 3.3 on pp. 57–63 in this double volume therein.
18.
S. O. Rice, in Selected Papers on Noise and Stochastic Processes, edited by N. Wax (Dover, New York, 1954), pp. 189–195 therein.
19.
L.
Gammaitoni
,
P.
Hänggi
,
P.
Jung
, and
F.
Marchesoni
,
Rev. Mod. Phys.
70
,
223
(
1998
).
20.
V. S.
Anishchenko
,
A. B.
Neiman
,
F.
Moss
, and
L.
Schimansky-Geier
,
Phys. Usp.
42
,
7
(
1999
).
21.
A.
Neiman
,
A.
Silchenko
,
V.
Anishchenko
, and
L.
Schimansky-Geier
,
Phys. Rev. E
58
,
7118
(
1998
).
22.
P.
Jung
,
Phys. Rep.
234
,
175
(
1995
).
23.
N. N. Bogoliubov and Y. A. Mitropolski, Asymptotic Methods in the Theory of Nonlinear Oscillations (Gordon and Breach, New York, 1961).
24.
The explicit expression for the phase involving the arctan has to be understood in the sense of adding multiples of π to make it a continuously growing function of time.
25.
P.
Hänggi
and
P.
Riseborough
,
Am. J. Phys.
51
,
347
(
1983
).
26.
The statement requires the function f defined in Eq. (2) to remain finite for φN=π/2+nπ,n∈N; this, however, is no severe restriction.
27.
A. T.
Winfree
,
J. Theor. Biol.
16
,
15
(
1967
);
J.
Guckenheimer
,
J. Math. Biol.
1
,
259
(
1975
).
28.
D.
Gabor
, J. IEE (London) 93 (III), 429 (1946).
29.
A. Pikovsky, M. Rosenblum, and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences (Cambridge University Press, Cambridge, 2001).
30.
L. A. Vainstein and D. E. Vakman, Frequency Analysis in the Theory of Oscillations and Waves (in Russian) (Nauka, Moscow, 1983).
31.
R. Carmona, W.-L. Hwang, and B. Torresani, Practical Time-Frequency Analysis (Academic, San Diego, 1998).
32.
J.-P.
Lachaux
,
E.
Rodriguez
,
M.
Le Van Quen
,
A.
Lutz
,
J.
Martinerie
, and
F.
Varela
,
Int. J. Bifurcation Chaos Appl. Sci. Eng.
10
,
2429
(
2000
);
M.
Le Van Quen
,
J.
Foucher
,
J.-P.
Lachaux
,
E.
Rodriguez
,
A.
Lutz
,
J.
Martinerie
, and
F.
Varela
,
J. Neurosci. Methods
111
,
83
(
2001
).
33.
D. J.
DeShazer
,
R.
Breban
,
E.
Ott
, and
R.
Roy
,
Phys. Rev. Lett.
87
,
044101
(
2001
).
34.
B.
McNamara
and
K.
Wiesenfeld
,
Phys. Rev. A
39
,
4854
(
1989
).
35.
D. R. Cox, Renewal Theory (Chapman and Hall, London, 1967).
36.
J. A. Freund, A. B. Neiman, and L. Schimansky-Geier, in Stochastic Climate Models: Progress in Probability, Vol. 49, edited by P. Imkeller and J. von Storch (Birkhäser, Boston, 2001).
37.
N. G. van Kampen, Stochastic Processes in Physics and Chemistry, revised and enlarged edition (North–Holland, Amsterdam, 1992).
38.
R.
Adler
,
Proc. IRE
34
,
351
(
1946
);
P.
Hänggi
and
P.
Riseborough
,
Am. J. Phys.
51
,
347
(
1983
).
39.
C.
Van den Broeck
,
Phys. Rev. E
47
,
4579
(
1993
);
U.
Zürcher
and
C. R.
Doering
,
Phys. Rev. E
47
,
3862
(
1993
).
40.
F.
Marchesoni
,
F.
Apostolico
, and
S.
Santucci
,
Phys. Rev. E
59
,
3958
(
1999
).
41.
A.
Neiman
,
L.
Schimansky-Geier
,
F.
Moss
,
B.
Shulgin
, and
J. J.
Collins
,
Phys. Rev. E
60
,
284
(
1999
).
42.
B.
Shulgin
,
A.
Neiman
, and
V.
Anishchenko
,
Phys. Rev. Lett.
75
,
4157
(
1995
).
43.
R.
Rozenfeld
,
J. A.
Freund
,
A.
Neiman
, and
L.
Schimansky-Geier
,
Phys. Rev. E
64
,
051107
(
2001
).
44.
P.
Hänggi
,
P.
Talkner
, and
M.
Borkovec
,
Rev. Mod. Phys.
62
,
251
(
1990
).
45.
P.
Hänggi
and
H.
Thomas
,
Phys. Rep.
88
,
207
(
1982
);
see Sec. 2.4 therein.
46.
P.
Hänggi
and
P.
Jung
,
Adv. Chem. Phys.
89
,
239
(
1995
);
P.
Hänggi
,
F.
Marchesoni
, and
P.
Grigolini
,
Z. Phys. B: Condens. Matter
56
,
333
(
1984
).
47.
R.
Benzi
,
G.
Parisi
, and
A.
Vulpiani
,
J. Phys. A
14
,
L453
(
1981
);
C.
Nicolis
,
Sol. Phys.
74
,
473
(
1981
).
48.
A.
Neiman
,
L.
Schimansky-Geier
,
F.
Moss
,
B.
Shulgin
, and
J. J.
Collins
,
Phys. Rev. E
60
,
284
(
1999
).
49.
B.
Lindner
,
M.
Kostur
, and
L.
Schimansky-Geier
,
Fluct. Noise Lett.
1
,
R25
(
2001
).
50.
J. A.
Freund
,
J.
Kienert
,
L.
Schimansky-Geier
,
B.
Beisner
,
A.
Neiman
,
D. F.
Russell
,
T.
Yakusheva
, and
F.
Moss
,
Phys. Rev. E
63
,
031910
(
2001
).
51.
L. Callenbach, Synchronization Phenomena in Chaotic and Noisy Oscillatory Systems (Logos Verlag, Berlin, 2001).
This content is only available via PDF.
You do not currently have access to this content.