Recently, the phase-flip bifurcation has been described as a fundamental transition in time-delay coupled, phase-synchronized nonlinear dynamical systems. The bifurcation is characterized by a change of the synchronized dynamics from being in-phase to antiphase, or vice versa; the phase-difference between the oscillators undergoes a jump of π as a function of the coupling strength or the time delay. This phase-flip is accompanied by discontinuous changes in the frequency of the synchronized oscillators, and in the largest negative Lyapunov exponent or its derivative. Here we illustrate the phenomenology of the bifurcation for several classes of nonlinear oscillators, in the regimes of both periodic and chaotic dynamics. We present extensive numerical simulations and compute the oscillation frequencies and the Lyapunov spectra as a function of the coupling strength. In particular, our simulations provide clear evidence of the phase-flip bifurcation in excitable laser and Fitzhugh–Nagumo neuronal models, and in diffusively coupled predator-prey models with either limit cycle or chaotic dynamics. Our analysis demonstrates marked jumps of the time-delayed and instantaneous fluxes between the two interacting oscillators across the bifurcation; this has strong implications for the performance of the system as well as for practical applications. We further construct an electronic circuit consisting of two coupled Chua oscillators and provide the first formal experimental demonstration of the bifurcation. In totality, our study demonstrates that the phase-flip phenomenon is of broad relevance and importance for a wide range of physical and natural systems.

1.
A.
Prasad
,
J.
Kurths
,
S. K.
Dana
, and
R.
Ramaswamy
,
Phys. Rev. E
74
,
035204
(R) (
2006
).
2.
A.
Prasad
,
Phys. Rev. E
72
,
056204
(
2005
).
3.
A.
Pikovsky
,
M.
Rosenblum
, and
J.
Kurths
,
Synchronization, A Universal Concept in Nonlinear Science
(
Cambridge University Press
,
Cambridge
,
2001
).
4.
L.
Pecora
and
T.
Carroll
,
Phys. Rev. Lett.
64
,
821
(
1990
).
5.
D. V. R.
Reddy
,
A.
Sen
, and
G. L.
Johnston
,
Phys. Rev. Lett.
80
,
5109
(
1998
);
S. H.
Strogatz
,
Nature (London)
394
,
316
(
1998
), and references therein.
6.
K.
Pyragas
,
Phys. Rev. E
58
,
3067
(
1998
);
E. M.
Shahverdiev
and
K. A.
Shore
,
Phys. Rev. E
71
,
016201
(
2005
);
S.
Zhou
,
H.
Li
, and
Z.
Wu
,
Phys. Rev. E
75
,
037203
(
2007
);
E.
Allaria
,
F. T.
Arecchi
,
A.
Di Garbo
, and
R.
Meucci
,
Phys. Rev. Lett.
86
,
791
(
2001
);
[PubMed]
A.
Uchida
,
K.
Mizumura
, and
S.
Yoshimori
,
Phys. Rev. E
74
,
066206
(
2006
);
E. M.
Shahverdiev
,
Phys. Rev. E
70
,
067202
(
2004
);
R.
Herrero
,
M.
Figueras
,
F.
Pi
, and
G.
Orriols
,
Phys. Rev. E
66
,
036223
(
2002
);
D. V.
Senthilkumar
and
M.
Lakshmanan
,
Phys. Rev. E
71
,
016211
(
2005
);
D. V.
Senthilkumar
,
M.
Lakshmanan
, and
J.
Kurths
,
Phys. Rev. E
74
,
035205
(
2006
);
D. V. R.
Reddy
,
A.
Sen
, and
G. L.
Johnston
,
Phys. Rev. Lett.
85
,
3381
(
2000
).
[PubMed]
7.
M.-Y.
Kim
, Ph.D. thesis,
University of Maryland
,
2005
;
M.-Y.
Kim
,
R.
Roy
,
J. L.
Aron
,
T. W.
Carr
, and
I. B.
Schwartz
,
Phys. Rev. Lett.
94
,
088101
(
2005
).
[PubMed]
8.
A. M.
Yacomotti
,
M. C.
Eguia
,
J.
Aliaga
,
O. E.
Martinez
, and
G. B.
Mindlin
,
Phys. Rev. Lett.
83
,
292
(
1999
);
F. C.
Hoppensteadt
and
E. M.
Izhikevich
,
Weakly Connected Neural Networks
(
Springer-Verlag
,
Berlin
,
1997
);
A. C.
Ventura
,
G. B.
Mindlin
, and
S.
Ponce Dawson
,
Phys. Rev. E
65
,
046231
(
2002
);
M. C.
Cross
and
P. C.
Hohenberg
,
Rev. Mod. Phys.
65
,
851
(
1993
);
B.
van der Pol
,
Philos. Mag.
3
,
65
(
1927
).
9.
M. C.
Eguia
,
G. B.
Mindlin
, and
M.
Giudici
,
Phys. Rev. E
58
,
2636
(
1998
).
10.
R.
Fitzhugh
,
Biophys. J.
1
,
455
(
1961
);
A. S.
Pikovsky
and
J.
Kurths
,
Phys. Rev. Lett.
78
,
775
(
1997
);
R.
Benzi
,
A.
Sutera
, and
A.
Vulpiani
,
J. Phys. A
14
,
L453
(
1981
).
11.
A.
Prasad
,
B.
Biswal
, and
R.
Ramaswamy
,
Phys. Rev. E
68
,
037201
(
2003
).
12.

We use a Runge–Kutta fourth order method to integrate the equations of motion, with step size Δt=τN; we take N=1000 for the excitable system Eq. (3), while N=100 suffices for the other cases.

13.
For calculating the Lyapunov exponents in time-delay systems, see
J. D.
Farmer
,
Physica D
4
,
366
(
1982
).
14.
X.
Li
,
X.
Yao
,
J.
Fox
, and
J. G.
Jefferys
,
J. Neurosci. Methods
160
,
178
(
2007
).
15.
M. L.
Rosenzweig
and
R. H.
MacArthur
,
Am. Nat.
97
,
209
(
1963
).
16.
B.
Blasius
,
A.
Huppert
, and
L.
Stone
,
Nature (London)
399
,
354
(
1999
);
B.
Blasius
and
L.
Stone
,
Int. J. Bifurcation Chaos Appl. Sci. Eng.
10
,
2361
(
2000
).
17.
D.
He
and
L.
Stone
,
Proc. R. Soc. London, Ser. B
270
,
1519
(
2003
).
18.
P. K.
Roy
,
S.
Chakraborty
, and
S. K.
Dana
,
Chaos
13
,
342
(
2003
).
19.
M. P.
Kennedy
,
IEEE Trans. Circuits Syst., I: Fundam. Theory Appl.
40
,
567
(
1993
).
20.
D.
Gabor
,
J. Inst. Electr. Eng., Part 3
93
,
429
(
1946
).
21.
R.
Karnatak
,
R.
Ramaswamy
, and
A.
Prasad
,
Phys. Rev. E
76
,
035201
(R) (
2007
).
22.
S. K.
Dana
,
B.
Blasius
, and
J.
Kurths
,
Chaos
16
,
023111
(
2006
).
23.
See, e.g.,
D. V.
Ramana Reddy
,
A.
Sen
, and
G. L.
Johnston
,
Phys. Rev. Lett.
85
,
3381
(
2000
);
[PubMed]
and
H.-J.
Wünsche
,
S.
Bauer
,
J.
Kreissl
,
O.
Ushakov
,
N.
Korneyev
,
F.
Henneberger
,
E.
Willie
,
E.
Erzgräber
,
M.
Peil
,
W.
Elsäßer
, and
I.
Fischer
,
Phys. Rev. Lett.
94
,
163901
(
2005
) for experiments where both in- and out-of-phase dynamics have been demonstrated.
[PubMed]
24.
R.
Karnatak
and
A.
Prasad
(preprint).
You do not currently have access to this content.