Systems of mutually coupled oscillators with delay coupling are of great interest for various applications in electronics, laser physics, biophysics, etc. Time delay usually originates from the finite speed of propagation of the coupling signal. In this paper, we present the results of detailed bifurcation analysis of two delay-coupled limit-cycle (Landau–Stuart) oscillators. First, we study the simplified case when the delay time is much smaller than the oscillation build-up time. When the coupling signal propagates between the two counterparts, it acquires a phase shift, which strongly affects the synchronization pattern. Depending on this phase shift, the system may demonstrate the behavior typical for either dissipative or conservative (reactive) coupling. We examine stability of the in-phase and anti-phase synchronous states and reveal the complicated pattern of the synchronization domains on the frequency mismatch—coupling strength parameter plane paying a special attention to the mechanisms of appearance and disappearance of the phase multistability. We demonstrate that taking into account reactive phase nonlinearity the coupling signal acquires an additional phase shift, which depends on the signal intensity. We also examine the more complicated case of finite delay time. The increase of the reactive nonlinearity parameter and the delay time leads to transformations of synchronization domains similar to those that occur when the phase shift increases. For the bifurcation analysis, we employ XPPAUT and DDEBifTool package and verify the results by direct numerical integration.

1.
P. S.
Landa
,
Nonlinear Oscillations and Waves in Dynamic Systems
(
Kluwer
,
1996
).
2.
A.
Pikovsky
,
M.
Rosenblum
, and
J.
Kurths
,
Synchronization: A Universal Concept in Nonlinear Science
(
Cambridge University Press
,
2001
).
3.
V. S.
Anischenko
,
V. V.
Astakhov
,
A. B.
Neiman
,
T. E.
Vadivasova
, and
L.
Schimansky-Geier
,
Nonlinear Dynamics of Chaotic and Stochastic Systems
(
Springer
,
2002
).
4.
A.
Balanov
,
N.
Janson
,
D.
Postnov
, and
O.
Sosnovtseva
,
Synchronization: From Simple to Complex
(
Springer
,
2009
).
5.
R. A.
York
and
R. C.
Compton
, “
Experimental observation and simulation of mode-locking phenomena in coupled-oscillator arrays
,”
J. Appl. Phys.
71
,
2959
(
1992
).
6.
J.
Benford
,
H.
Sze
,
W.
Woo
,
R. R.
Smith
, and
B.
Harteneck
, “
Phase locking of relativistic magnetrons
,”
Phys. Rev. Lett.
62
,
969
(
1989
).
7.
W.
Woo
,
J.
Benford
,
D.
Fittinghoff
,
G.
Harteneck
,
D.
Price
,
R.
Smith
, and
H.
Sze
, “
Phase locking of high power microwave oscillators
,”
J. Appl. Phys.
65
,
861
(
1989
).
8.
H.
Sze
,
D.
Price
, and
B.
Harteneck
, “
Phase locking of two strongly coupled vircators
,”
J. Appl. Phys.
67
,
2278
(
1989
).
9.
P.
Pengvanich
,
V. B.
Neculaes
,
Y. Y.
Lau
,
R. M.
Gilgenbach
,
M. C.
Jones
,
W. M.
White
, and
R. D.
Kowalczyk
, “
Modeling and experimental studies of magnetron injection locking
,”
J. Appl. Phys.
98
,
114903
(
2005
).
10.
P.
Pengvanich
,
Y.
Lau
,
E.
Cruz
,
R. M.
Gilgenbach
,
B.
Hoff
, and
J. W.
Luginsland
, “
Analysis of peer-to-peer locking of magnetrons
,”
Phys. Plasmas
15
,
103104
(
2008
).
11.
E. J.
Cruz
,
B. W.
Hoff
,
P.
Pengvanich
,
Y. Y.
Lau
,
R. M.
Gilgenbach
, and
J. W.
Luginsland
, “
Experiments on peer-to-peer locking of magnetrons
,”
Appl. Phys. Lett.
95
,
191503
(
2009
).
12.
R. M.
Rozental
,
N. S.
Ginzburg
,
M. Y.
Glyavin
,
A. S.
Sergeev
, and
I. V.
Zotova
, “
Mutual synchronization of weakly coupled gyrotrons
,”
Phys. Plasmas
22
,
093118
(
2015
).
13.
V. V.
Klinshov
and
V. I.
Nekorkin
, “
Synchronization of delay-coupled oscillator networks
,”
Phys. Usp.
56
,
1323
(
2013
).
14.
S. A.
Usacheva
and
N. M.
Ryskin
, “
Phase locking of two limit cycle oscillators with delay coupling
,”
Chaos
24
,
023123
(
2014
).
15.
A. B.
Adilova
,
S. A.
Gerasimova
, and
N. M.
Ryskin
, “
Bifurcation analysis of mutual synchronization of two oscillators coupled with delay
,”
Rus. J. Nonlinear Dyn.
13
(
1
),
3
(
2017
).
16.
See http://www.math.pitt.edu/∼bard/xpp/xpp.html for information about XPPAUT.
17.
K.
Engelborghs
,
T.
Luzyanina
, and
D.
Roose
, “
Numerical bifurcation analysis of delay differential equations using DDE-BIFTOOL
,”
ACM Trans. Math. Software
28
(
1
),
1
(
2002
).
18.
S.
Wirkus
and
R.
Rand
, “
The dynamics of two coupled van der Pol oscillators with delay coupling
,”
Nonlinear Dyn.
30
,
205
(
2002
).
19.
A. P.
Kuznetsov
,
N. V.
Stankevich
, and
L. V.
Turukina
, “
Coupled van der Pol–duffing oscillators: Phase dynamics and structure of synchronization tongues
,”
Physica D
238
,
1203
(
2009
).
20.
M. V.
Ivanchenko
,
G. V.
Osipov
,
V. D.
Shalfeev
, and
J.
Kurths
, “
Synchronization of two non-scalar-coupled limit-cycle oscillators
,”
Physica D
189
,
8
(
2004
).
21.
A. G.
Balanov
,
N. B.
Janson
,
V. V.
Astakhov
, and
P. V. E.
McClintock
, “
Role of saddle tori in the mutual synchronization of periodic oscillations
,”
Phys. Rev. E
72
,
026214
(
2005
).
22.
The XPPAUT allows plotting minimum and maximum of the corresponding variable for a limit cycle, which are shown by circles in Fig. 2. However, we cannot draw a similar plot for the stable limit cycle CS since it is non-contractible and, thus, the phase difference φ(t) exhibits the unbounded phase drift.
23.
V. S.
Tiberkevich
,
R. S.
Khymyn
,
H. X.
Tang
, and
A. N.
Slavin
, “
Sensitivity to external signals and synchronization properties of a non-isochronous auto-oscillator with delayed feedback
,”
Sci. Rep.
4
,
3873
(
2014
).
24.
M.
Wolfrum
and
S.
Yanchuk
, “
Eckhaus instability in systems with large delay
,”
Phys. Rev. Lett.
96
,
220201
(
2006
).
25.
M. I.
Balakin
and
N. M.
Ryskin
, “
Bifurcational mechanism of formation of developed multistability in a van der Pol oscillator with time-delayed feedback
,”
Rus. J. Nonlinear Dyn.
13
,
151
(
2017
).
You do not currently have access to this content.