Modelling the phase synchronization in systems of two and three coupled oscillators
We obtain regions of synchronization of two and three globally coupled oscillators, and describe the main mechanisms and bifurcations through which the systems synchronization is lost.
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Інститут математики НАН України
2004
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| Цитувати: | Modelling the phase synchronization in systems of two and three coupled oscillators / A.A. Vasylenko, Y.L. Maistrenko, M. Hasler // Нелінійні коливання. — 2004. — Т. 7, № 3. — С. 311-327. — Бібліогр.: 8 назв. — англ. |
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irk-123456789-1770192021-02-10T01:26:13Z Modelling the phase synchronization in systems of two and three coupled oscillators Vasylenko, A.A. Maistrenko, Y.L. Hasler, M. We obtain regions of synchronization of two and three globally coupled oscillators, and describe the main mechanisms and bifurcations through which the systems synchronization is lost. Отримано областi синхронiзацiї двох та трьох глобально зв’язаних осциляторiв. Описано основнi механiзми та бiфуркацiї, через якi втрачається синхронiзацiя в розглянутих системах. 2004 Article Modelling the phase synchronization in systems of two and three coupled oscillators / A.A. Vasylenko, Y.L. Maistrenko, M. Hasler // Нелінійні коливання. — 2004. — Т. 7, № 3. — С. 311-327. — Бібліогр.: 8 назв. — англ. 1562-3076 http://dspace.nbuv.gov.ua/handle/123456789/177019 517.9 en Нелінійні коливання Інститут математики НАН України |
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We obtain regions of synchronization of two and three globally coupled oscillators, and describe the main
mechanisms and bifurcations through which the systems synchronization is lost. |
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Article |
| author |
Vasylenko, A.A. Maistrenko, Y.L. Hasler, M. |
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Vasylenko, A.A. Maistrenko, Y.L. Hasler, M. Modelling the phase synchronization in systems of two and three coupled oscillators Нелінійні коливання |
| author_facet |
Vasylenko, A.A. Maistrenko, Y.L. Hasler, M. |
| author_sort |
Vasylenko, A.A. |
| title |
Modelling the phase synchronization in systems of two and three coupled oscillators |
| title_short |
Modelling the phase synchronization in systems of two and three coupled oscillators |
| title_full |
Modelling the phase synchronization in systems of two and three coupled oscillators |
| title_fullStr |
Modelling the phase synchronization in systems of two and three coupled oscillators |
| title_full_unstemmed |
Modelling the phase synchronization in systems of two and three coupled oscillators |
| title_sort |
modelling the phase synchronization in systems of two and three coupled oscillators |
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Інститут математики НАН України |
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2004 |
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http://dspace.nbuv.gov.ua/handle/123456789/177019 |
| citation_txt |
Modelling the phase synchronization in systems of two and three coupled oscillators / A.A. Vasylenko, Y.L. Maistrenko, M. Hasler // Нелінійні коливання. — 2004. — Т. 7, № 3. — С. 311-327. — Бібліогр.: 8 назв. — англ. |
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Нелінійні коливання |
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| first_indexed |
2025-07-15T14:58:46Z |
| last_indexed |
2025-07-15T14:58:46Z |
| _version_ |
1837725412748165120 |
| fulltext |
UDC 517.9
MODELLING THE PHASE SYNCHRONIZATION IN SYSTEMS OF TWO
AND THREE COUPLED OSCILLATORS
МОДЕЛЮВАННЯ ФАЗОВОЇ СИНХРОНIЗАЦIЇ В СИСТЕМАХ ДВОХ
I ТРЬОХ ЗВ’ЯЗАНИХ ОСЦИЛЯТОРIВ
A. Vasylenko
Inst. Math. Nat. Acad. Sci. Ukraine
Ukraine, 01601, Kyiv 4, Tereshchenkivska Str., 3
e-mail: Ann@imath.kiev.ua
Yu. Maistrenko
Inst. Math. Nat. Acad. Sci. Ukraine
Ukraine, 01601, Kyiv 4, Tereshchenkivska Str., 3
M. Hasler
Swiss Federal Institute Technol.
CH-1015, Lausanne, Switzerland
We obtain regions of synchronization of two and three globally coupled oscillators, and describe the main
mechanisms and bifurcations through which the systems synchronization is lost.
Отримано областi синхронiзацiї двох та трьох глобально зв’язаних осциляторiв. Описано
основнi механiзми та бiфуркацiї, через якi втрачається синхронiзацiя в розглянутих системах.
1. Introduction. Phase synchronization is one of the fundamental and universal nonlinear
phenomena in physics, engineering, biology, economics and other fields of science and techno-
logy [1 – 4]. Synchronization occurs if some oscillatory systems interact and adjust their behavi-
ours to one another in order to achieve a state where they oscillate in unison, the state of
synchronization. Well-known examples of the synchronization are the example of two clocks
or pendulums hanging on a wall which come to the synchronized state due to the interaction
through the wall, the synchronization of the rotation of the moon with its orbital motion, so
that the moon always shows the same side towards the earth.
In the present paper we investigate the phase synchronization of two and three globally
coupled nonlinear oscillators (or clocks) whose individual dynamics are described by the shift
circle maps. Ensembles of the coupled oscillators model a wide variety of different phenomena.
Investigation of their dynamics is particularly of interest in connection with different engineeri-
ng and communication problems, for example transmitting information [5].
The paper is organized as follows. In Section 2 first we introduce the two-dimensional
discrete-time system of the globally coupled circle maps coupled with the sine-coupling, and bri-
efly describe the dynamics of analogue continuous-time model emphasizing some differences
and common features of these two systems. Then we describe the dynamics of the discrete-
time system of the coupled oscillators. We outline the regions of the phase synchronization and
explain the bifurcations through which the synchronization is lost. In Section 3 we carry out a
c© A. Vasylenko, Yu. Maistrenko, and M. Hasler, 2004
ISSN 1562-3076. Нелiнiйнi коливання, 2004, т . 7, N◦ 3 311
312 A. VASYLENKO, YU. MAISTRENKO, AND M. HASLER
similar analysis for three globally coupled circle maps, starting from several particular cases and
coming to their generalization. In particular we outline the synchronization regions for three
identical oscillators and then for the system of two identical oscillators and the third one being
different from them. For this latter case we also investigate in detail how the system equilibria
change with varying the parameter values. The paper ends with the section of conclusions.
2. Synchronization in the system of two coupled oscillators. 2.1. The model. Let us consider
the system of two coupled oscillators or clocks whose dynamics are described by the Arnold
circle maps
xn+1 = xn + 2πν1 +
K
2
sin (yn − xn) ,
(1)
yn+1 = yn + 2πν2 +
K
2
sin (xn − yn) ,
where both the variables x and y are calculated modulo 2π. System (1) describes the dynamics of
two phase oscillators with the corresponding frequencies ν1 and ν2, which are mutually coupled
with the sine coupling and K is the coupling strength.
The synchronization of the oscillators can be expressed in terms of the difference between
the phase variables xn and yn. After subtracting the second equation of the system from the
first one the following equation for the phase difference is obtained:
xn+1 − yn+1 = xn − yn + 2π (ν1 − ν2)−K sin (xn − yn)
or after introducing the new variable zn = xn − yn,
zn+1 = zn + 2π (ν1 − ν2)−K sin zn, (2)
where z is calculated modulo 2π. Therefore we can conclude that the dynamics of the phase di-
fference are described by the well known sine circle map, and the synchronization for the system
(1) can be achieved, e.g. with the parameter values for which the map (2) has an attracting fixed
point.
2.2. Continuous-time Kuramoto system. It is worth emphasizing that the system (1) is a
discrete analog of a so called Kuramoto system [6],
ψ̇i = ωi +
K
N
N∑
j=1
sin (ψj − ψi) . (3)
The simplest example N = 2,
ψ̇1 = ω1 +
K
2
sin (ψ2 − ψ1) ,
(4)
ψ̇2 = ω2 +
K
2
sin (ψ1 − ψ2) ,
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MODELLING THE PHASE SYNCHRONIZATION IN SYSTEMS OF TWO AND THREE COUPLED OSCILLATORS 313
Fig. 1. Stability diagram for the equation (5). For the parameter
values from the shaded region a stable equilibrium exists,
and therefore the Kuramoto system (4) is synchronized.
where 0 ≤ ψi ≤ 2π. If we consider the difference between the equations of this system, then
we obtain the following differential equation:
ϕ̇ = ω −K sinϕ, (5)
where ϕ = ψ1 − ψ2 is the difference of the phase variables and ω = ω1 − ω2 the difference
of the angular frequencies. The fixed point ϕ∗ = arc sin
ω
K
corresponds to the constant phase
difference ψ1 − ψ2 and therefore is a state of synchronization. This fixed point exists and is
stable for the values of ω and K such that
∣∣∣ ω
K
∣∣∣ ≤ 1, as shown in Fig. 1, and there are no other
periodic orbits for this equation.
2.3. Synchronization regions for the discrete-time model. An analysis of the dynamics of
the discrete Kuramoto model (1) can be performed following the ideology of the paper [7].
The Arnol’d sine circle map (2) describes the dynamics of the phase difference for our system
of coupled oscillators, and depending on the parameter values this map exhibits periodic or
quasiperiodic behaviour. While considering two identical oscillators in (1), i. e., ν1 = ν2, the
map (2) has the fixed point z∗ = 0 which corresponds to the full synchronization in (1),
|xn − yn| → 0 as n → ∞. For the oscillators with different frequencies ν1 and ν2 the fixed
point is z∗ = arc sin
2π (ν1 − ν2)
K
, and its stability, if it takes place, means that the oscillators
are synchronized but with a nonzero phase difference |xn − yn| → z∗ as n → ∞. Unlike the
case of the continuous-time Kuramoto system, the map (2) can also exhibit synchronized peri-
odic orbits of higher periods as well as synchronized chaotic motion, when there exist C > 0:
|zn| < C ∀n. Coming back to the variables xn and yn, existence and stability of the q-periodic
orbit of the map (2) means that the oscillators are synchronized so that their phase difference
zn = xn − yn changes between q constant values. Therefore we have 1:1 synchronization here.
In general k : m synchronization can be defined as the following phase-locking:
|kxn −myn| < const
for all integers n. For stable periodic orbits of the map (2) the following phase difference is
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314 A. VASYLENKO, YU. MAISTRENKO, AND M. HASLER
bounded:
|xn − yn| < const
and, for the corresponding parameter values, 1 : 1 synchronization of the oscillators (1) takes
place. Nevertheless one should distinguish between the periodic orbits with zero and nonzero
rotation numbers. Only the former correspond to the synchronization in the system, as such
trajectories are bounded and they exist inside the Arnol’d tongue for the rotation number
zero. Stable periodic orbits that exist inside all other phase-locking regions of nonzero rotation
numbers do not lead to a synchronization in the system since such trajectories tend to infinity
in the corresponding lift from the circle [0; 2π) to R1.
Figure 2 shows phase-locking regions, the so-called Arnol’d tongues of the following rotati-
on numbers: ρ = 0,
1
3
,
1
2
,
1
1
. Figure 3 gives an enlargement with some very narrow tongues.
Each tongue emanates from a point (p/q, 0) for integer p and q, and corresponds to the exi-
stence of stable and unstable periodic orbits of period q and of the rotation number p/q. Rotati-
on number is a number between 0 and 1 [8], which measures the average amount points are
rotated by an iteration of the map. It is defined as follows:
ρ =
1
2π
lim
n→∞
zn − z0
n
,
where zn in (2) is calculated without the mod 2π term.
For the parameter values in between the Arnol’d tongues we have quasiperiodic behaviour,
and the phase difference zn varies in the interval [0, 2π) but never returns to its original value,
therefore there is no synchronization of the oscillators (1) for these values of the parameters.
For the values of K such that K < 1 the map (2) is invertible, the Arnol’d tongues do
not intersect, and the system can exhibit only periodic or quasiperiodic dynamics. We are also
interested in larger values ofK since forK > 1 significant regions of the phase synchronization
can be found. Moreover, for these parameters values some interesting phenomena which do not
occur for the invertible as well as time-continuous systems can be observed.
The widest phase-locking region for the rotation number ρ = 0 is bounded by the line
K = 2π (ν1 − ν2) which is a saddle-node bifurcation line for the stable and unstable fixed
points existing inside the tongue. It is the lower desynchronization line in Fig. 2, and for the
parameter values such that K < 2π (ν1 − ν2) no fixed point exists.
Beyond the line K = 1 the map (2) becomes noninvertible. With an increase in K insi-
de the tongue, a period-doubling cascade occurs until the dynamics become chaotic and the
synchronization is lost at the upper desynchronization line.
The graphs of the sine circle map (2) in Fig. 4 can explain different types of the system
behaviour in more details. Fig. 4 (a) shows the simplest situation, when all the trajectories are
attracted to the stable fixed point. The first boundary crisis line BC1 in Fig. 2 corresponds to
the graph in Fig. 4 (b), when the maximum value of the map becomes equal to the value of the
unstable fixed point z(u) +2π. The second boundary crisis line BC2 represents the analogous si-
tuation but for the minimum value which becomes equal to z(u)−2π. Fig. 4 (d) shows the graph
of the map at the point where these two lines cross. The line BC3 corresponds to the situation
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MODELLING THE PHASE SYNCHRONIZATION IN SYSTEMS OF TWO AND THREE COUPLED OSCILLATORS 315
Fig. 2. Region of the phase synchronization of the system (1) and bifurcation curves for the sine
circle map (2): SN saddle-node, PD period-doubling, BC boundary crisis; the fractions
show the rotation numbers for the corresponding Arnol’d tongues; the region of the
rectangle is enlarged in Fig. 3; the points a – g show the parameter values for the graphs
of the map (2) in Figs. 4 and 5.
Fig. 3. Enlargement of the rectangle from Fig. 2.
when the minimum value reaches the values of the channel appearing after the first boundary
crisis (see Fig. 4 (f)). Then, the trajectories come to the unstable fixed point z(u) + 2π after the
second iteration.
As shown in Fig. 4 (c) after the first crisis BC1, depending on initial conditions a trajectory
of the map (2) can be immediately attracted to the stable fixed point z(s), or it can make one or
more revolutions around the circle and eventually be attracted by z(s). Both these possibilities
ISSN 1562-3076. Нелiнiйнi коливання, 2004, т . 7, N◦ 3
316 A. VASYLENKO, YU. MAISTRENKO, AND M. HASLER
a b
c d
e f
Fig. 4. Appearance of the desynchronous orbits in the map (2); the parameter values for the graphs of the
map correspond to the points a – f from Fig. 2.
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MODELLING THE PHASE SYNCHRONIZATION IN SYSTEMS OF TWO AND THREE COUPLED OSCILLATORS 317
Fig. 5. The superstable period-2 orbit of the map (2) with the pa-
rameter values at the point g from the Fig. 2: ν1−
−ν2 = 0, 33, K = 2, 4; the rotation number ρ = 1/2.
lead clearly to zero rotation number for the trajectory examined. Furthermore the trajectory
can make infinitely many rotations, i. e., tend to infinity in the corresponding lift from the circle
[0; 2π) to R1. This behaviour can lead to a nonzero rotation number and, hence, desynchroni-
zation. As an example, in Fig. 5 we showed the superstable trajectory with rotation number
ρ =
1
2
, which is a period-2 orbit, attracting all the trajectories that are not attracted to any of
the fixed points. Therefore for the same parameter values beyond BC1-crisis one can observe
trajectories with different rotation numbers depending on the original point from which a
trajectory is calculated. Such a situation means a coexistence of synchronization and desynchroni-
zation.
Finally, after the boundary crisis BC3, almost all trajectories do not exhibit zero rotation
numbers, and consequently, there is no synchronization for these parameter values. Fig. 4 (e, f)
show trajectories and the graphs of (2) before and after the crisis BC3, correspondingly. Before
the crisis the fixed point has already lost its stability but there exist other attracting states as
well as desynchronized trajectories which have arisen after the first BC1 crisis. But after the
BC3 crisis no more synchronized orbits exist and almost all trajectories tend to infinity.
Figure 6 shows that for many different pairs of K and ν1 − ν2 one can observe at least two
different rotation numbers if the trajectories are calculated from different initial conditions. To
obtain this figure, for every pair of the parameter values we considered about 300 initial condi-
tions uniformly distributed in the interval [0, 2π) and obtained either zero or some particular
nonzero rotation number. Figures 2 and 3 show several of the widest and most well defined
regions of such a coexistence where two different Arnol’d tongues intersect.
For small values of the difference ν1 − ν2 the first boundary crisis line coincides with the
upper desynchronization line, the line beyond which almost all trajectories do not exhibit zero
rotation numbers (see Fig. 2). In general for the values of K beyond this line one can find di-
fferent multistability regions as well as some trajectories with rotation number zero. To illustrate
the loss of the synchronization with increasing K, in Fig. 7 we plotted the rotation numbers ρ
of the trajectories for different nonlinearities K while ν1 − ν2 were fixed with the values 0,005,
ISSN 1562-3076. Нелiнiйнi коливання, 2004, т . 7, N◦ 3
318 A. VASYLENKO, YU. MAISTRENKO, AND M. HASLER
Fig. 6. Rotation numbers of the trajectories of the map (2) starting
from different initial conditions. The values of ν1 − ν2 and
K are changed inside the Arnol’d tongue for ρ = 0, along the
right boundary.
Fig. 7. Rotation numbers ρ for the trajectories of the map (2).
0,05 and 0,1. For every pair of the parameter values (ν1 − ν2,K) we calculated ρ for trajectories
with about 300 different initial conditions. As shown in Fig. 7 for these parameter values multi-
stability is not typical as there are almost no intersections of the Arnol’d tongues. One can see
that for smaller values of ν1 − ν2 after desynchronization the rotation numbers change from
zero to very small positive or negative values and a number of synchronized states can also
be observed. With an increase in the difference of frequencies, the rotation numbers become
nonzero more abruptly and take larger values. Figure 8 shows an overview of the parameter
region beyond our desynchronization line for which one can observe the trajectories with rotati-
on number smaller than 0,001 in absolute value.
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MODELLING THE PHASE SYNCHRONIZATION IN SYSTEMS OF TWO AND THREE COUPLED OSCILLATORS 319
Fig. 8. Black points denote the values of the parameters for which a
trajectory exists with rotation number smaller than 0,001 in
absolute value.
3. Dynamics of the three-dimensional system. Now let us continue our analysis for the
following system of three globally coupled phase oscillators:
xn+1 = xn + 2πν1 +
K
3
[
sin (yn − xn) + sin (zn − xn)
]
,
yn+1 = yn + 2πν2 +
K
3
[
sin (xn − yn) + sin (zn − yn)
]
, (6)
zn+1 = zn + 2πν3 +
K
3
[
sin (xn − zn) + sin (yn − zn)
]
,
where the phase variables are calculated modulo 2π. After successive subtracting the second
and the third equations from the first one, our system can be reduced to the following two-
dimensional system:
un+1 = un + 2π∆1 +
K
3
[
−2 sinun + sin vn − sin (un + vn)
]
,
(7)
vn+1 = vn + 2π∆2 +
K
3
[
−2 sin vn + sinun − sin (un + vn)
]
,
where un = xn − yn, vn = zn − xn, ∆1 = ν1 − ν2, ∆2 = ν3 − ν1.
Let us consider different cases of the system (7).
3.1. Symmetric case ∆1 = ∆2 = 0. This is the case of three identical phase oscillators,
i. e., their frequencies are equal, ν1 = ν2 = ν3. The system (7) has the following six fixed points
with corresponding eigenvalues:
O (0, 0), λ1 = λ2 = 1−K, which is an attracting node for K < 2,
S1 (π, 0), S2 (π, π), S3 (0, π), λ1 = 1− K
3
, λ2 = 1 +K, saddles with |λ1| < 1 for K < 6 and
λ2 > 1 for all positive K,
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320 A. VASYLENKO, YU. MAISTRENKO, AND M. HASLER
Fig. 9. Phase portrait of the system (7) with
∆1 = ∆2 = 0.
R1
(2π
3
,
2π
3
)
, R2
(4π
3
,
4π
3
)
, λ1 = λ2 = 1 +
K
2
, repelling nodes for all positive K.
Phase portrait of the system with ∆1 = ∆2 = 0 is shown in Fig. 9.
3.2. ∆1 = ∆2 > 0. Denote ∆ df= ∆1 = ∆2. Then for the system (7) which is defined on
the two-dimensional torus T2 = [0, 2π)2, the diagonal D =
{
(un, vn) ∈ T2 : un = vn
}
is an
invariant manifold. This implies that the points from D are again mapped into some points on
this line, while for any nonvanishing difference between the values of ∆1 and ∆2 the dynamics
becomes two dimensional and the diagonal is not invariant anymore.
In the manifold D, system (7) is reduced to the following one-dimensional map:
un+1 = un + 2π∆− K
3
[
sin(un) + sin(2un)
]
. (8)
The map has two pairs of fixed points O, R1 and S2, R2, which are born in the saddle-
node bifurcations at K ≈ 1, 7 · 2π∆ and K ≈ 8, 1 · 2π∆. Figure 10 shows the corresponding
saddle-node bifurcation lines as well as several major Arnol’d tongues. Both stable fixed points
O and S2 lose their stabilities through period-doubling bifurcations. The bifurcation line for O
is denoted as PD in Fig. 10 (for S2 the bifurcation occurs for K > 6). As shown in Fig. 10 other
Arnol’d tongues cross the widest tongue of the rotation number ρ = 0 beyond the boundary
crisis line, and this implies coexistence of the trajectories with different rotation numbers, as
was described for the sine circle map, see Section 2.3.
As was described above for the sine circle map (2), appearance of the desynchronous orbi-
ts can be caused by a boundary crisis, when the maximum or the minimum value of the map
becomes equal to the corresponding value of the unstable fixed point. The map (8) is bimodal
and has two pairs of the fixed points. Therefore for this map a few additional boundary cri-
ses may occur as compared with the map (2). Fig. 11 shows the regions of synchronization
and coexistence of synchronous and desynchronous trajectories for the map (8) and Fig. 12
illustrates appearance of the desynchronous trajectories after the crises BC1 – BC5 as denoted
in Fig. 11.
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MODELLING THE PHASE SYNCHRONIZATION IN SYSTEMS OF TWO AND THREE COUPLED OSCILLATORS 321
Fig. 10. Arnol’d tongues for the system (6) with ∆1 = ∆2 = ∆ > 0 and
un = vn; the bifurcation curves for the map (8) are denoted as
follows: SN O&R1, SN S2&R2 the saddle-node bifurcations for
the corresponding fixed points, BC1, BC2 are boundary crises.
Fig. 11. Regions of the synchronization and coexistence of synchronous and
desynchronous trajectories for the map (8), the points a – h corres-
pond to parameter values for the graphs in Fig. 12, BC1 – BC5
denote the boundary crisis lines.
ISSN 1562-3076. Нелiнiйнi коливання, 2004, т . 7, N◦ 3
322 A. VASYLENKO, YU. MAISTRENKO, AND M. HASLER
a b
c d
e f
g h
Fig. 12. Appearance of desynchronous orbits for the map (8) with the parameter values at the points a – h
from Fig. 11, the designations BC1 – BC5 correspond to the boundary crises as in Fig. 11.
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MODELLING THE PHASE SYNCHRONIZATION IN SYSTEMS OF TWO AND THREE COUPLED OSCILLATORS 323
Coming back to the two-dimensional system (7), first we shall look at the transverse stabi-
lity of the fixed points O and S2 or, more generally, the transverse stability of the invariant
manifold D. For any point on the diagonal, the system (7) has two eigendirections: (1, 1) along
the diagonal and (1,−1) perpendicular to it. The eigenvalue that corresponds to the transversal
direction is 1 −K cosun and for any orbit {u}∞n=1 in the diagonal D the transversal Lyapunov
exponent λ⊥ is equal to
λ⊥ = lim
N→∞
1
N
N∑
n=1
ln |1−K cosun| .
The transversal Lyapunov exponent λ⊥ gives an insight into stability of the periodic orbits
or chaotic one-dimensional attractors in the plane. If λ⊥ for an attractor existing on the di-
agonal is negative then this attractor attracts almost all trajectories from its two-dimensional
neighborhood. In Fig. 10 we outlined the parameters regions for which the fixed points and
some periodic orbits attract trajectories from the diagonal D, and in Fig. 13 the curves wi-
th hachures separate regions where the transversal Lyapunov exponents of the corresponding
attractors are negative (area with hachures) and positive (the other side). Thus the fixed point
O and the higher-periodic orbits existing in the main Arnol’d tongues are transversally stable
for smaller values of coupling K, while the periodic orbits born in the period-doubling cascade
from the fixed point O remain transversally stable for larger K but lose their stability with an
increase in ∆. In Fig. 14 we plotted the graphs of λ⊥ of the points O and S2 as functions of K
for ∆ = 0, 001. They show that the fixed point S2 is always transversally unstable, while O loses
its transverse stability after K = 2, at the corresponding bifurcation line with hachures which
is shown in Fig. 13.
Figure 13 also shows the line of the transverse pitchfork bifurcation denoted anP S1&R1&S3.
In the bifurcation the saddle point R1 transforms to an unstable node giving rise to two saddles
S1 and S3. The phase portrait of the system (7) after this bifurcation is shown in Fig. 15.
3.3. ∆1 = 0, ∆2 ∈ [0, 1]. Let us fix the first difference ∆1 = 0 and consider the parameter
plane (∆2;K). In this case the vn-axis V0 =
{
(un, vn) ∈ T2 : un = 0
}
is an invariant manifold,
and the dynamics in the manifold is governed by the sine circle map (2). Therefore the regions of
existence and stability of the fixed point and other periodic orbits inside the manifold coincide
with the Arnol’d tongues of the one-dimensional sine circle map (2), some of these regions
are shown in Fig. 16. Apart from the manifold, other attractors exist outside the vn-axis. As
shown in Fig. 9, in the case ∆1 = ∆2 = 0, there exist 6 fixed points. Four of them, S1, S2,
R1 and R2, are located outside the manifold V0. We find that with increasing ∆2 they disappear
simultaneously through the saddle-node bifurcations atK ≈ 4·2π∆2. The corresponding phase
portraits before and after the bifurcations are shown in Fig. 17. With further increase in ∆2 the
fixed points O and S3 collide and disappear in the saddle-node bifurcation at K = 2π∆2, and
this implies desynchronization of the system.
With an increase in the coupling K, the stable fixed point O loses its transverse stability
through a period-doubling bifurcation (PD1 curve in Fig. 16) and the stability in V0 through the
bifurcation PD2. With more increase in K, the period-2 orbit which was born in the bifurcation
PD1 undergoes a Naimark – Sacker bifurcation (NS curve in Fig. 16), while after the bifurcation
PD2, which takes place in the manifold V0, a period-doubling cascade occurs. All the points of
the periodic orbits of this cascade have un = 0 and therefore remain on the vn-axis.
3.4. General case ∆1 6= ∆2. Let us consider several nonzero values of ∆1 in the system
(7) and investigate how the shape of the phase-locking tongues in the plane (∆2;K) changes
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324 A. VASYLENKO, YU. MAISTRENKO, AND M. HASLER
Fig. 13. Regions of the transverse stability of several higher-periodic or-
bits, the curves with hachures denote the boundaries of the
regions where the transversal Lyapunov exponents of the corres-
ponding attractors are negative; P S1&R1&S3 is the transverse
pitchfork bifurcation.
Fig. 14. Variation of the transversal Lyapunov exponents of the fixed
points O and S2 with the coupling parameter K for ∆ = 0, 001
in the map (8).
for different fixed values of ∆1. Figure 18 shows several major Arnol’d tongues for ∆1 = 0, 1.
One can see that their symmetry with respect to the line ∆2 = 0, 5 is lost, as we have the
interaction of two nontrivial frequencies in the system. Moreover the synchronization regions
do not emerge from the points on the ∆2-axis and do not have the exact shape of the tongues
any more. This means that for small values of K ≤ 0, 5 the system (6) cannot be synchroni-
zed. With further increase in ∆1 the synchronization regions move away from the ∆2-axis and
become smaller, since for larger ∆1 it becomes more difficult for the system to achieve the
synchronization.
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MODELLING THE PHASE SYNCHRONIZATION IN SYSTEMS OF TWO AND THREE COUPLED OSCILLATORS 325
Fig. 15. Phase portrait of the system (7) with the
values of ∆ and K after the pitchfork
bifurcation S1&R1&S3 has occurred.
Fig. 16. Arnol’d tongues for the system (6) when the difference ∆1 = 0. The
dashed lines correspond to the bifurcations of the attractors existing
or emerging outside the manifold V0; the bifurcation lines are denoted
as follows: SN saddle-node, 2 SNs two simultaneous saddle-node bi-
cations, PD1 transverse period-doubling, PD2 period-doubling in the
manifold V0, NS Naimark – Sacker bifurcations for the period-2
orbit born in the transverse period-doubling bifurcation.
Fig. 17. Phase portraits of the system (7) with ∆1 = 0 before and after the saddle-
node bifurcations S1&R1 and S2&R2.
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326 A. VASYLENKO, YU. MAISTRENKO, AND M. HASLER
Fig. 18. Regions of stability of higher-periodic orbits of the system (6) when
∆1 = 0, 1. The bifurcation lines: SN saddle-node, PD the first peri-
od-doubling, NS the Naimark – Sacker bifurcation for the period-2
orbit born in the transverse period-doubling bifurcations.
4. Conclusions. In this paper we considered the dynamics of two- and three-dimensional
systems of globally coupled shift circle maps with sine-coupling. To investigate the phase synch-
ronization in the systems we considered the corresponding maps for the differences of our ori-
ginal phase variables and studied stability and bifurcations of the equilibria. We also compared
the discrete-time model with the continuous-time Kuramoto system and concluded that in
the case of the difference equations the system exhibits many more equilibria and dynamical
phenomena which cannot be observed for the Kuramoto system.
In the case N = 2 the dynamics of the difference xn − yn are described by the well-known
Arnol’d sine circle map. We plotted several Arnol’d tongues for the higher-periodic orbits and
explained how the loss of synchronization occurs with varying the parameters. We also emphasi-
zed that for values of the coupling constantK larger than 1, when Arnol’d map is noninvertible,
for the same parameter values one can observe trajectories with different rotation numbers in
the regions of intersection of different Arnol’d tongues, and hence coexistence of synchronous
and desynchronous orbits. We described in detail the boundary crises and bifurcations occurri-
ng in our system.
When N = 3 the system for the differences xn − yn and zn − xn is two-dimensional and
more complicated than the single equation in the caseN = 2. Therefore for the system of three
coupled oscillators we considered several particular cases. In the case when all three oscillators
have the same frequencies we found the exact coordinates of the fixed points and plotted the
phase portrait of the system. Then we considered the case when the frequencies of two of our
oscillators differ from the frequency of the third one by some quantities that are opposite in
sign but equal in absolute value. Under such a condition the problem can be reduced to the
one-dimensional equation for the difference xn − yn. This map describes the dynamics on the
diagonal which is an invariant manifold in the case. We explained the trajectories behaviour in
the manifold as well as the transversal stability of the fixed points and higher-periodic orbits
of the map. Furthermore we considered a two-dimensional case when the difference between
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MODELLING THE PHASE SYNCHRONIZATION IN SYSTEMS OF TWO AND THREE COUPLED OSCILLATORS 327
the frequencies of two of the oscillators is fixed to be zero and the third oscillator differs from
the two others. In this case the vn-axis is an invariant manifold and the dynamics on the mani-
fold are described by the one-dimensional sine circle map. Apart from the attractors on the
manifold several other fixed points exist, and we plotted the corresponding phase portraits and
described the bifurcations for these equilibria. Finally we considered the general case when all
the frequencies of our oscillators are different, and we explained briefly how the phase-locking
regions change with changing the differences of the frequencies.
5. Acknowledgements. A. Vasylenko and Yu. Maistrenko are grateful for the hospitality
of the Laboratory of Nonlinear Systems, Swiss Federal Institute of Technology. A. Vasylenko
acknowledges the scholarship of the Swiss Government.
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336 p.
Received 27.04.2004
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