Stability of periodic clusters in globally coupled maps
The phenomenon of partial synchronization, — or clustering, — in a system of globally coupled C 1 - smooth maps is analyzed. We prove stability of equally populated K-clustered states with period-n temporal dynamics, referred to as PnCK-states. For this, we first obtain formulas giving relation b...
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irk-123456789-1758372021-02-03T01:28:17Z Stability of periodic clusters in globally coupled maps Panchuk, A.A. Maistrenko, Y.L. The phenomenon of partial synchronization, — or clustering, — in a system of globally coupled C 1 - smooth maps is analyzed. We prove stability of equally populated K-clustered states with period-n temporal dynamics, referred to as PnCK-states. For this, we first obtain formulas giving relation between longitudinal and transverse nultipliers of the in-cluster periodic orbits and then, using these formulas, find exact parameter intervals for the transverse stability. We conclude that typically, for the symmetric PnCK-states, in-cluster stability implies transverse stability. Moreover, transverse stability can take place even if the incluster dynamics is unstable. Проводиться аналiз явища часткової синхронiзацiї, або кластеризацiї, в системi глобально зв’язаних вiдображень гладкостi C 1 . Розглядаються K-кластернi стани з n-перiодичною динамiкою, якi називаються PnCK-станами, i доводиться їх стiйкiсть. Для цього спочатку отримано формули, що пов’язують поздовжнi та трансверсальнi мультиплiкатори кластеризованих перiодичних орбiт, а потiм з допомогою цих формул знайдено точнi межi iнтервалiв для трансверсальної стiйкостi. Зроблено висновок, що для симетричних PnCK-станiв iз стiйкостi всерединi кластера випливає стiйкiсть трансверсальна. Бiльше того, навiть у випадку, коли динамiка всерединi кластера нестiйка, трансверсальна стiйкiсть може мати мiсце 2002 Article Stability of periodic clusters in globally coupled maps / A.A. Panchuk, Y.L. Maistrenko // Нелінійні коливання. — 2002. — Т. 5, № 3. — С. 334-345. — Бібліогр.: 15 назв. — англ. 1562-3076 http://dspace.nbuv.gov.ua/handle/123456789/175837 517.9 en Нелінійні коливання Інститут математики НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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English |
| description |
The phenomenon of partial synchronization, — or clustering, — in a system of globally coupled C
1
-
smooth maps is analyzed. We prove stability of equally populated K-clustered states with period-n temporal dynamics, referred to as PnCK-states. For this, we first obtain formulas giving relation between longitudinal and transverse nultipliers of the in-cluster periodic orbits and then, using these formulas, find exact
parameter intervals for the transverse stability. We conclude that typically, for the symmetric PnCK-states,
in-cluster stability implies transverse stability. Moreover, transverse stability can take place even if the incluster dynamics is unstable. |
| format |
Article |
| author |
Panchuk, A.A. Maistrenko, Y.L. |
| spellingShingle |
Panchuk, A.A. Maistrenko, Y.L. Stability of periodic clusters in globally coupled maps Нелінійні коливання |
| author_facet |
Panchuk, A.A. Maistrenko, Y.L. |
| author_sort |
Panchuk, A.A. |
| title |
Stability of periodic clusters in globally coupled maps |
| title_short |
Stability of periodic clusters in globally coupled maps |
| title_full |
Stability of periodic clusters in globally coupled maps |
| title_fullStr |
Stability of periodic clusters in globally coupled maps |
| title_full_unstemmed |
Stability of periodic clusters in globally coupled maps |
| title_sort |
stability of periodic clusters in globally coupled maps |
| publisher |
Інститут математики НАН України |
| publishDate |
2002 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/175837 |
| citation_txt |
Stability of periodic clusters in globally coupled maps / A.A. Panchuk, Y.L. Maistrenko // Нелінійні коливання. — 2002. — Т. 5, № 3. — С. 334-345. — Бібліогр.: 15 назв. — англ. |
| series |
Нелінійні коливання |
| work_keys_str_mv |
AT panchukaa stabilityofperiodicclustersingloballycoupledmaps AT maistrenkoyl stabilityofperiodicclustersingloballycoupledmaps |
| first_indexed |
2025-07-15T13:17:02Z |
| last_indexed |
2025-07-15T13:17:02Z |
| _version_ |
1837719013109530624 |
| fulltext |
UDC 517. 9
STABILITY OF PERIODIC CLUSTERS IN GLOBALLY COUPLED MAPS
СТIЙКIСТЬ ПЕРIОДИЧНИХ КЛАСТЕРIВ
У СИСТЕМI ГЛОБАЛЬНО ЗВ’ЯЗАНИХ ВIДОБРАЖЕНЬ
A. A. Panchuk∗, Yu. L. Maistrenko∗∗
Inst. Math. Nat. Acad. Sci. Ukraine
Tereshchenkivs’ka St., 3, Kyiv, 01601, Ukraine
The phenomenon of partial synchronization, — or clustering, — in a system of globally coupled C1-
smooth maps is analyzed. We prove stability of equally populated K-clustered states with period-n tempo-
ral dynamics, referred to as PnCK-states. For this, we first obtain formulas giving relation between longi-
tudinal and transverse nultipliers of the in-cluster periodic orbits and then, using these formulas, find exact
parameter intervals for the transverse stability. We conclude that typically, for the symmetric PnCK-states,
in-cluster stability implies transverse stability. Moreover, transverse stability can take place even if the in-
cluster dynamics is unstable.
Проводиться аналiз явища часткової синхронiзацiї, або кластеризацiї, в системi глобально зв’я-
заних вiдображень гладкостi C1. Розглядаються K-кластернi стани з n-перiодичною динамi-
кою, якi називаються PnCK-станами, i доводиться їх стiйкiсть. Для цього спочатку отрима-
но формули, що пов’язують поздовжнi та трансверсальнi мультиплiкатори кластеризованих
перiодичних орбiт, а потiм з допомогою цих формул знайдено точнi межi iнтервалiв для транс-
версальної стiйкостi. Зроблено висновок, що для симетричних PnCK-станiв iз стiйкостi все-
рединi кластера випливає стiйкiсть трансверсальна. Бiльше того, навiть у випадку, коли ди-
намiка всерединi кластера нестiйка, трансверсальна стiйкiсть може мати мiсце.
1. Introduction. The highly complex nature of different dynamical systems in various areas of
science, such as physics, biology, and other natural sciences, has attracted recently a growing
interest. In this context, new interesting phenomena, such as partial synchronization, have been
discovered. This new properties are investigated from both practical and theoretical viewpoints
[1 – 9]. The effects of synchronization and partial synchronization could be observed in a great
variety of applied problems, such as pattern formation, Josephson junction arrays, multimode
lasers, charge-density waves, insulin secretion, oscillatory neuronal systems, and so on [10 – 13].
The paper presented has an aim to investigate the stability of partially synchronized states
in a system of globally coupled maps. Together with the “ordinary” stability, the notion of
transverse stability is also of great importance for partially synchronized systems [1, 2]. An
attractor of such a clustered system could be represented as a stable set that belongs to a mani-
fold having lower dimension than the whole initial space. Transverse stability means that the
attractor is stable not only inside the manifold, but also from outside, i.e., along all of the basic
vectors of the whole space. And such an attractor loses its stability when at least one of the
clusters has split up, i.e., when the attractor spreads over the manifold of higher dimension.
One of the main results of the work presented is the analytical proof of the fact that the
∗ A. A. Panchuk acknowledges the INTAS organization for the grant YSF 01 – 165.
∗∗ This work has been financially portially supported by the Swiss National Science Foundation SCOPES
(project N◦ 7SUPJ062310).
c© A. A. Panchuk, Yu. L. Maistrenko, 2002
334 ISSN 1562-3076. Нелiнiйнi коливання, 2002, т . 5, N◦ 3
STABILITY OF PERIODIC CLUSTERS IN GLOBALLY COUPLED MAPS 335
transverse stability results from the longitudinal stability for absolutely symmetric clustered
periodic states of different periods.
2. The model. In this paper we study the dynamics of a system of N globally coupled one-
dimensional maps, which initially was introduced by K. Kaneko [14, 15],
Fε : xti 7→ (1− ε)f
(
xti
)
+
ε
N
N∑
j=1
f
(
xtj
)
, i = 1, N, (1)
where xt = {xti ∈ R}Ni=1 is an N -dimensional state vector, t = 0, 1, . . . represents a discrete
time index, and f : R 7→ R is a C1-smooth one-dimensional map. In the numerical experi-
ments the one-dimensional map f is taken in the form of the logistic map f(x) = ax(1− x).
Thus, the behaviour of the system (1) is ruled by its two parameters, the nonlinearity parameter
a ∈ [0, 4] and the coupling parameter ε ∈ [0, 1]. When ε = 0 the system (1) represents N
noncoupled one-dimensional maps. Every coordinate of the state vector xt = (xt1, x
t
2, . . . , x
t
N )
moves independently according to the map f. When ε = 1, after the first iteration all coordi-
nates xti, i = 1, N, become identical and continue to move synchronously according to the
map f.
For the other values of ε ∈ (0, 1), in particular, the system (1) can fall into the state of
so-called partial synchronization or clustering [1 – 5], when the coordinates of the state vector
xt unite into several groups, called clusters, producing the same behaviour within each group.
Then, each cluster can be considered as one dynamical element [1, 3, 4],
xt1 = xt2 = . . . = xtN1
df
= yt1,
xtN1+1 = xtN1+2 = . . . = xtN1+N2
df
= yt2, (2)
....................................................................................
xtN1+...+NK−1+1 = xtN1+...+NK−1+2 = . . . = xtN
df
= ytK ,
where Ni is the number of elements in the i-th cluster, N1 +N2 + . . .+NK = N.
Relations (2) define a K-dimensional manifold M (K) ⊂ R(N) of the form
M (K) =
{
(x1, x2, . . . , xN ) ∈ RN : x1 = x2 = . . . = xN1 ,
xN1+1 = xN1+2 = . . . = xN1+N2 , . . . , xN1+...+Nk−1+1 = . . . = xN ;
N1 + . . .+NK = N
}
. (3)
It is easy to see, that, by virtue of the complete symmetry of the system (1), for any set {Ni ∈
∈ N :
∑K
i=1Ni = N} the manifold M (K) is invariant with respect to the map Fε, i.e., ∀ x ∈
∈ M (K) its image Fε(x) ∈ M (K). So it is possible to consider the restriction Gε
df
= Fε
∣∣
M(K)
as a dynamical system on M (K) :
Gε : yti 7→ (1− ε)f(yti) + ε
K∑
j=1
pjf(y
t
j), i = 1,K. (4)
ISSN 1562-3076. Нелiнiйнi коливання, 2002, т . 5, N◦ 3
336 A.A. PANCHUK, YU.L. MAISTRENKO
Here yt = {yti}Ki=1 is a K-dimensional state vector, and pi
df
= Ni/N,
∑K
i=1 pi = 1, is
referred to as population of the i-th cluster given by Eq. (2).
Note that system (4) has also the form of a globally coupled map system but, in contrast
with system (1), different coupling weights pi as a measure for the contribution of the i-th
coordinate yi to the global coupling term. Therefore system (4) is asymmetric; it becomes
symmetric only when p1 = p2 = . . . = pK = 1/K (then the dimension N of the initial
system (1) must be divisible by the dimension K of the system (4), i.e., N = k1K for some
k1 ∈ N ).
Suppose that the K-dimensional map Gε has an n-periodic cycle P
(K)
n =
{
y1,y2, . . . ,yn
}
such that
yi+1 = Gε(yi), i = 1, n− 1,
(5)
y1 = Gε(yn),
where yj = (y1j , y2j , . . . , yKj) ∈ RK , j = 1, n. Then the original N -dimensional system (1)
has the corresponding n -periodic cycle P
(N)
n =
{
x1,x2, . . . ,xn
}
of the form
xi+1 = Fε(xi), i = 1, n− 1,
(6)
x1 = Fε(xn),
which belongs to the manifold M (K) ⊂ R,
xj =
(
y1j , . . . , y1j︸ ︷︷ ︸
N1
, y2j , . . . , y2j︸ ︷︷ ︸
N2
, . . . , yKj , . . . , yKj︸ ︷︷ ︸
NK
)
∈ RN , j = 1, n. (7)
For system (1) a cycle P
(N)
n of the form (7) is referred to as n-periodic K-clustered state, or
simply PnCK-state. Our goal is to investigate the stability of these periodic states in the whole
N -dimensional space RN .
Definition 1. Consider the PnCK-state of system (1), i.e., a period-n cycle P
(N)
n = {x1,
x2, . . . ,xn} of the map Fε of the form (7). Let ν(N)
i , i = 1, N, be eigenvalues of the Jacobian
matrix DFn
ε (x1) = DFε(xn)DFε(xn−1) . . . DFε(x1), where Fn
ε is the n-th iteration of the
map Fε. Then, the values µ
(N)
i =
n
√
ν
(N)
i , i = 1, N, are called multipliers of PnCK-state, i.e.,
of the cycle P
(N)
n .
Definition 2. The PnCK-state of the system (1), i.e., the cycle P
(N)
n =
{
x1,x2, . . . ,xn
}
of
the map Fε is called Lyapunov stable if all its multipliers
{
µ
(N)
i
}N
i=1
lie inside the unit circle,
i.e., |µ(N)
i | < 1, i = 1, N.
Any PnCK-state has N multipliers. K of them, {µ(N)
‖,i }
K
i=1, correspond to the eigenvec-
tors of the matrix DFn
ε lying in the K-clustered manifold M (K) of the form (3). Hence, they
control stability of the PnCK-state inside M (K). They coincide with the multipliers {µ(K)
i }Ki=1
ISSN 1562-3076. Нелiнiйнi коливання, 2002, т . 5, N◦ 3
STABILITY OF PERIODIC CLUSTERS IN GLOBALLY COUPLED MAPS 337
of the corresponding cycle P
(K)
n of the form (5) for the K-dimensional system (4). We will
call them longitudinal multipliers of the PnCK-state. The other N −K multipliers correspond
to the eigenvectors of DFn
ε transverse to the manifold M (K) (i.e., lying in the supplement
RN\M (K) of M (K)). They will be referred to as transverse multipliers of PnCK .
As it was shown [1], if all the numbers Ni > 1, i = 1,K (see (3)), then there are K disti-
nct transverse multipliers
{
µ
(N)
⊥,i
}K
i=1
each having multiplicity Ni − 1. Otherwise, the number
K1 of distinct transverse multipliers
{
µ
(N)
⊥,i
}K1
i=1
equals the number of clusters having more
than 1 element, i.e., K1 = card
{
i = 1,K : Ni > 1
}
. The transverse multipliers control out-of-
cluster stability of the PnCK-state in RN space, i.e., stability with respect to small perturbations
beyond the clustered manifold M (K). To ensure the transverse stability of PnCK we demand
the transverse multipliers to lie in the unit circle,
∣∣µ(N)
⊥,i
∣∣ < 1, i = 1,K1, 0 ≤ K1 ≤ K.
Lemma 1. The transverse multipliers
{
µ
(N)
⊥,i
}K1
i=1
for the PnCK-state of the form (6), (7) are
equal to
µ
(N)
⊥,i = (1− ε)
n∏
j=1
f ′(yij)
1
n
, i = 1,K1. (8)
Proof. The proof follows from formula (7) in the paper [1].
Remark 1. As it is easy to see, both longitudinal and transverse multipliers µ
(N)
‖,i , and µ
(N)
⊥,i ,
of the PnCK-state do not depend on the space dimension N. Therefore we can omit the upper
index (N) writing simply µ‖,i and µ⊥,i.
3. Relations between the transverse and longitudinal multipliers. Let the coupling weights
{pi}Ki=1 in the map Gε : RK 7→ RK of the form (4) be equal, p1 = p2 = . . . = pK = 1/K,
and the map f ∈ C1. Consider a period-n cycle P
(K)
n of the map Gε. For any N = k1K
( k1 > 1 is an integer), this cycle generates a PnCK-state of system (1) in the N-dimensional
phase space (in accordance with formulas (6) and (7), where Ni = k1, i = 1,K ). The PnCK-
state has K longitudinal multipliers
{
µ‖,i
}K
i=1
(which coincide with the multipliers
{
µ
(K)
i
}K
i=1
of the cycle P
(K)
n ) and K transverse multipliers
{
µ⊥,i
}K
i=1
(which are given by formula (7)
of the Lemma 1).
Theorem 1. For the transverse and longitudinal multipliers of the PnCK-state as above the
following relation holds:
K∏
i=1
µ⊥,i = (1− ε)
K∏
i=1
µ‖,i. (9)
To prove the theorem, the following lemma is needed.
ISSN 1562-3076. Нелiнiйнi коливання, 2002, т . 5, N◦ 3
338 A.A. PANCHUK, YU.L. MAISTRENKO
Lemma 2. The determinant of the Jacobian matrix DGε can be represented in the form
detDGε(y) = (1− ε)K−1
K∏
i=1
f ′(yi) (10)
for any y = (y1, y2, . . . , yK) ∈ RK .
Proof. The Jacobian matrix of the map Gε is
DGε(y) =
(
1− ε+ ε
K
)
f ′(y1)
ε
K
f ′(y2) . . .
ε
K
f ′(yK)
ε
K
f ′(y1)
(
1− ε+ ε
K
)
f ′(y2) . . .
ε
K
f ′(yK)
...
... . . .
...
ε
K
f ′(y1)
ε
K
f ′(y2) . . .
(
1− ε+ ε
K
)
f ′(yK)
.
For any j = 1, . . . ,K consider the (j × j)-matrix
Aj =
(1− ε)f ′(yK−j+1) 0 . . . −(1− ε)f ′(yK)
0 (1− ε)f ′(yK−j+2) . . . −(1− ε)f ′(yK)
...
... . . .
...
ε
K
f ′(ykj+1)
ε
K
f ′(ykj+2) . . .
(
1− ε+ ε
K
)
f ′(yK)
. (11)
It is easy to show that
detDGε(y) = detAK . (12)
The determinant of Aj can be represented recursively as
detAj = (1− ε)f ′(yK−j+1) detAj−1 + (−1)j+1 ε
K
f ′(yK−j+1) detBj−1, (13)
where Bj−1 is a (j − 1)× (j − 1)-matrix of the form
Bj−1 =
0 0 . . . 0 −(1− ε)f ′(yK)
(1− ε)f ′(yK−j+2) 0 . . . 0 −(1− ε)f ′(yK)
0 (1− ε)f ′(yK−j+3) . . . 0 −(1− ε)f ′(yK)
...
... . . .
...
...
0 0 . . . (1− ε)f ′(yK−1) −(1− ε)f ′(yK)
.
ISSN 1562-3076. Нелiнiйнi коливання, 2002, т . 5, N◦ 3
STABILITY OF PERIODIC CLUSTERS IN GLOBALLY COUPLED MAPS 339
Its determinant equals
detBj−1 = −(−1)j(1− ε)f ′(yK)(1− ε)j−2
K−1∏
i=K−j+2
f ′(yi) = (−1)j+1(1− ε)j−1
K∏
i=K−j+2
f ′(yi).
Then, as it follows from (13),
detAj = (1− ε)f ′(yK−j+1) detAK−1+
+ (−1)j+1 ε
K
f ′(yK−j+1)(−1)j+1(1− ε)j−1
K∏
i=K−j+2
f ′(yi) =
= (1− ε)f ′(y1) detAK−1 +
ε
K
(1− ε)j−1
K∏
i=K−j+1
f ′(yi). (14)
Let us prove that
detAj = (1− ε)j−1
(
1− ε+ jε
K
) K∏
i=K−j+1
f ′(yi)
using the method of mathematical induction. For j = 2 it is easy to see that
detA2 = det
(1− ε)f ′(yK−1) −(1− ε)f ′(yK)
ε
K
f ′(yK−1)
(
1− ε+ ε
K
)
f ′(yK)
= (1− ε)
(
1− ε+ 2ε
K
)
f ′(yK−1)f
′(yK).
Suppose that detAj−1 = (1− ε)j−2
(
1− ε+ (j − 1)ε
K
)
K∏
i=K−j+2
f ′(yi). Substituting the latter
expression into Eq. (14) we obtain
detAj = (1− ε)f ′(yK−j+1)(1− ε)j−2
(
1− ε+ (j − 1)ε
K
) K∏
i=K−j+2
f ′(yi)+
+
ε
K
(1− ε)j−1
K∏
i=K−j+1
f ′(yi) =
= (1− ε)j−1
K∏
i=K−j+1
f ′(yi)
(
1− ε+ (j − 1)ε
K
+
ε
K
)
=
= (1− ε)j−1
(
1− ε+ (j − 1)ε
K
) K∏
i=K−j+1
f ′(yi).
ISSN 1562-3076. Нелiнiйнi коливання, 2002, т . 5, N◦ 3
340 A.A. PANCHUK, YU.L. MAISTRENKO
From Eq. (12) it follows that
detDGε(y) = detAK = (1− ε)K−1
(
1− ε+ Kε
K
) K∏
i=1
f ′(yi) = (1− ε)K−1
K∏
i=1
f ′(yi).
The lemma is proved.
Proof of Theorem 1. The longitudinal multipliers
{
µ‖,i
}K
i=1
of the PnCK-state equal to
the multipliers of the cycle P
(K)
n of the K-dimensional map Gε of the form (4). They can be
obtained as the n-th roots of the eigenvalues of the Jacobian matrix for the n-th iteration of
the map Gε,
DGn
ε (y1) = DGε(yn)DGε(y2) . . . DGε(y1).
By Vieta’s theorem, the product of all eigenvalues of a K ×K matrix equals its determinant,
i.e.,
K∏
i=1
µ‖,i = (detDGn
ε (y1))
1
n = (detDGε(yn) . . . detDGε(y1))
1
n .
Then using Lemma 2 we obtain
K∏
i=1
µ‖,i = (1− ε)K−1
n∏
j=1
K∏
i=1
f ′(yij)
1
n
. (15)
On the other hand, due to Lemma 1, the transverse multipliers µ⊥,i, i = 1,K, of the PnCK-
state can be represented in the form (8). Therefore, it follows that the product of the multipliers
is equal to
K∏
i=1
µ⊥,i = (1− ε)K
K∏
i=1
n∏
j=1
f ′(yij)
1
n
. (16)
Finally, Eqs. (15) and (16) imply Eq. (9).
4. Cyclicity condition. For the map Gε of the form (4) with the equal populations pi =
= 1/K, i = 1,K, consider the cycle P
(K)
mK of the period mK for some m ≥ 1. Put ε = 0.
Then the coupling term vanishes and the K-dimensional map G0 is a direct product of K
one-dimensional maps f : x 7→ f(x), x ∈ R. Therefore, the dynamics of G0 is defined by
the dynamics of f. In particular, if the one-dimensional map f has a period-mK cycle
PmK =
{
y1, y2, . . . , ymK
}
,
then for the map G0 there exists a corresponding period-mK cycle P
(K)
mK of the form
P
(K)
mK =
{
y1,y2, . . . ,ymK
}
(17)
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STABILITY OF PERIODIC CLUSTERS IN GLOBALLY COUPLED MAPS 341
such that
yj =
(
yj , ym+j , . . . , ym(K−1)+j), j = 1,m,
yim+j = πiK(yj), i = 1,K − 1, j = 1,m,
(18)
where πK is a cyclical permutation, e.i.,
πK(s1, s2, . . . , sK) = (s2, s3, . . . , sK , s1)
for any set of K real numbers si, i = 1,K, and πiK is the i-th iteration of πK . We will call
relations (18) the cyclicity condition for the cycle P
(K)
mK . Obviously, for m = 1 the cyclicity
condition for the cycle
P
(K)
K =
{
y1,y2, . . . ,yK
}
(17′)
becomes
y1 =
(
y1, y2, . . . , yK),
yi+1 = πiK(y1), l = 1,K − 1.
(18′)
Due to the symmetry and smooth dependence of the map Gε, we expect that the cycles
P
(K)
mK (ε) =
{
y1,y2, . . . ,ymK
}
satisfying the cyclicity condition can also exist for ε > 0.
Indeed, due to the implicit function theorem, if detDG
(n)
0 (y1) 6= 0 there exists ε0 >
> 0 such that ∀ε ∈ [0, ε0] for the map Gε, there exists a cycle P
(K)
mK = P
(K)
mK (ε) that is a
continuation of the cycle P
(K)
mK (0), i.e.,
lim
ε→0
P
(K)
mK (ε) = P
(K)
mK (0).
Sufficient conditions for the cycle P
(K)
mK = P
(K)
mK (ε) to satisfy the cyclicity condition (18) (or
(18′) for the case m = 1 ) are given by the following Lemmas 3 and 4.
Lemma 3. Suppose that the one-dimensional map
g : x 7→ (1− ε)f(x) + εh, h ≡ const, (19)
has a period-K cycle PK = {y1, y2, . . . , yK} such that
1
K
K∑
j=1
yj = h.
Then the K-dimensional map Gε has a period-K cycle P
(K)
K satisfying the cyclicity condition
(18 ′).
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342 A.A. PANCHUK, YU.L. MAISTRENKO
Proof. Consider any point y1 = (y1, y2, . . . , yK) ∈ RK . The i-th coordinate of its Gε
image is equal to
(Gε(y1))i = (1− ε)f ((y1)i) +
ε
K
K∑
j=1
f ((y1)j) , i = 1,K.
Since PK is a cycle for the map g of the form (19), we have
yi+1 = (1− ε)f(yi) + εh, i = 1,K − 1,
y1 = (1− ε)f(yK) + εh.
Adding all these K equations and then dividing the result by K we obtain
1
K
K∑
j=1
yj = (1− ε) 1
K
K∑
j=1
f(yj) + εh ⇔
⇔ h(1− ε) = (1− ε) 1
K
K∑
j=1
f(yj) ⇔
⇔ h =
1
K
K∑
j=1
f(yj).
This implies
(Gε(y1))i = (1− ε)f(yi) +
ε
K
K∑
j=1
yj = (1− ε)f(yi) + εh, i = 1,K,
and, therefore,
(Gε(y1))i = yi+1, i = 1,K − 1,
(Gε(y1))K = y1,
i.e., Gε(y1) = πK(y1). Obviously, applying the map Gε K times we obtain
GK
ε (y1) = y1,
which means that P (K)
K =
{
y1, Gε(y1), G
2
ε(y1), . . . , G
K−1
ε (y1)
}
is a K-cycle of Gε, satisfying
the cyclicity condition.
Lemma 4. Let m > 1 and let the m-dimensional map
g : xi 7→ (1− ε)f(xi) + εhi, hi = const, i = 1,m, (20)
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STABILITY OF PERIODIC CLUSTERS IN GLOBALLY COUPLED MAPS 343
have a period-mK cycle PmK = {y1, y2, . . . , ymK} such that
1
K
K∑
j=1
y(j−1)m+i = hi, i = 1,m.
Then the map Gε has a period-mK cycle P
(K)
mK satisfying the cyclicity condition (17), (18).
The proof of Lemma 4 is analogous to the one of the Lemma 3.
5. Stability of PnCK-states. Formula (9) obtained in Section 3 enables to prove the transver-
se stability of PnCK-states, in the case n = mK, in terms of stability of the cycle P
(K)
n inside
the cluster manifold M (K).
Consider the PmKCK-state in the N-dimensional phase space generated by a period- mK
cycle P
(K)
mK of the K-dimensional map Gε (m ∈ N is any integer). Suppose the PmKCK-
state to be symmetric, i.e.,
1) Ni = k1, k1 > 1, for all i = 1,K (see Eq. (7)),
2) the cycle P
(K)
mK satisfies the cyclicity condition (17), (18) (or (17′), (18′) resp.).
Theorem 2. Let the conditions 1 and 2 be satisfied. Then all transverse multipliers {µ⊥,i}Ki=1
of the PmKCK-state are equal and can be represented as
µ⊥,i = (1− ε) K
√√√√ K∏
j=1
f ′(yj)
df
= µ⊥, i = 1,K.
Proof. Directly from the Eq. (8) we derive that the transverse multipliers for the PmKCK-
state of the form (6), (7) are
µ⊥,i = (1− ε)
K∏
j=1
f ′(yj)
1
mK
, i = 1,K.
Theorem 2 results in the following corollaries.
Theorem 3. If the cycle P
(K)
n , n = mK, satisfying conditions 1 and 2 above, is Lyapunov
stable inside the manifold M (K), then the clustered PmKCK-state is Lyapunov stable for any
ε ∈ [0, 2].
Let us prove the theorem only for the case m = 1. For m > 1 the proof is analogous.
Proof. Since the longitudinal multipliers {µ‖,i}Ki=1 of the PKCK-state are the multipliers
{µ(K)
i }Ki=1 of the K-dimensional cycle P
(K)
K , we have
∣∣µ‖,i∣∣ = ∣∣∣µ(K)
i
∣∣∣ < 1, i = 1,K.
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344 A.A. PANCHUK, YU.L. MAISTRENKO
Then, by Theorem 2,
µ⊥,i = µ⊥, i = 1,K,
which implies
K∏
i=1
µ⊥,i = µK⊥ .
Therefore, using Theorem 1,
|µ⊥,i|K =
∣∣∣∣∣(1− ε)
K∏
i=1
µ‖,i
∣∣∣∣∣ = |1− ε|
K∏
i=1
∣∣µ‖,i∣∣ < 1
for any ε ∈ [0, 2]. Finally, this gives
|µ⊥| < 1,
which means transverse stability of the PKCK-state. The theorem is proved.
Denote the product of the multipliers of the cycle P
(K)
mK by
σ
df
=
K∏
i=1
µ
(K)
i
and call σ the generalized saddle value of the cycle P
(K)
mK .
Theorem 4. Let the cycle P
(K)
n , n = mK, satisfy the conditions 1 and 2. If the saddle value
of the cycle P
(K)
n lies inside the unit circle, i.e.,
|σ| < 1,
then the corresponding PmKCK-state is transversally stable in the whole k1K-dimensional phase
space, for any ε ∈ [0, 2].
Proof. As in the previous proof, due to Theorems 1 and 2,
|µ⊥|K = |µ⊥,i|K =
∣∣∣∣∣(1− ε)
K∏
i=1
µ‖,i
∣∣∣∣∣ = |1− ε|
∣∣∣∣∣
K∏
i=1
µ‖,i
∣∣∣∣∣ = |1− ε||σ|.
Since |σ| < 1 and ε ∈ [0, 2], we have
|µ⊥|K = |µ⊥,i|K < 1.
Theorem 5. Let the cycle P
(K)
n , n = mK, satisfy the conditions 1 and 2. Then the correspon-
ding PmKCK-state is transversally stable for any
ε ∈
[
1− 1
|σ|
, 1 +
1
|σ|
]
.
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STABILITY OF PERIODIC CLUSTERS IN GLOBALLY COUPLED MAPS 345
Proof. As in the previous proof, we have
|µ⊥|K = |1− ε||σ|.
For the PmKCK-state to be transversally stable all the transverse multipliers must lie in the
unit circle,
|µ⊥|K < 1.
Therefore,
|1− ε||σ| < 1 ⇔ |ε− 1| < 1
|σ|
⇔ 1− 1
|σ|
< ε < 1 +
1
|σ|
,
which completes the proof.
Acknowledgement. We also thank M. Hasler and E. Mosekilde for a number of fruitful
discussions.
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Received 23.04.2002
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