Stability of synchronized and clustered states in coupled piecewise linear maps
Parameter regions for different types of stability of synchronized and clustered states are obtained for two interacting ensembles of globally coupled one-dimensional piecewise linear maps. We analyze strong (asymptotic) and weak (Milnor) stability of the synchronized state, as well as its instabi...
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irk-123456789-1770062021-02-10T01:26:05Z Stability of synchronized and clustered states in coupled piecewise linear maps Matskiv, I.V. Parameter regions for different types of stability of synchronized and clustered states are obtained for two interacting ensembles of globally coupled one-dimensional piecewise linear maps. We analyze strong (asymptotic) and weak (Milnor) stability of the synchronized state, as well as its instability. We found that the stability and instability regionsin the phase space depend only on parameters of the individualskew tent map, and do not depend on the ensembles size. In the simplest nontrivial case of four coupled chaotic maps we obtain stability regions for coherent and two-cluster states. The regions appear to be large enough to provide an effective control of coherent and clustered chaotic regimes. Transition from desynchronization to synchronization is identified to be qualitatively different in smooth and piecewise linear models. Знайдено параметричнi областi для рiзних типiв стiйкостi синхронiзованих та кластерних станiв для двох взаємодiючих ансамблiв глобально зв’язаних одновимiрних кусково-лiнiйних вiдображень. Дослiджено сильну (асимптотичну) та слабку (за Мiлнором) стiйкiсть та нестiйкiсть синхронiзованого стану системи. Визначено, що областi стiйкостi та нестiйкостi у просторi параметрiв залежать лише вiд коефiцiєнтiв кусково-лiнiйного вiдображення i не залежать вiд розмiру ансамблiв. Для найпростiшого нетривiального випадку чотирьох зв’язаних вiдображень отримано областi стiйкостi для когерентного та двокластерних станiв. Досить великi розмiри областей стiйкостi у просторi параметрiв дають можливiсть проводити ефективне керування когерентним та кластерними режимами у системi. Крiм цього, виявлено якiсно рiзнi способи десинхронiзацiї у системах кусково-лiнiйних та гладких вiдображень. 2004 Article Stability of synchronized and clustered states in coupled piecewise linear maps / I.V. Matskiv // Нелінійні коливання. — 2004. — Т. 7, № 2. — С. 217-228. — Бібліогр.: 20 назв. — англ. 1562-3076 http://dspace.nbuv.gov.ua/handle/123456789/177006 517.9 en Нелінійні коливання Інститут математики НАН України |
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English |
| description |
Parameter regions for different types of stability of synchronized and clustered states are obtained for
two interacting ensembles of globally coupled one-dimensional piecewise linear maps. We analyze strong
(asymptotic) and weak (Milnor) stability of the synchronized state, as well as its instability. We found that
the stability and instability regionsin the phase space depend only on parameters of the individualskew tent
map, and do not depend on the ensembles size. In the simplest nontrivial case of four coupled chaotic maps
we obtain stability regions for coherent and two-cluster states. The regions appear to be large enough to
provide an effective control of coherent and clustered chaotic regimes. Transition from desynchronization
to synchronization is identified to be qualitatively different in smooth and piecewise linear models. |
| format |
Article |
| author |
Matskiv, I.V. |
| spellingShingle |
Matskiv, I.V. Stability of synchronized and clustered states in coupled piecewise linear maps Нелінійні коливання |
| author_facet |
Matskiv, I.V. |
| author_sort |
Matskiv, I.V. |
| title |
Stability of synchronized and clustered states in coupled piecewise linear maps |
| title_short |
Stability of synchronized and clustered states in coupled piecewise linear maps |
| title_full |
Stability of synchronized and clustered states in coupled piecewise linear maps |
| title_fullStr |
Stability of synchronized and clustered states in coupled piecewise linear maps |
| title_full_unstemmed |
Stability of synchronized and clustered states in coupled piecewise linear maps |
| title_sort |
stability of synchronized and clustered states in coupled piecewise linear maps |
| publisher |
Інститут математики НАН України |
| publishDate |
2004 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/177006 |
| citation_txt |
Stability of synchronized and clustered states in coupled piecewise linear maps / I.V. Matskiv // Нелінійні коливання. — 2004. — Т. 7, № 2. — С. 217-228. — Бібліогр.: 20 назв. — англ. |
| series |
Нелінійні коливання |
| work_keys_str_mv |
AT matskiviv stabilityofsynchronizedandclusteredstatesincoupledpiecewiselinearmaps |
| first_indexed |
2025-07-15T14:57:55Z |
| last_indexed |
2025-07-15T14:57:55Z |
| _version_ |
1837725359422832640 |
| fulltext |
UDC 517.9
STABILITY OF SYNCHRONIZED AND CLUSTERED STATES
IN COUPLED PIECEWISE LINEAR MAPS
СТIЙКIСТЬ СИНХРОНIЗОВАНИХ ТА КЛАСТЕРНИХ СТАНIВ
У СИСТЕМI ЗВ’ЯЗАНИХ КУСКОВО-ЛIНIЙНИХ ВIДОБРАЖЕНЬ
I. V. Matskiv
Inst. Math. Nat. Acad. Sci. Ukraine
Ukraine, 01601, Kyiv 4, Tereshchenkivska Str., 3
e-mail: matskiv@imath.kiev.ua
Parameter regions for different types of stability of synchronized and clustered states are obtained for
two interacting ensembles of globally coupled one-dimensional piecewise linear maps. We analyze strong
(asymptotic) and weak (Milnor) stability of the synchronized state, as well as its instability. We found that
the stability and instability regions in the phase space depend only on parameters of the individual skew tent
map, and do not depend on the ensembles size. In the simplest nontrivial case of four coupled chaotic maps
we obtain stability regions for coherent and two-cluster states. The regions appear to be large enough to
provide an effective control of coherent and clustered chaotic regimes. Transition from desynchronization
to synchronization is identified to be qualitatively different in smooth and piecewise linear models.
Знайдено параметричнi областi для рiзних типiв стiйкостi синхронiзованих та кластерних
станiв для двох взаємодiючих ансамблiв глобально зв’язаних одновимiрних кусково-лiнiйних вi-
дображень. Дослiджено сильну (асимптотичну) та слабку (за Мiлнором) стiйкiсть та нестiй-
кiсть синхронiзованого стану системи. Визначено, що областi стiйкостi та нестiйкостi у
просторi параметрiв залежать лише вiд коефiцiєнтiв кусково-лiнiйного вiдображення i не за-
лежать вiд розмiру ансамблiв. Для найпростiшого нетривiального випадку чотирьох зв’язаних
вiдображень отримано областi стiйкостi для когерентного та двокластерних станiв. Досить
великi розмiри областей стiйкостi у просторi параметрiв дають можливiсть проводити ефек-
тивне керування когерентним та кластерними режимами у системi. Крiм цього, виявлено якiс-
но рiзнi способи десинхронiзацiї у системах кусково-лiнiйних та гладких вiдображень.
1. Introduction. Network organizations appear in a wide variety of phenomena in physics, engi-
neering, biology, medicine and other fields. That is why investigation of the network formations
and dependence of their properties on the network structure has become an important appli-
ed problem. There are a lot of different mathematical models describing netwok organizati-
ons, developed to understand the collective behavior of their elements. Examples of network
formations can be presented as ensembles of globally (mean field) coupled oscillators
xt+1
i = (1 − ε)f(xt
i) +
ε
N
N
∑
j=1
f(xt
j), (1)
which have been first suggested and intensively studied by Kaneko [1, 2]. If not all oscillators
in the ensemble interact, we get a so-called ’fractally’ coupled system,
xt+1
i = (1 − ε)f(xt
i) +
ε
Ai
∑
j∈con n
f(xt
j), (2)
c© I. V. Matskiv, 2004
ISSN 1562-3076. Нелiнiйнi коливання, 2004, т . 7, N◦ 2 217
218 I. V. MATSKIV
where Ai is the number of connections at the ith element, and j runs over all elements that are
connected to the element i. An interesting example of such networks is given by ’small-world’
systems, introduced by Watts and Strogatz [3, 4] and interpreted as mathematical models of
many biologic and social networks. Another example is given by neural network, where groups
of peripheral oscillators interact with only one leading oscillator [5].
Investigation of such systems is typically concerned with studying their dynamics properties
such as full and partial synchronization [6].
In the present paper we consider a 2N -dimensional system of coupled maps which models
the situation when there are two globally coupled Kaneko systems of the mean field type (1),
and their elements are pairwisely coupled with each others,
xt+1
i =
(
1 − ε −
δ
2
)
f(xt
i) +
ε
N
N
∑
j=1
f(xt
j) +
δ
2
f(yt
i),
(3)
yt+1
i =
(
1 − ε −
δ
2
)
f(yt
i) +
ε
N
N
∑
j=1
f(yt
j) +
δ
2
f(xt
i), i = 1, . . . , N.
Here {xt
i}
N
i=1 and {yt
i}
N
i=1 are N -dimensional state vectors; t = 0, 1, . . . is a discrete time index,
f : R → R is a one-dimensional map.
A model of the form (3), in the case of smooth, logistic map f(x) = ax(1 − x) was studied
in [7]. In the present work, f is chosen as a piecewise linear, skew tent map of the form
f(x) = fl,p(x) =
lx + 1 − l −
l
p
, x ≤ 1 +
1
p
;
px − p, x > 1 +
1
p
.
(4)
Piecewise linear maps systems are widely spread in a variety of technical, engineering, and
electronic applications. Depending on parameters, such systems are characterized by regular or
complex chaotic dynamics [8, 9]. In many cases piecewise linear dynamics appear to be rather
different from smooth ones, especially as for different types of bifurcations and their [10 – 12].
The behavior of 2N -dimensional system (3) is controlled by four parameters, l > 0 and
p < −1, which are the coefficients of linear parts of the individual skew tent map f = fl,p, and
coupling parameters ε, δ.
Model (3) can be interpreted as two groups of interacting oscillators (i. e. neurons), as well
as the simplest example of a deterministic complex ’small-world’ network having two groups of
elements with a strong coupling inside and weaker coupling between the groups.
In the present paper we obtain parameter regions for strong and weak stability and instabi-
lity of the synchronizing chaotic set on the main diagonal of the 2N -dimensional phase space
of system (3) in terms of the coefficients of the individual skew tent map fl,p. In the parameter
(δ, ε)-plane we delineate schematically stability regions and identify variation of their border
with change of l and p. As the simplest nontrivial example we consider a system of four coupled
skew tent maps and delineate parameter regions for stability of synchronized (coherent) and
two-cluster states.
ISSN 1562-3076. Нелiнiйнi коливання, 2004, т . 7, N◦ 2
STABILITY OF SYNCHRONIZED AND CLUSTERED STATES . . . 219
2. Stability of chaotic synchronizing set. Consider a 2N -dimensional map F defined by the
system (3), where N is a number of elements in each globally coupled ensemble. The Jacobian
matrix of this system is
DF
(
X
Y
)
=
(
G(X) L(Y )
L(X) G(Y )
)
,
where
X =
x1
...
xN
, Y =
y1
...
yN
,
G(Z) =
(
1 −
(N − 1)ε
N
−
δ
2
)
f ′(z1) · · ·
ε
N
f ′(zN )
...
. . .
...
ε
N
f ′(z1) · · ·
(
1 −
(N − 1)ε
N
−
δ
2
)
f ′(zN )
and
L(Z) =
δ
2
f ′(z1) 0 · · · 0
0
δ
2
f ′(z2) · · · 0
...
...
. . .
...
0 0 · · ·
δ
2
f ′(zN )
.
We denote by D2N = {z ∈ R
2N |z1 = . . . = z2N} the main diagonal of the 2N -dimensional
phase space. It is not difficult to obtain that the Jacobian matrix DF (x), taken in the point
x = (x, . . . , x) ∈ D2N , has only four different eigenvalues,
ν‖(x) = f ′(x), ν⊥,1(x) = f ′(x)(1 − ε),
(5)
ν⊥,2(x) = f ′(x)(1 − δ), ν⊥,3(x) = f ′(x)(1 − ε − δ).
Note that ν⊥,2(x) and ν⊥,3(x) have multiplicity N − 1.
The eigenvector v‖ corresponding to the eigenvalue ν‖ is directed along the main diagonal
D2N , hence this eigenvalue is responsible for ’dispersion’ of the trajectories along D2N . Three
other eigenvectors are transverse to D2N . Hence, the corresponding eigenvalues (ν⊥,1, ν⊥,2, and
ν⊥,3), named usually transverse, control the attracting (repelling) strength for the trajectories
in the corresponding transverse directions.
The following consideration is concerned with the next definitions.
ISSN 1562-3076. Нелiнiйнi коливання, 2004, т . 7, N◦ 2
220 I. V. MATSKIV
Let AD ⊂ D2N be an invariant set for the system (3). We denote by B(AD) the basin
of attraction of AD. Recall that B(AD) = {z ∈ R
2N |ω(z, F ) ⊂ AD}, where ω(z, F ) =
=
⋂
n∈N
{F k(z) : k > n}.
The set AD is called strongly or asymptotically stable if for any its neighborhood U(AD) ⊂
⊂ R
2N there exists another neighborhood V (AD) ⊂ R
2N such that for any z ∈ V (AD):
1) Fn(z) ∈ U(AD) for all n ∈ N;
2) ρ(Fn(z), AD) → 0 for n → ∞, ρ(·, ·) being the distance between a point and a set
in R
2N .
Let Pre (AD) = {z ∈ R
2N |∃n ∈ N : Fn(z) ∈ AD} be a set of preimages for AD.
Obviously Pre (AD) ⊆ B(AD). The set AD is called strongly or asymptotically unstable if
B(AD) = Pre (AD), which means that AD attracts only its preimages.
The other type of stability considered in the present paper was introduced by Milnor [13].
Typically, it is referred to as weak stability. The set AD is called weakly or Milnor stable if its
basin of attraction B(AD) has a positive Lebesgue measure in R
2N . Alternatively, AD is called
weakly or Milnor unstable in the opposite case, i. e., if mes B(AD) = 0. Note that strong stability
implies weak stability, but not vice versa.
Let AD ⊂ D2N be a chaotic synchronizing set for the system (3). Depending on the coupling
parameters values, it can be characterized by the defined above types of stability or instability.
Being strongly stable for certain parameter values, when ε and δ change, the set AD typically
loses its strong stability through a riddling bifurcation [14]. After the riddling bifurcation, the
synchronizing set AD can still be weakly stable.
Further change of the coupling parameters can cause a loss of weak stability of AD through
blowout bifurcation [15]. After blowout bifurcation there still can exist trajectories attracted by
the synchronizing set AD, but the Lebesgue measure of their initial points is equal to zero.
In the present paper such a sequence of parameter regions for strong and weak stability, as
well as instability, of the chaotic synchronizing set AD for the system (3) are obtained analyti-
cally.
As shown above, the eigenvalues (5) do not depend on the system size 2N . So the parameter
regions for different types of stability for system (3) do not depend on the number of oscillators,
too.
Denote
b
(1)
1 = 1 −
(
1
lk−1|p|
)1/k
, b
(2)
1 = 1 +
(
1
lk−1|p|
)1/k
,
b
(1)
2 = 1 −
1
|p|
, b
(2)
2 = 1 +
1
|p|
,
and k =
[
2 −
ln l + p(l − 1)
ln l
]
, [·] being the integer part of a real number.
Lemma 1. For the skew tent map fl,p (4) the most unstable among all existing cycles are:
the chaotic cycle of the maximal period k, if l > |p|;
the fixed point of the map fl,p, if l < |p|.
ISSN 1562-3076. Нелiнiйнi коливання, 2004, т . 7, N◦ 2
STABILITY OF SYNCHRONIZED AND CLUSTERED STATES . . . 221
Proof. (1) Let l > |p|. Suppose for fl,p there exists a cycle
γk = {(x1, . . . , x1); . . . ; (xk, . . . , xk) | xi+1 = fl,p(xi), i = 1, . . . , k − 1; k > 1;
x1 = fl,p(xk), }
that belongs to the chaotic synchronizing set AD. The multiplier of γk is
ν(k) = ν(x1) . . . ν(xk) = |f ′
l,p(x1)| . . . |f
′
l,p(xk)| = lm|p|k−m,
where 0 < m ≤ k. The most unstable cycle is the first to lose its stability that is why the value
of ν(k) for this cycle is maximal among multipliers of all existing cycles. ν(k) reaches its maxi-
mal value if the product includes as more multipliers l as possible. A cycle with the kneading
sequence LL . . . L does not exist, that is why ν(k) reaches its maximal value at ν(k) = lk−1|p|
in the case of a cycle of maximal period k, the kneading of which is Lk−1R. This means that
the cycle of period higher than k does not exist. The cycle of period k + 1 exists when p ≤
≤ −
1 − lk−1
lk−2(1 − l)
. From this condition we get the value of the k-maximal period of the existing
cycle in terms of the coefficients l and p of the map fl,p,
k =
[
2 −
ln l + p(l − 1)
ln l
]
, (6)
[·] being the integer part of a real number.
(2) Let l < |p|. In this case the value of ν(k) is maximal if the product includes as more
multipliers |p| as possible. This corresponds to the fixed point ν(k) = |p|k with the kneading
sequence (R)k.
Lemma 2. For the skew tent map fl,p (4) the most stable among all existing cycles are:
the fixed point of the map fl,p, if l > |p|;
the chaotic cycle of maximal period k, if l < |p|.
Proof. By analogy to the proof of Lemma 1, the most stable cycle is the last to lose its
stability, its multiplier is less than multipliers of the other cycles.
In the case l > |p|, ν(k) reaches its minimal value if the product includes as more multipliers
|p| as possible, ν(k) = |p|k. This means that the most stable is the fixed point with the kneading
(R)k.
If l < |p|, the multiplier of γk is minimal at ν(k) = lk−1|p| and the cycle of the maximal
period k (Lk−1R) is the most stable. k can be found, as in Lemma 1, from the condition that the
cycle of higher period k + 1 does not exist.
Theorem 1. In the system (3), where the individual map f = fl,p has form (4), the chaotic
synchronizing set AD ⊂ D2N is strongly stable, if
ε, δ, (ε + δ) ∈ (b
(1)
1 , b
(2)
1 ) for l > |p|;
ε, δ, (ε + δ) ∈ (b
(1)
2 , b
(2)
2 ) for l < |p|,
and strongly unstable if
ε, δ, (ε + δ) ∈
(
−∞; b
(1)
2
)
⋃
(
b
(2)
2 ,+∞
)
for l > |p|;
ε, δ, (ε + δ) ∈
(
−∞; b
(1)
1
)
⋃
(
b
(2)
1 ,+∞
)
for l < |p|.
ISSN 1562-3076. Нелiнiйнi коливання, 2004, т . 7, N◦ 2
222 I. V. MATSKIV
Proof. The synchronizing chaotic set AD loses its strong stability at the moment when the
most unstable periodic cycle loses its stability in the transverse direction. We will consider two
cases, l > |p| and l < |p|.
1) l > |p|.
Lemma 1 shows that the most unstable is the chaotic cycle with the maximal period k. It is
transversely stable if its transverse multiplier
ν⊥,1 = |lk−1p(1 − ε)k| < 1,
(7)
1 −
(
1
lk−1|p|
)1/k
< ε < 1 +
(
1
lk−1|p|
)1/k
.
By analogy, for the synchronizing chaotic set AD to be strongly stable in two other transverse
directions, the inequalities
ν⊥,2 = |lk−1p(1 − δ)k| < 1, ν⊥,3 = |lk−1p(1 − ε − δ)k| < 1,
should be satisfied. From the inequalities we get that
δ, (ε + δ) ∈ (b
(1)
1 , b
(2)
1 ). (8)
The region of strong stability of AD is the intersection of the regions in the parameter plane,
defined by (8) and (9).
2) l < |p|.
From Lemma 1 we get that the most unstable cycle is the fixed point of the individual map
fl,p. Its multiplier in the first transverse direction is
ν⊥,1 = |pk(1 − ε)k| < 1,
1 −
1
|p|
< ε < 1 +
1
|p|
.
From the condition of stability in two other transverse directions, we find parameter regions for
strong stability of the synchronizing chaotic set AD,
δ, (ε + δ) ∈ (b
(1)
2 , b
(2)
2 ).
Transition to strong instability of the chaotic synchronizing set AD is the moment when the
most stable cycle which belongs to AD loses its stability in the transverse direction.
By analogy to the case of strong stability, using results of Lemma 2 we get that AD is strongly
unstable if
ε, δ, (ε + δ) ∈
(
−∞, 1 −
(
1
lk−1|p|
)1/k
]
⋃
[
1 +
(
1
lk−1|p|
)1/k
, +∞
)
for l < |p|. We note that a k can be found from the condition that a cycle of a period bigger
than k does not exist.
ISSN 1562-3076. Нелiнiйнi коливання, 2004, т . 7, N◦ 2
STABILITY OF SYNCHRONIZED AND CLUSTERED STATES . . . 223
In the case l > |p|, the synchronizing chaotic set AD is strongly unstable if
ε, δ, (ε + δ) ∈
(
−∞, 1 −
1
|p|
]
⋃
[
1 +
1
|p|
,+∞
)
.
The theorem is proved.
Consider case where the skew tent map f(x) = fl,p(x) has a chaotic interval I = [0, 1].
According to the theorem of Lasota and Yorke [16], for the map fl,p there exists a unique
probability invariant measure µ = µl,p, absolutely continuous with respect to the Lebesgue
measure. Denote m = µl,p ({x ∈ [1 + 1/p; 1]}), i. e., m =
∫ 1
1+1/p
ρ(x)dx, where ρ(x) is a density
function of the invariant measure.
Theorem 2. For the system (3), where the individual map f = fl,p has form (4), the chaotic
synchronizing set AD ⊂ D2N is weakly stable if
ε, δ, (ε + δ) ∈
(
1 −
1
l1−m|p|m
, 1 +
1
l1−m|p|m
)
,
and weakly unstable if
ε, δ, (ε + δ) ∈
(
−∞, 1 −
1
l1−m|p|m
)
⋃
(
1 +
1
l1−m|p|m
, +∞
)
.
Proof. Transition from weak (Milnor) stability to instability occurs in the moment when the
Lyapunov exponent of the typical trajectory of the map fl,p becomes equal to zero.
Let {fn
l,p(x)} be a typical trajectory of the piecewise linear map fl,p(x). Since m is a measure
of the interval
[
1 +
1
p
, 1
]
, (1 − m) is the measure of the interval
[
0, 1 +
1
p
]
. Let m be known,
then
λA
⊥,1 =
∫
A
ln |f ′(x)(1 − ε)|dµ(x) = ln |(l(1 − ε))1−m(p(1 − ε))m|.
For the synchronizing set to be Milnor stable the condition λA
⊥,1 < 0 must be satisfied. It means
that
|(l(1 − ε))1−m(p(1 − ε))m| < 1,
|1 − ε| <
1
l1−m|p|m
, (9)
1 −
1
l1−m|p|m
< ε < 1 +
1
l1−m|p|m
.
From analogous conditions for the transverse directions v⊥,2 and v⊥,3 we get identical stability
regions for parameter values δ and (ε + δ).
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224 I. V. MATSKIV
Fig. 1. Parameter regions for different types of stability of
the set AD . In the labeled regions, AD is: asymptotically
stable (AS), Milnor (weakly) stable (MS), Milnor
unstable (MU), strongly unstable (SU).
The synchronizing set is Milnor unstable in every transverse direction when λA
⊥,1 > 0,
λA
⊥,2 > 0, and λA
⊥,3 > 0. From these conditions we get
ε, δ, (ε + δ) ∈
(
−∞, 1 −
1
l1−m|p|m
)
⋃
(
1 +
1
l1−m|p|m
,+∞
)
. (10)
The theorem is proved.
The invariant measure µ can be found in an implicit form only in special cases, in particular
when the trajectory of the extreme point x0 = 1−1/p becomes, eventually, an unstable periodic
orbit.
Suppose that the parameters l and p are such that the map fl,p is chaotic and its invari-
ant measure µ is found, being distributed in the interval [0, 1]. Then one can find borders of
the parameter region of weak stability for the synchronizing chaotic set AD using results of
Theorem 2, as well as regions of strong stability and instability from results of Theorem 1.
All stability regions are shown in Fig. 1. Triangle colored with dark gray and labeled AS is
the region of strong (asymptotic) stability of AD. The region colored with light gray and labeled
MS is the region of Milnor stability. Lines which bound the AS region, divide the MS region
into six parts, marked on the figure with numbers from I to VI. In these parameter regions, the
diagonal still attracts trajectories from a set of initial points of positive Lebesque measure.
The synchronizing chaotic set AD on the main diagonal D2N can lose its stability in three
transverse directions which correspond to crossing each of three sides of the stability triangle
AS in Fig. 1.
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STABILITY OF SYNCHRONIZED AND CLUSTERED STATES . . . 225
Fig. 2. A dependence of the values of b: the border of strong sta-
bility region of the chaotic synchronizing set AD as the
coefficients l and p of the single skew tent map fl,p change.
After the loss of the stability, a transition to a cluster state takes place. Clustering in the
system of coupled oscillators occurs when the population of oscillators splits into subgroups
(clusters) with different dynamics, but such that all oscillators within each cluster asymptotically
move in synchrony.
When AD loses stability entering region I, a transition to the two-cluster state C
(1)
2 =
= {x1 = . . . = xN ; y1 = . . . = yN} takes place. When entering region II, we get transi-
tion to N -cluster state CN = {x1 = y1; . . . ;xN = yN}. Finally, when the loss of stability
happens through the border of region III, then for even N = 2n it results in the two-cluster
state C
(3)
2 = {x1 = . . . = xn = yn+1 = . . . = yN ; xn+1 = . . . = xN = y1 = . . . = yn}.
Coexistence of different clusters is possible in the parameter regions IV, V and VI.
The set AD is Milnor unstable in the blank region labeled MU. For the parameter values
outside this region MU, AD is strongly unstable (SU).
To obtain an analytically asymptotic stability region AS, it is enough to know the value
b =
{
b
(1)
1 , l > |p|;
b
(1)
2 , l < |p|,
which is the left corner point of AS in Fig. 1. Indeed, as it follows from
the Theorem 1, the strong stability region is bounded by the straight lines ε = b, δ = b, and
ε + δ = 2 − b. The value of b varies from 0 to 0, 5, with the change of coefficients of the
single map fl,p, b = 0 corresponds to the strong stability of the synchronizing set AD for all
0 ≤ ε, δ ≤ 1. Figure 2 shows a plot of values b versus l and p. Blank regions in the (l, p)-plane
correspond to periodic windows of the piecewise linear map fl,p. For coefficient pairs (l, p)
belonging to these regions, the diagonal is strongly stable in the whole unit square [0, 1] × [0, 1]
of the (δ, ε)-parameter plane. Figure 3 shows the variation of b versus l for four fixed values of
p = −10; −50; −100; −200.
3. An example: a system of four coupled maps. As the simplest nontrivial example, consider
system (3) with 2N = 4 coupled skew tent maps. The coupling parameters are supposed to be
0 ≤ ε, δ ≤ 1.
A partial case of system (3) with four coupled maps is an example of much interest. Its
structure is similar to the nearest-neighboring systems, where every element is coupled only
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226 I. V. MATSKIV
Fig. 3. A dependence of b for fixed values of the parameter p of a
single skew tent map. In the periodic windows of the skew tent
map, b is equal to zero, which means that the region of strong
stabilitycovers the whole unit square [0, 1] × [0, 1] of the
parameter plane.
with two neighboring elements. In the present system we have two couplings with strength ε and
two with strength δ. Such an ensemble of four coupled maps can be interpreted as a description
of quadrupeds legs locomotion. This problem was studied in [17 – 20].
Figure 4(a) shows a stability diagram for the system of four coupled skew tent maps fl,p
with the coefficients l = 0, 6, p = −10. The region of strong synchronization (coherence) is
shown in gray. Stability regions of two-cluster states C
(1)
2 = {x1 = x2; y1 = y2}, C
(2)
2 =
= {x1 = y1; x2 = y2} and C
(3)
2 = {x1 = y2; x2 = y1} are hatched. The coherent state
can lose its stability in three transverse directions, which correspond to the transition to a two-
cluster state. Stable C
(1)
2 and C
(2)
2 clusters coexist in the doubly hatched region near the left
lower corner of the coherence region.
Figure 4(b) shows another example; here l = 0, 85, p = −50. Note that the stability region
for the coherent state is wider in comparison with the previous case, and for the two-cluster
states, the stability regions are smaller. Moreover, there is no coexistence of two-cluster states
C
(1)
2 and C
(2)
2 .
From the obtained stability diagrams we conclude that for small values of the coupling
parameters ε and δ, in the blank regions in Fig. 4(a) and 4(b) there is no clustering in the system
(3). Then, increasing and varying the parameters ε and δ, one or even two cluster states C
(1)
2 or
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STABILITY OF SYNCHRONIZED AND CLUSTERED STATES . . . 227
Fig. 4. (a) Stability diagram for the system (3) with 2N = 4, l = 0,6, p = −10.
The region of stability of the coherent state is shown in gray; the regi-
ons of stability of two-cluster states are denoted by C
(i)
2 , i = 1, 2, 3.
There is a doubly hatched region of coexistence of C
(1)
2 and C
(2)
2 .
(b) A stability diagram for the system of four coupled maps with l =
= 0, 85, p = −50.
C
(2)
2 can stabilize, followed by stabilization of the coherent state. Moreover, for some values of
the coefficients l and p, the coherent state can be the first to stabilize as the coupling parameters
grow.
4. Conclusions. To investigate stability properties of the chaotic synchronizing set AD ⊂
⊂ D2N for 2N -dimensional system (3) we found analytically parameter regions for strong and
weak stability of AD as well as both types of instability. Borders of stability regions are obtained
in terms of the coefficients l and p of individual skew tent map fl,p and do not depend on the
ensemble size 2N .
Supposing the coefficients l and p to be fixed, we have delineated schematically stability
regions in the (δ, ε)-parameter plane and got a sequence of ’embedded’ triangles. In the case
when the pair of coefficients (l, p) corresponds to a periodic window of the individual skew tent
map fl,p, the synchronizing chaotic set AD is strongly stable in the unit square [0, 1] × [0, 1] of
the coupling parameter plane.
The parameter b that defines borders of the strong stability region of AD has been introduced,
and a dependence of b on the coefficients l and p was analyzed.
As an example, results of numerical experiments for the system (3) with 2N = 4 oscillators
were shown. In the unit square 0 ≤ ε, δ ≤ 1 we have delineated regions of stability of strongly
synchronized (coherent) and partially synchronized (clustered) states.
The type of the synchronization transition in the case of coupled piecewise linear maps,
where for small values of the coupling parameters, as they increase, a stabilization of the two-
cluster states C
(1)
2 or C
(2)
2 takes place, is essentially different from the case of the coupled
smooth maps fa(x) = ax(1 − x) [7]. In the smooth case, with an increase of small values of
the coupling parameters ε and δ, the two-cluster state C
(3)
2 is the first to stabilize, only after we
get a stabilization of C
(1)
2 or C
(2)
2 , followed by a transition to coherence.
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228 I. V. MATSKIV
Moreover, from the obtained stability diagrams it is obvious that parameter regions of stabi-
lity for synchronized and clustered states are rather large, which makes it possible to provide
an effective control of the system dynamics varying the coupling coefficients.
Acknowledgements. The author is grateful to Yu. L. Maistrenko and S. I. Popovych for a
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Received 20.02.2004
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