Chaos, Solitons and Fractals 34 (2007) 1082–1104

www.elsevier.com/locate/chaos

Physical interpretation and theory of existence ofcluster structures in lattices of dynamical systems

Nikolai N. Verichev a, Stanislav N. Verichev c, Marian Wiercigroch b,*

a Mechanical Engineering Institute, Russian Academy of Sciences, Belinskogo 85, 603024 Nizhny Novgorod, Russiab Centre for Applied Dynamics Research, Department of Engineering, University of Aberdeen, King’s College,

Aberdeen AB24 3UE, Scotland, UKc Technology Company Schlumberger, Respubliki 59, 625000 Tyumen, Russia

Accepted 22 May 2006

Abstract

The alternative theory of existence of cluster structures in lattices of dynamical systems (oscillators) is proposed. Thistheory is based on representation of structures as a result of classical (full) synchronization of structural objects calledcluster oscillators (C-oscillators). Different types of C-oscillators, whose number is defined by the geometrical propertiesof lattices (dimensions and forms) are introduced. The completeness of all types of C-oscillators for lattices of differentdimensions is proven. This fact provides a full set of types of cluster structures that can be realized in a given lattice. Thesolution is derived without the necessity to verify the existence of invariant (cluster) manifolds. The principles ofcoupling of C-oscillators into the cluster structures and principles of transformations of such structures are described.Having interpreted the processes of structuring in terms of the classical synchronization of C-oscillators, one can solvethe problem of fusion of lattices with pre-described properties at the engineering level.� 2006 Elsevier Ltd. All rights reserved.

1. Introduction

The history of experimental observations and theoretical studies of the effect of self-organization consisting in theappearance of stationary or multistable spatio-temporal structures in homogeneous active media spans for at least halfa century. The history and the state-of-the art of this subject, which is also referred as synergetics are covered in [1–3].Nevertheless, the problems concerned with the structuring of dynamical processes, being one of the main subjects ofnonlinear physics and oscillation’s theory, are still actively investigated. This is the primary concern of new problemsin different fields such as radiophysics, electronics, chemistry and, especially, in biological and technical applications [4–12]. Note that during last years, studies of structures are generally stimulated by the discovery of the effect of chaoticsynchronization [13,14], which has fundamentally broaden conventional outlook on the synchronization and dynamicsof structures.

Lattices of oscillators (rotators) belong to a special branch in investigations of the dynamics of structures. On theone hand, such lattices represent adequate models of active media. On the other hand, they are of potential importance

0960-0779/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.doi:10.1016/j.chaos.2006.05.062

* Corresponding author. Tel.: +44 1224 272509; fax: +44 1224 272497.E-mail address: M.Wiercigroch@eng.abdn.ac.uk (M. Wiercigroch).

N.N. Verichev et al. / Chaos, Solitons and Fractals 34 (2007) 1082–1104 1083

for practical applications. There exist a number of papers dealing with the dynamics of lattices of oscillators including[15–33].

In cluster dynamics of homogeneous lattices, the group of identically synchronized oscillators of a lattice is called ascluster. The conventional theory of existence of cluster structures has been built as a theory of existence of invariantmanifolds of differential equations governing a lattice [22–28]. According to this theory, some sub-manifold of aninvariant manifold corresponds to each cluster of the structure. Therefore, the problem of existence of cluster structuresin a lattice is reduced to the problem of finding the different invariant manifolds. Oscillators that belong to differentclusters are not synchronized and due to this fact the process of the formation of cluster structure has been called aspartial synchronization or cluster synchronization as an alternative to the full synchronization corresponding to thespatio-temporal state of a lattice. According to this interpretation, cluster synchronization is considered as an indepen-dent physical effect. Interpretation of cluster structures from the standpoint of invariant manifolds is not devoid ofdisadvantages.

Firstly, the problem of finding the invariant manifolds of differential equations requires considerable efforts, evensome ‘‘art’’ and generally is a matter of luck, especially for the lattices of irregular geometries. Secondly, having foundthe set of manifolds for a given lattice, one has to prove the fullness of this set. That is, to prove that there are no othermanifolds and therefore, no cluster structures except this set. The authors of this paper have never seen such proves everbeen published. Thirdly, invariant manifolds exist independently of the dynamical properties of elementary systems.Such systems can be so that some or even all cluster structures, corresponding to the existing invariant manifolds wouldnot exist (existence of a manifold does not presume the existence of cluster structure). Fourthly, interpretation of clusterstructures in terms of invariant manifolds does not have any physical background that obviously complicates the under-standing of cluster dynamics of lattices by the physical audience. Fifthly, such interpretation does not provide any algo-rithm of the fusion of lattices of complicated geometrical forms with pre-described cluster structures containingarbitrary couplings. This list is by no means exhaustive.

This paper is aimed to outline the general approach for constructing (coupling) one-dimensional (chain) and two-dimensional (lattice) structures from the building blocks (basic nonlinear oscillators). For the purpose of clarity, at thisstage we do not intend to discuss the stability of cluster synchronization (stability of cluster structures) and to presentthe full mathematical description of the considered global (chain, lattice) and local (nonlinear oscillator) dynamical sys-tems and their properties, in particular the smoothness of the right-hand parts of differential equations describing them.All these aspects will be discussed in the next paper, where the stability of the cluster dynamics will be analysed. Inshort, the main motto of this paper is ‘‘cluster structures – it is simple!’’ In particular, it will be

• shown that the cluster synchronization is a special case of the classical synchronization and therefore it is not anindependent physical effect;

• demonstrated that the theory of existence of cluster structures can be built without any appealing to the invariantmanifolds of the differential equations governing systems of lattices;

• proven the fullness of the set of types of cluster structures in a chain and in a two-dimensional lattice of oscillatorsusing the elementary tools;

• listed the properties of cluster structures responsible for answering the question ‘May a given structure be the clusterstructure or may not?’

• explained how lattices of complicated forms with local and non-local couplings containing pre-described dynamicalcluster structure can be constructed.

The paper is organized as follows: In the next section cluster oscillators and how they can form structures in a chainof coupled oscillators are introduced and analysed. Symmetrical and asymmetrical clusters of oscillators are consideredin detail where the Chua’s oscillator is used as an archetype. Section 3 defines and analyses lattice structures of coupledoscillators. An exhaustive list of cluster oscillators is developed. Section 4 gives brief comments on cluster oscillatorsand structures in three-dimensional lattice of coupled oscillators. Section 4.2 touches upon a cluster formation ofdynamical processes in irregular homogeneous lattices. The summarising remarks conclude the paper.

2. Chain of coupled systems

The simplest example of a coupled system is a chain of diffusively coupled oscillators with Neiman boundary con-ditions of the form

_X i ¼ FðX iÞ þ eCðX i�1 � 2X i þ X iþ1Þ; ð1Þ

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where

i ¼ 1::N ; X i 2 Rm; FðX iÞ : Rm ! Rm;

e P 0; C ¼ diagðc1; c2; . . . ; cmÞ; ci P 0;

X0 � X1:

Here, matrix C defines the group of parameters, through which the coupling is realized. Elements of ci are equal either 0or 1, e is a scalar strength of coupling. As soon as it will be necessary, we will define the boundary conditions at theother end of the chain. Particularly, the chain can be considered as infinite. Generally speaking, it is not necessaryto provide the dynamical system of the form

_X ¼ FðXÞ;

with any special properties in order to solve the problem of existence of cluster structures. This system may be dissipa-tive or not dissipative, may be smooth or piecewise smooth, may be gradient, etc. However, for a better understandingof the arguments and terminology given below, we will assume that an elementary dynamical system represents an oscil-lator with regular or chaotic dynamics. As an illustration, we will show the dynamics of coupled Chua’s chaotic oscil-lators [5]. Dynamical properties of this classical oscillator are governed by the equations of the form

X ¼ ðx; y; zÞT;_x ¼ aðy � hðxÞÞ;_y ¼ x� y þ z;

_z ¼ �by � cz; hðxÞ ¼ m1xþ 1

2ðm0 � m1Þðjxþ 1j � jx� 1jÞ:

Hereinafter, the following set of parameters is used

fa; b; c; m0; m1; eg ¼ 9:5; 14; 0:1;� 1

7;2

7; e

� �:

The value of the parameter of coupling strength e will be indicated at the left upper corner of each figure showing aphase portrait. We assume that the dynamical properties of an elementary oscillator are completely known. In partic-ular, we suppose that maximum Lyapunov exponent k(1) of attractor A(1) shown in Fig. 1 is known.

2.1. Symmetrical cluster oscillators

Let us start with the simplest example and we will consider the system of two dissipative- and symmetrically coupledoscillators of the form

_X1 ¼ FðX1Þ þ eCðX2 � X1Þ;_X2 ¼ FðX2Þ þ eCðX1 � X2Þ:

ð2Þ

One can pose a quite fundamental question ‘What is known about possible stationary regimes of two coupled oscilla-tors irrespective of the number of their degrees-of-freedom and other individual dynamical properties’? Firstly, for somevalues of coupling strength e, which depends on maximum Lyapunov exponent k(1) of the attractor A(1) for certain

Fig. 1. Chua’s attractor A(1).

Fig. 2. Simple chaotic synchronization of two Chua’s oscillators.

V V V V

1 2

Fig. 3. Schematic representation of the system of two coupled Chua’s oscillators in a regime of synchronization.

Fig. 4. Attractor As(2), corresponding to the regime of stationary chaotic beatings in a system of two oscillators in projections on co-ordinate planes.

N.N. Verichev et al. / Chaos, Solitons and Fractals 34 (2007) 1082–1104 1085

initial conditions, a regime of simple (by means of the rotation number [14]) mutual chaotic synchronization of self-oscillations will be observed [13,14]: X1 = X2 for t!1 (see Fig. 2).

After some time, oscillations of both oscillators become identical. Suppose that the regime of synchronization is real-ized exactly. Since later on we will emphasize on the existence but not on the stability of structures, this supposition isfully justified. Moreover, in this case we may suppose that initial conditions are realized exactly at the necessary attrac-tor. A schematic representation of this system in a regime of synchronization is depicted in Fig. 3.

Note that voltages at the oscillators (at the points of connection) are equal to each other at any moment of time.Points of connections 1 and 2 are equipotential, and if one ‘‘cuts’’ them, the regime of synchronization will not be dis-turbed. In general case, besides the regime of synchronization, a regime of stationary regular or stationary chaotic beat-ings can exist. Fig. 4 shows an example of such a regime. Both regimes are realized depending on the different initialconditions.

In a regime of stationary beatings, the values of the system parameters for two oscillators (red and yellow)1 are dif-ferent and this system can be represented schematically in a following form (see Fig. 5).

1 For interpretation of colour in Fig. 4, the reader is referred to the web version of this article.

V1 V2

Fig. 5. Schematic representation of the system of two coupled Chua’s oscillators in a regime of stationary beatings.

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In a regime of stationary beatings system (2) is inseparable into two oscillators and represents a single object. If thereexists attractor As(2), then we will call system (2) a cluster oscillator (C-oscillator) of the form Os(2). Let us rewrite sys-tem (2) in a vector form:

_X ¼ FðXÞ; ð3Þ

where

X ¼ ðX1;X2ÞT; FðXÞ ¼ ðFðX1Þ;FðX2ÞÞT þ eB � CX ;

B ¼�1 1

1 �1

� �:

� is the symbol of direct multiplication of matrices. Index s in denotation for C-oscillator means the symmetry of thecluster matrix B.

Now we will couple two oscillators of the type Os(2) and consider the following system:

_X ¼ FðXÞ þ e�C�ðY � XÞ;_Y ¼ FðYÞ þ e�C�ðX � YÞ;

ð4Þ� �

where C� ¼0 00 1

� C .

System (4) represents a pair of interacting oscillators and in relation to the previous case, these oscillators just havedoubled the number of degrees-of-freedom. This does not change the stationary dynamics of the system as for somevalue of the coupling strength e* that depends on the maximum Lyapunov exponent k(2) of the attractor As(2), for cer-tain initial conditions a simple classical synchronization of oscillators on this attractor will occur: X = Y for t!1. Aschematic representation of this system in a regime of cluster synchronization is depicted in Fig. 6. Hereinafter, we willsuppose that the synchronization regime is realized exactly.

On the other hand, if e* = e, and Y = (X4,X3)T then system (4) becomes system (1) for N = 4 and the Neiman bound-ary conditions at the end. Thus, in the regime of simple synchronization in a chain of C-oscillators a so-called ‘‘central’’cluster structure occurs, Sc

sð2Þ: X1 = X4, X2 = X3 [24]. Note that chain can be cut into a pair of identical C-oscillatorswithout disturbing the regime of synchronization. Let us connect one more C-oscillator to the pair of C-oscillators:

_X ¼ FðXÞ þ e�C�ðY � XÞ;_Y ¼ FðYÞ þ e�C�ðX � YÞ þ e�C �ðZ � YÞ;_Z ¼ FðZÞ þ e�C�ðY � ZÞ;

ð5Þ

� �

where C� ¼ 1 0

0 0� C .

In the regime of a simple synchronization X = Y = Z, for which a corresponding schematic representation isdepicted in Fig. 7.

On the other hand, if one will put e* = e, and Z = (X5,X6)T in the system (5) then for N = 6 this system will becomesystem (1). Thus, simple synchronization of three C-oscillators of the type Os(2) produces in a so-called ‘‘alternative’’cluster structure Salt

s ð2Þ : X1 ¼ X4 ¼ X5;X2 ¼ X3 ¼ X6 [24]. As an example, the alternative cluster structure in a chainof N = 6 Chua’s oscillators is depicted in Fig. 8. The cluster attractor As(2) depicted in Fig. 4, and the attractor of the

Fig. 6. Schematic representation of the system of four coupled oscillators in a regime of cluster synchronization.

Fig. 7. Schematic representation of the system (5) in the regime of simple classical synchronization.

Fig. 8. Alternative cluster structure produced by the simple classical synchronization of three C-oscillators Os(2) in a chain of N = 6elementary Chua’s oscillators.

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structure is the same. Again, we draw the attention to the property of splitting of a chain into identical C-oscillators inthe regime of synchronization.

The chain of cluster oscillators can be extended further by adding and coupling C-oscillators. We will call the systemof three symmetrically coupled oscillators of the form

_X1 ¼ FðX1Þ þ eð�X1 þ X2Þ;_X2 ¼ FðX2Þ þ eðX1 � 2X2 þ X3Þ;_X3 ¼ FðX3Þ þ eðX2 � X3Þ

ð6Þ

as cluster oscillator Os(3) if there exists a cluster attractor As(3) corresponding to the regime of stationary beatings of allthese three oscillators. An example of such cluster attractor As(3) is depicted in Fig. 9, where a schematic representationof C-oscillator Os(3) in its ‘‘operating’’ regime is shown in Fig. 10. That is, the significant dynamics of this C-oscillatoroccurs at the attractor As(3).

Now we rewrite system (5) as one vector equation

_X ¼ FðXÞ;X ¼ ðX1;X2;X3ÞT; FðXÞ ¼ ðFðX1Þ;FðX2Þ;FðX3ÞÞT þ eB � CX ;

B ¼�1 1 0

1 �2 1

0 1 �1

0B@

1CA:

Fig. 9. Attractor As(3) of C-oscillator Os(3) in projections on co-ordinate planes.

Fig. 10. Schematic representation of C-oscillator Os(3).

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Then we couple a pair of such C-oscillators, each one of which is in its operating regime and consider the followingsystem:

_X ¼ FðXÞ þ e�C�ð�X þ YÞ;_Y ¼ FðYÞ þ e�C�ðX � YÞ;

where

C� ¼0 0 0

0 0 0

0 0 1

0B@

1CA� C :

Again we have obtained a system of two interacting oscillators. The interpretation of the stationary dynamicsremains the same. In the regime of simple synchronization X = Y. If e* = e, and Y = (X6,X5,X4)T then for N = 6this system is identical to system (1). In other words, a central cluster structure occurs in the chainSc

sð3Þ : X1 ¼ X6;X2 ¼ X5;X3 ¼ X4. As an example, the central cluster structure in a chain of N = 6 elementary Chua’soscillators is depicted in Fig. 11. A schematic representation of this structure is shown in Fig. 12.

Attaching a third C-oscillator to the pair of C-oscillators of the type Os(3) in the same way, we obtain the followingsystem:

_X ¼ FðXÞ þ e�C�ð�X þ YÞ;_Y ¼ FðYÞ þ e�C�ðX � YÞ þ e�C �ð�Y þ ZÞ;_Z ¼ FðZÞ þ e�C�ðY � ZÞ;

where

C� ¼1 0 0

0 0 0

0 0 0

0B@

1CA� C ; Z ¼ ðX7;X8;X9Þ:

In the regime of simple synchronization X = Y = Z that corresponds to the alternative cluster structureSalt

s ð3Þ : X1 ¼ X6 ¼ X7;X2 ¼ X5 ¼ X8;X3 ¼ X4 ¼ X9 in a chain of N = 9 oscillators. Fig. 13 shows the schematic rep-resentation of this structure.

By adding new C-oscillators one can proceed further with up-building of the chain. For even numbers of C-oscil-lators one will obtain central structures and for the odd numbers of C-oscillators one will get alternative three-clusterstructures.

Fig. 11. Central cluster structure caused by the simple synchronization of two C-oscillators Os(3) in a chain of N = 6 Chua’soscillators.

Fig. 12. Schematic representation of central cluster structure in a chain of N = 6 Chua’s oscillators.

Fig. 13. Schematic representation of the alternative cluster structure in a chain of N = 9 oscillators.

N.N. Verichev et al. / Chaos, Solitons and Fractals 34 (2007) 1082–1104 1089

Definition 1. A system of n oscillators of type (1) with the boundary condition Xn � Xn+1 is called as a symmetric cluster

oscillator of the type Os(n) if there exists attractor As(n) corresponding to the regime of stationary beatings of allcomprising oscillators. Suppose there is a C-oscillator Os(n), which is set into its ‘‘operational’’ regime, i.e., its motionsoccur at the attractor As(n). Suppose also that we have an infinite number of copies of this C-oscillator in suchdynamical regime. In this case, the principle of coupling of C-oscillators in a chain the formation of the cluster structureis shown in Fig. 14.

The simple classical synchronization of m coupled C-oscillators of the type Os(n) for even m defines central n-clusterstructure Sc

sðnÞ in a chain of N = mn elementary oscillators with the Neiman boundary conditions at the end and definesthe alternative cluster structure Salt

s ðnÞ for the odd m.Consider the system of two asymmetrically coupled elementary oscillators of the form

_X1 ¼ FðX1Þ þ eð�X1 þ X2Þ;_X2 ¼ FðX2Þ þ 2eðX1 � X2Þ:

ð7Þ

1 2 . . . n-1 n n+1 n+2 . . . 2n-1 2n 2n+1 . . .

Fig. 14. Schematic representation of the coupling in a chain of cluster oscillators.

Fig. 15. Attractor Aa(2) corresponding to the regime of stationary beatings of two asymmetrically coupled Chua’s oscillators inprojections on co-ordinate planes.

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In this system, besides the regime of synchronization, a regime of stationary beatings (attractor Aa(2)) does also exist.The example of attractor Aa(2) for two asymmetrically coupled Chua’s oscillators is depicted in Fig. 15.

If there exists a regime of stationary beatings, then we will call the system (7) as C-oscillator of the type Oa(2). Thissystem is defined by the following equation:

_X ¼ FðXÞX ¼ ðX1;X2ÞT; FðXÞ ¼ ðFðX1Þ;FðX2ÞÞT þ eB � CX ;

B ¼�1 1

2 �2

� �

Consider a system of two coupled C-oscillators of the type Oa(2)

_X ¼ FðXÞ þ e�Dð�X þ YÞ;_Y ¼ FðYÞ þ e�DðX � YÞ;

ð8Þ� �

where D ¼ 0 01 0

� C and X = (X1,X2)T, Y = (Y1,Y2)T, X2 = Y2 the additional internal coupling. This system can

be physically represented by the circuit that is depicted in Fig. 16.

In this circuit, elementary oscillators are denoted by encircled crosses. Two oscillators in the middle are coupled atthe same points and thereby, they are always synchronized. Being coupled in a chain, asymmetric C-oscillators alwaysco-exist by pairs.

R R

1X 2X 2Y 1Y

Fig. 16. Physical representation of two coupled C-oscillators of the type Oa(2).

Fig. 17. Schematic representation of a central cluster structure in the regime of simple synchronization in a system of two coupledC-oscillators of the type Oa(2) with an additional internal coupling.

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In the regime of simple synchronization of these oscillators X = Y. On the other hand, for e* = e, Y = (X3,X2)T andfor N = 3 this system becomes system (1). A central cluster structure Sc

að2Þ will be realized in the chain. Fig. 17 shows aschematic representation of this structure.

Consider the system of two coupled pairs of C-oscillators, each one is in its operating regime:

Fig.

_X ¼ FðXÞ þ e�Dð�X þ YÞ;_Y ¼ FðYÞ þ e�DðX � YÞ þ e�C�ð�Y þ ZÞ;_Z ¼ FðZÞ þ e�Dð�Z þWÞ þ e�C �ðY � ZÞ;_W ¼ FðWÞ þ e�DðZ �WÞ;

with additional internal couplings X2 = Y2, Z2 = W2.In the regime of simple synchronization X = Y = Z = W. On the other hand, if e* = e, X = (X1,X2)T, Y = (X3,X2)T,

Z = (X4,X5)T, W = (X6,X5)T then this system transfers to system (1) for N = 6. A central cluster structure Scað2Þ is real-

ized in the chain, which schematic representation is depicted in Fig. 18.Fig. 19 shows the central cluster structure in a chain of N = 6 Chua’s oscillators. One can compare this structure to

attractor Aa(2) of Oa(2), which is depicted in Fig. 15. It is seen that they are the same.

Fig. 18. Central cluster structure Scað2Þ in a chain of two dissipative-coupled pairs of C-oscillators.

19. Central cluster structure based on the C-oscillator of the type Oa(2) in a chain of N = 6 elementary Chua’s oscillators.

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Definition 2. We will call a sub-system of n elementary oscillators of the system (1) with the boundary conditionXn�1 � Xn+1 as asymmetric cluster oscillator of the type Oa(n) if there exists attractor Aa(n) corresponding to the regimeof stationary beatings of all these elementary oscillators.

Asymmetric C-oscillators always co-exist by pairs. Suppose we have a pair of C-oscillators Oa(n) with exactly real-ized synchronization regime, i.e., the motions of the pair occur at the attractor Aa(n). Suppose we have an infinite num-ber of copies of this pair. In this case, the principle of coupling of pairs in a chain and the formation of the clusterstructure is shown in Fig. 20.

Simple synchronization of m pairs of cluster oscillators of the type Oa(n) defines central cluster structure ScaðnÞ in a

chain of N = (2n � 1)m elementary oscillators.

Remark. Indexes s and a in notations for cluster oscillators denote symmetry or asymmetry of the cluster matrix B.Hereinafter, these indexes will not be concerned with symmetry or asymmetry of cluster structures, produced by thesynchronization of C-oscillators.

Theorem 1. Cluster oscillators Os(n) and Oa(n) represents the full set of types of cluster oscillators in a homogeneous chain

of oscillators with the Neiman boundary conditions.

Consequence. The central and the alternative cluster structure constitutes the full set of types of cluster structures in achain of diffusively coupled oscillators with the Neiman boundary conditions.

Proof. Suppose that some cluster structure is realized in a chain and there is a segment of this chain consisting of first n

not synchronized oscillators (cluster oscillator). This means that oscillator with number (n + 1) is synchronized withsome of other oscillators of the considered segment. In this case, values of all variables of these oscillators (currents,voltages) are equal to each other at any moment of the time. Points at the inputs of oscillators are equipotential so thereis no current trough the resistances connecting them (see the circuit depicted in Fig. 21).

Currents at inputs of synchronized oscillators fulfill the following equations:

Fig. 2oscilla

Vn�1 � V ¼ IR; Vnþ2 � V ¼ IR ) Vn�1 ¼ Vnþ2:

It means that oscillators with numbers (n � 1) and (n + 2) are also synchronized by pairs. Writing down equations forthe voltages of these oscillators, we obtain

Vn�2 þ V � 2Vn�1 ¼ In�1R; Vnþ3 þ V � 2Vnþ2 ¼ Inþ2R ) Vn�2 ¼ Vnþ3:

Continuing this we finally obtain that the condition Xn = Xn+1 corresponds to the coupling of C-oscillators of the typeOs(n). Suppose now that (n + 1)th oscillator is synchronized with (n � 1)th one. Considering equations for voltages ofsynchronized oscillators, we obtain

Vn�2 � 2Vn�1 þ Vn ¼ In�1R; Vnþ2 � 2Vnþ1 þ Vn ¼ Inþ1R ) Vn�2 ¼ Vnþ2:

-1nV V V 2nV +

1 2 . . . 1n − n 1n + 2n + . . .

I I

Fig. 21. Physical representation of a chain of oscillators.

1 2 . . . n-1 n n+1 2n-2 2n-1 2n . . .

0. Schematic representation of a cluster structure of a chain of asymmetric based on the synchronization of asymmetric C-tors.

n 1n + 2n +1k + k 1k −

n 1n + 2n +1k + k 1k −

n 1n + 1k −1k + k 2n +

Fig. 22. A set of circuit transformations.

N.N. Verichev et al. / Chaos, Solitons and Fractals 34 (2007) 1082–1104 1093

Continuing doing this we obtain that condition Xn = Xn+1 corresponds to the coupling of C-oscillators of the typeOa(n). Finally, we suppose that oscillator with number (n + 1) is synchronized with some other k-th oscillator of thecluster oscillator (k 6 n � 2). By coupling the appropriate equipotential points and by making further transformationsof the circuit (which do not harm the dynamical regime of the chain), we obtain a sequence of equivalent circuits rep-resented in Fig. 22.

Note that in the first and the third circuits there is no current through the connector. We obtain

Vn � 2Vnþ1 þ Vnþ2 ¼ IR; Vkþ1 � 2Vk þ Vnþ2 ¼ IR ) Vn ¼ Vkþ1:

Here, comes the contradiction as the system of first n elementary oscillators contains the synchronized oscillators andtherefore is not a cluster oscillator. Thus, all C-oscillators of the types Os(n) and Oa(n) constitute a full set of types ofcluster oscillators in a chain. Now we conclude the following:

1. The effect of cluster synchronization consists in a simple synchronization of cluster oscillators (‘‘full classical syn-chronization of the regimes of stationary beatings’’).

2. Any stationary cluster structure that exists in a chain of elementary oscillators is in the univocal correspondence tothe simple synchronization of certain number of equal cluster oscillators of one of these two types: Os(n) or Oa(n).

3. Each cluster structure can be ‘‘cut’’ through the connections of equipotential points into a certain number of equalsynchronized C-oscillators of the type Os(n) or into a certain number of equal pairs of C-oscillators of the type Oa(n)with their further convolution into one cluster oscillator (procedure of connection of the equipotential points).

4. Each structure that cannot be ‘‘cut’’ and is convoluted is not a cluster structure (the necessary condition).5. The problem of existence of different types of cluster structures for assigned ‘‘length’’ of a chain N can be reduced to

the elementary problem of covering a segment of oscillators by equal cluster oscillators with different number ofclusters n.

6. Let p be the number of all odd multipliers of the number N and q is that of the even ones (simple and composite, notequal to 1 and N). If the number N is the odd number, then there exist p alternative cluster structures based on C-oscillators Os(n) and p + 1 central cluster structure based on C-oscillators Oa(n) in a chain of N elementary oscilla-tors with the Neiman boundary conditions. If the number N is the even number, then there exist p alternative clusterstructures based on C-oscillators Os(n), p central cluster structure based on C-oscillators Oa(n) and q central clusterstructure based on C-oscillators Os(n) in a chain of N elementary oscillators with the Neiman boundary conditions.The full number of structures including one-cluster (trivial one) and N-cluster structures is equal to 2p + 3 for theodd N and 2p + q + 2 for the even N.

7. During the construction of cluster structures in chains, the attachment of C-oscillators can be done endlessly in bothdirections (see Figs. 14 and 20). On the other hand, we did not use the condition of limitation of a chain. Thus, C-oscillators Os(n) and Oa(n) are the only structure forming objects for infinite in both directions chains. In this case,we do not discuss the stationarity or non-stationarity of the corresponding cluster structures. h

Remark 1. All known theorems concerning existence of alternative invariant manifolds in the system (1) [22,24,25] arenow just a simple consequence of the aforementioned theorem. Moreover, it follows from this theorem that there are noother types of integral manifolds in the system (1). However, in the course of the proposed interpretation of structures,this information is redundant.

Fig. 23. The complex of cluster structures in a chain of N = 12 oscillators.

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Remark 2. According to the proposed interpretation of cluster structures, any cluster structure realized in a chain, canbe enclosed (through a resistor) into a ring (connection of the equipotential points). On the contrary, if a cluster struc-ture containing two neighbouring synchronized elementary oscillators is realized in a ring, then by cutting the equipo-tential points, we obtain the cluster structure in a chain. Moreover, according to the aforementioned theorem, thisstructure has to be a central structure. It means that C-oscillators Os(n) and Oa(n) enter the number of basic C-oscil-lators of a ring. Finally, before the enclosure of a chain into a ring, we can preliminary cut out a number of C-oscillatorsOs(n) with a certain number of blocks of the C-oscillators Oa(n). That is, the ‘‘length’’ of C-oscillator or that of theblock is a period of the structure. Now everything is already to formulate the statements about existence of a so-calledcut cluster structures (and, if necessary, existence of the corresponding integral manifolds) in a ring. Note, that for thering there exists one more type of C-oscillators defining uncut cluster structures. The full set of the types of C-oscillatorsin a ring consists of three C-oscillators and will be presented by us later as an independent study.

Example. Fig. 23 shows a complex of cluster structures in a chain of N = 12 elementary oscillators. According to theaforementioned theorem, there are no other cluster structures in a chain of N = 12 oscillators.

3. Two-dimensional lattices of coupled oscillators

Consider a homogeneous rectangular lattice of coupled oscillators with Neiman boundary conditions

_X ij ¼ FðX ijÞ þ eCðX i�1j þ X iþ1j þ X ij�1 þ X ijþ1 � 4X ijÞ;i ¼ 1;N1; j ¼ 1;N2;

X0j � X1j; X i0 � X i1:

ð9Þ

As soon as necessary, we will define the boundary conditions at the other ends of the lattice. Again we will solve theproblem of existence of different cluster structures by finding a set of basic types of cluster oscillators and by followingthe elementary idea of covering of a lattice by equal cluster oscillators and by blocks of cluster oscillators.

Definition 3. We will call a lattice of the size p · q a simple cell, whose nodes are filled up by the elements of one clusteroscillator or by the elements of a block of equal cluster oscillators, which cannot be cut.

In the case of a chain, such cells are segments, filled up by elements of C-oscillator of type Os(n) or by elements of pair ofC-oscillators of type Oa(n). In a two-dimensional case, such cells will be filled up by elements of one C-oscillator ore byelements of a pair, four, eight but not more C-oscillators. Note that the entire lattice N1 · N2 under some conditions canbe considered as a simple cell. This simple fact allows us to suppose that simple cells have a ‘‘rectangular’’ form. We willuse different types of the symmetry of plane figures; understanding the symmetry more in physical than in geometricalsense as equality of potentials of certain points of simple cell with respect to its axes and centres of symmetry.

3.1. Cluster oscillators and simple cells

The simplest type of C-oscillators are the one-dimensional oscillators of the type Os(1 · n),Os(n · 1),Oa(1 · n),Oa(n · 1) that are inheritable by a two-dimensional lattice from a chain. A simple synchronizationof such oscillators realizes ‘‘band-like’’ cluster structures. An example of such a structure is depicted in Fig. 24.

Fig. 24. Band-like cluster structures based on oscillators of types Os(1 · 3),Oa(1 · 2),Os(3 · 1),Oa(2 · 1), respectively.

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C-oscillators of the type Os(Æ) and blocks of oscillators of the type Oa(Æ) (simple cells) can be coupled. ‘‘Two-dimen-sional’’ oscillators in a regime of simple synchronization define mosaic cluster structures.

C-oscillator of the type Oss(m · n) is a sub-system of mn elementary oscillators of the system (9) with boundary con-ditions Xmj � Xm+1j, Xin � Xin+1 if there exists attractor Ass(m · n). C-oscillators of the type Oss(m · n) are the simplecells that can be dissipative-coupled as by co-ordinate i as by co-ordinate j as shown in Fig. 25.

Let us list C-oscillators that are related to vertical and horizontal axes of symmetry of a rectangle. Suppose that asimple cell represents a block of two C-oscillators. Therefore, reduction of a chain into one oscillator (corresponding tothe connection of equipotential points) results in bending of a cell through one of the axes of symmetry; vertical(Fig. 26) or horizontal (Fig. 27).

Cluster oscillator Oas(m · n) is a sub-system of mn oscillators of system (9) with boundary conditions Xm�1j � Xm+1j,Xin � Xin+1 if there exists attractor Aas(m · n). C-oscillators of this type always co-exist by pairs. A building of a simple

Fig. 25. Example of coupling of C-oscillators of the type Oss(2 · 3).

Fig. 26. Example building of a simple cell and posterior cluster structure based on C-oscillator Oas(2 · 2).

Fig. 27. Example of building of cell and lattice based on C-oscillator Osa(2 · 2).

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cell and of a posterior cluster structure into a ‘‘large’’ lattice can be done in the way that is depicted in Fig. 26. The sizeof a simple cell of this type is (m · 2n � 1).

C-oscillator Osa(m · n) is a sub-system of mn elementary oscillators of system (9) with boundary conditionsXmj � Xm+1j, Xin�1 � Xin+1 if there exists attractor Asa(m · n). C-oscillators of this type also always co-exist by pairs.Coupling of C-oscillators in a cell and coupling of cells in a lattice can be done in the way that is depicted inFig. 27. The size of a simple cell of this type is (2m � 1 · n).

Suppose that simple cell represents a block of four C-oscillators. In this case, reduction of the cell in one C-oscillatorcauses by double sequential bending through both axes. C-oscillator Oaa(m · n) is a sub-system of mn elementary oscil-lators of system (9) with boundary conditions Xm�1j � Xm+1j, Xin�1 � Xin+1 under condition of existence of attractorAaa(m · n) corresponding to the regime of stationary beatings of all these elementary oscillators. C-oscillators of thistype also always co-exist by fours and can be coupled in a simple cell as depicted in Fig. 28. The size of a simple cellof this type is (2m � 1 · 2n � 1).

There are no other bends of a simple cell allowed. This fact can be easily proven by means of Kirchhoff laws. Simplecell cannot contain more than for C-oscillators of such configuration. For oscillators with diagonal symmetry of asquare we assume that a simple cell is square-like and has a size (n · n). Let us make the following transformationsof such a cell concerned with the connection of possible equipotential points. We combine a lattice by bending itthrough one of diagonals and by bending all consequent figures through their axes of symmetry. By performing thesetransformations we obtain the sequences of figures shown in Fig. 29.

At the second iteration of the lattice transformation with even number of elements we cut figure through its axis ofsymmetry. All further transformations can be done in the same way as in the case of lattice with odd number of ele-ments. C-oscillators are the systems of not synchronized oscillators placed at the joints of latticing figures. UnfoldingC-oscillators into a corresponding square (a simple cell), the order of their coupling goes in the reverse sequence. Duringthe inverse transformations, the following conditions have to be fulfilled: (a) the number of coupled cluster oscillatorsforming a simple cell should not exceed eight, (b) elementary oscillators from one cluster cannot be both on a boundaryof a square and inside it. These conditions can be elementary proven within the framework of the Kirchhoff laws as ithas been done for a chain. Examples of construction of simple cells and corresponding cluster structures are shown inFig. 30. Further cells can be coupled in a ‘‘large’’ lattice.

It is easy to see that there are six qualitatively different types of cluster oscillators concerned with diagonal symmetryof a square. We will skip determining the set of equations for C-oscillators, but we will make their ‘‘geometrical’’description.

Fig. 29. Sequences of figures concerned with connection of possible equipotential points.

Fig. 28. An example building of cell and lattice based on C-oscillator Oaa(2 · 2).

Fig. 30. Examples of construction of simple cells and corresponding cluster structures.

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1. ‘‘Flags-2’’ are the C-oscillators of the type O2a(n(n + 1)/2) and O�2aðnðnþ 1Þ=2Þ unfolding to n–square in one itera-tion as shown in Fig. 30(b) and (e). O�2að�Þ results in rotating O2a(Æ) into 90�. In a simple cell, these oscillators definestructures with diagonal symmetry.

2. ‘‘Pyramid’’ is a cluster oscillator O4a((n + 1)2/4) unfolding to n = 2k + 1 –square in two iterations as depicted inFig. 30(d) and (g). This C-oscillator and all the further ones define structures with central symmetry.

3. ‘‘Mausoleum’’ is a cluster oscillator O4a((n2 + 2n)/4) unfolding to n = 2k–square (simple cell) in two iterations asshown in Fig. 30(a) and (h).

Fig. 31. Example of the rule of formation of a simple cell and further cluster structures based on Op2að6Þ.

Fig. 32. Example of the rule of formation of a simple cell and further cluster structures based on Op2að5Þ.

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4. ‘‘Flag-8’’ is a cluster oscillator O8a((n + 1)(n + 3)/8) unfolding to n = 2k + 1–square in three iterations as shown inFig. 30(c) and (f).

5. ‘‘p-rectangle’’, a cluster oscillator Op2sðmn=2Þ is a rectangular lattice of the size (m/2 · n) for m = 2k or of the size

(m · n/2) for n = 2k. Simple cell of the size (m · n) can be derived by coupling C-oscillator and its image. The latterresults in rotation of C-oscillator into 180�. Such C-oscillators co-exist by pairs. The rule of formation of a simplecell and further cluster structures is shown in Fig. 31.

6. ‘‘p-flag’’ is a C-oscillator of the type Op2aððmnþ 1Þ=2Þ. Simple cell of the size (m · n), m = 2k + 1, n = 2p + 1 that can

be derived by connecting C-oscillator and its image. The latter results in rotation of C-oscillator into 180�. Such C-oscillators co-exist by pairs. The rule of formation of a simple cell and further cluster structures is shown in Fig. 32.

Theorem 2. All aforementioned 15 types of C-oscillators (Os(1 · n), Os(n · 1), Oa(1 · n), Oa(n · 1), Oss(m · n) ,Oas(m · n),

Osa(m · n), Oaa(m · n), flags-2: O2a(n(n + 1)/2) and O�2aðnðnþ 1Þ=2Þ, pyramid, mausoleum, flags-8, p-rectangle, p-flag).

Consequence. Classical synchronization of the one-type C-oscillators from all the 15 aforementioned types, defines thefull set of types of cluster structures that can be realized in two-dimensional homogeneous lattice of elementary oscil-lators with the Neiman boundary conditions for arbitrary sizes of a lattice. If the sizes of a lattice are known, then independence of the numbers N1 and N2 the certain part of these structures can be realized.

Remark 3. The coupling of simple cells into a one large lattice can be infinitely extended in all directions (see, for exam-ple, Figs. 14–20, 31, 32) covering the entire plane. Thus, the aforementioned types of C-oscillators and corresponding tothem simple cells are the structure forming objects also for the infinite lattices.

Remark 4. Suppose that having coupled simple cells, we have obtained a cluster structure in a rectangular lattice withthe identical series of synchronized elementary oscillators at the opposite ends (see Figs. 24–32). In this case, by cou-pling (through resistance) the corresponding oscillators (connection of equipotential points), we transform the latticeinto two-dimensional ring or Moebius loop. Thus, all aforementioned types of C-oscillators are the structure formingobjects also for two-dimensional cyclic lattices. They define so-called cut cluster structures. In addition to these types ofC-oscillators, for cyclic lattices other types of C-oscillators also exist.

Remark 5. The problem of existence of different types of integral manifolds in system (9) can be solved based on the setof types of C-oscillators and simple cells in combination of properties of numbers N1 and N2 (sizes of lattice).

Example. Fig. 33 shows the full set of cluster oscillators and structures in the lattice (3 · 6). Let us discuss briefly someconsequences of the proposed theory below.

Fig. 33. Cluster oscillators and structures in homogeneous lattice (3 · 6) of coupled oscillators with Neiman boundary conditions.

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4. On cluster oscillators and structures in three-dimensional lattice of coupled oscillators

The list of C-oscillators and cluster structures, generated by them in a spatial lattice is too large to be covered in thispaper. Therefore, we will limit ourselves to short remarks.

The simplest are the cluster oscillators of the type Os(n · 1 · 1), Oa(n · 1 · 1),Os(1 · n · 1), Oa(1 · n · 1). . . that areinherited from two-dimensional lattice. Simple synchronization of these oscillators defines layered cluster structures:band-like structure that is made of elementary oscillators of certain layer, which can be coupled to each other. As aresult one has layered spatial cluster structure. There are only 12 such C-oscillators.

All two-dimensional mosaic cluster oscillators (together with their equations) represent themselves C-oscillators ofthree-dimensional arrays. In particular, such systems are the C-oscillators concerned with axial symmetry of rectanglesof the type Oss(m · n · 1), Oss(m · 1 · n), Oss(1 · m · n), Osa(m · n · 1). . .. Altogether, there are 33 such oscillators.Simple synchronization of two-dimensional mosaic C-oscillators in a three-dimensional lattice in colour representationgives parallel multicoloured ‘‘threads’’ – mosaic layer couples dissipative with the identical one (colour to colour). Inresult one has aforementioned cluster structure.

Spatial mosaic cluster structures are defined by simple synchronization, in particular by synchronization of C-oscil-lators of types Osss(m · n · p),Ossa(m · n · p), Osas(m · n · p). . . Altogether, there are 27 such C-oscillators.

All other spatial mosaic C-oscillators are concerned with different ‘‘equipotential’’ transforms of parallelepiped andcube that concerned with different types of symmetry of these figures.

4.1. Cluster trees of coupled oscillators

Consider C-oscillator Os(n) in the operating regime, i.e., its motions occur at the attractor As(n) and suppose that wehave an infinite number of copies of this oscillator including the dynamical regime. In this case, using C-oscillators asbuilding blocks, we can compose various lattices of most miraculous forms, such as cluster ‘‘trees’’ having as‘‘branches’’ as rings – ‘‘branches that are grown together’’ (see Fig. 36). A symbolic representation of cluster tree isdepicted in Fig. 34.

1 2 3 4

6

5

Fig. 34. Symbolic representation of cluster tree.

1 2 3 4

6

5

Fig. 35. Block diagram of the simplest tree.

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In this paper, we will not pay attention to the stability of cluster structures in such systems and to physical interpre-tation or any other interpretation of cluster trees. We will just pay attention to the properties of differential equationsthat govern their dynamics based on an example of a simplest tree (see Fig. 35). These equations have the form

_X1 ¼ FðX1Þ þ eCðX2 � X1Þ;_X2 ¼ FðX2Þ þ eCðX1 � 2X2 þ X3Þ;_X3 ¼ FðX3Þ þ eCðX2 þ X4 þ X5 � 3X3Þ;_X4 ¼ FðX4Þ þ eCðX3 � X4Þ;_X5 ¼ FðX5Þ þ eCðX3 þ X6 � 2X5Þ;_X6 ¼ FðX6Þ þ eCðX5 � X6Þ:

ð10Þ

This system has the following properties. First, system (10), as it has been done above, can be rewritten as system ofthree interacting dissipative-coupled symmetric C-oscillators of the type Os (2). Simple synchronization of these oscil-lators causes cluster structure that is depicted in Fig. 34. Second, it is easy to establish that system (10) has integralmanifold M = {X2 = X3 = X5, X1 = X4 = X6}. Note, that it is not an easy task to find this manifold directly fromthe differential equations for the simplest cluster tree (Eq. 10) (not to mention the corresponding task in the case ofa lattice, which structure is shown in Fig. 36).

4.2. Cluster formation of dynamical processes in irregular homogeneous lattices

Suppose we have homogeneous lattice that is large enough in all directions and a cluster structure based on certaintype of cluster oscillator is realized in this lattice. By cutting the equipotential points we can obtain any number of C-oscillators from this chain. Such a cutting can be done as for boundary C-oscillators as for the internal ones. After thisprocedure, the border of the lattice becomes irregular and, if necessary, multivariable. At the same time, its physicalproperties as a ‘‘formative’’ basic cluster oscillator remain the same as for the initial ‘‘regular’’ lattice (see Fig. 37).If initial lattice ‘‘physically’’ is rather small-sized and cluster oscillator contains small number of elements than theboundary of ‘‘new’’ lattice can be formed to the shape, defined beforehand as close as one wants. The border couldbe adjusted as almost smooth and lattice itself could have a shape of a disc or a ring.

Fig. 36. An example of cluster tree built based on C-oscillator Os(2).

Fig. 37. Example of cluster structure in a homogeneous lattice of irregular form (parts of the figure coded by the black colourcorrespond to the cutout C-oscillators and do not enter the lattice – holes in the lattice).

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4.3. Cluster formation of dynamical processes in ordered inhomogeneous coupled systems

Always, during the definition of cluster in homogeneous systems as a group of synchronized oscillators, the obviousfact is not discussed. This fact is that synchronization is not ‘‘identical’’: Xi = Xj for t!1 with Xi and Xj the variablesof elementary oscillators from one cluster or, as it was mentioned above, the variables of the self-named elementaryoscillators entering different cluster oscillators of the same type (the core of cluster synchronization). It is possible, beingin the frames of this definition including identity of synchronization to violate the condition of homogeneity of a ‘‘med-ium’’. Namely, if a lattice is built from a specific cluster oscillator (!) then there is no necessity for elementary oscillatorsforming C-oscillator to be identical (in particular, they can be of different types). Moreover, in case of not identity ofelements a cluster oscillator, the main requirement consisting in the absence of synchronization between its elements willbe fulfilled even better if one would account for influence of parameters of coupling on the ‘‘duration of life’’ of clusterattractors of C-oscillators. By coupling C-oscillators with non-identical elements according to aforementioned princi-ples we will get a ‘‘large’’ ordered-inhomogeneous lattice, which dynamics will be clusterized as in a homogeneousmedium.

5. Conclusions

Firstly, we would like to make a remark concerning the investigation of system cluster dynamics. In systems (1) and(9), the intervals of existence of cluster attractors for typical elementary oscillators (Chua’s oscillator, Lorenz oscillator,etc.) are the finite intervals depending on the parameter of coupling. There are no cluster attractors outside these inter-vals and corresponding C-oscillators cease to be cluster oscillators. It is natural that the appropriate cluster structureswill not be observed since they even do not exist independently on the stability of corresponding integral manifold.Thus, studying the conditions of existence of cluster attractors is the main problem of cluster dynamics.

In this paper, we have focussed on the problems concerned with the existence of cluster structures and the main con-clusions are:

1. Structuring of the dynamical processes in lattices of oscillators is an ordinary process based of the synchronization ofelementary dynamical systems that are integrated in groups called as cluster oscillators (C-oscillators). Due to thisreason, cluster dynamics cannot be considered as a separate part of the nonlinear physics and oscillation’s theoryand represents the classical part of these subjects.

2. Having interpreted the cluster structure as a result of classical synchronization of structure forming objects, we haveobtained that the number of types of possible cluster structures in lattices of different geometries is relatively small

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and such structures can be built without appealing to the theory of integral manifolds by coupling simple cells of thedifferent types.

3. The correctness of building the cluster structures can be checked out by the following criteria: the possibility to becut and to be convoluted.

4. Using simple cells as building blocks, one can built a lattice of pre-described geometry and cluster structure, possess-ing as local as non-local couplings of elementary oscillators.

Finally, we would like to comment the stability of cluster structures. In this work we have understood that C-oscil-lator exists when the attractor corresponding to the regime of stationary beatings of elementary oscillators comprisingthis C-oscillator also exists. If there is no such attractor, then the corresponding system of elementary oscillators is not aC-oscillator and therefore, there is no corresponding cluster structure. Thus, studying the conditions of existence ofcluster attractors is the main problem in structures stability analysis. In the next paper, we will show that for fixedparameters of elementary oscillators, the domains of existence of cluster attractors represent intervals, limited by thecoupling strength. Having chosen the coupling strength outside of those intervals, one will never observe the corre-sponding cluster structures despite the existing invariant manifolds including stable ones. Cluster structures can be onlylocally stable. We will show in the next paper that it is impossible to define the conditions of the stability of clusterstructures based on the conditions of global stability of integral manifolds.

Acknowledgement

We would like to thank Dr. Valery V. Matrosov from the Radiophysics Faculty, Nizhny Novgorod State University,for discussion and allowing us to use his custom made software to carry out the numerical study.

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