International Journal of Bifurcation and Chaos, Vol. 12, No. 8 (2002) 1895–1907c© World Scientific Publishing Company

DEVELOPING CHAOS ON BASE OFTRAVELING WAVES IN A CHAIN OF COUPLED

OSCILLATORS WITH PERIOD-DOUBLING.SYNCHRONIZATION AND HIERARCHY OF

MULTISTABILITY FORMATION

A. SHABUNIN, V. ASTAKHOV and V. ANISHCHENKOPhysics Department, Saratov State University, Astrakhauskaya Str. 83,

Saratov, 410026, Russia

Received April 9, 2001; Revised September 14, 2001

The work is devoted to the analysis of dynamics of traveling waves in a chain of self-oscillatorswith period-doubling route to chaos. As a model we use a ring of Chua’s circuits symmetricallycoupled via a resistor. We consider how complicated are temporal regimes with parameterschanging influences on spatial structures in the chain. We demonstrate that spatial periodicityexists until transition to chaos through period-doubling and tori birth bifurcations of regularregimes. Temporal quasi-periodicity does not induce spatial quasi-periodicity in the ring. Aftertransition to chaos exact spatial periodicity is changed by the spatial periodicity in the average.The periodic spatial structures in the chain are connected with synchronization of oscillations.For quantity researching of the synchronization we propose a measure of chaotic synchronizationbased on the coherence function and investigate the dependence of the level of synchronizationon the strength of coupling and on the chaos developing in the system. We demonstrate thatthe spatial periodic structure is completely destroyed as a consequence of loss of coherence ofoscillations on base frequencies.

Keywords: Chaos; chaotic synchronization; coupled oscillators; traveling waves.

1. Introduction

In recent years a great interests has been placedon problems of collective dynamics of simpleoscillators which demonstrate different kinds ofbehavior from complete spatio-temporal synchro-nization to the formation of dissipative struc-tures. These investigations have both theo-retical and applied interests in different fieldsof physics [Kaneko, 1989a; Fisher, 1985] chem-istry [Kuramoto, 1984] and biology [Mirollo &Strogatz, 1990]. A series of works has been de-voted to consideration of coupled maps arrays mod-eling different physical phenomena [Kaneko, 1989a,1989b; Kuznetzov & Kuznetzov, 1990]. Lattices

of maps with chaotic behavior have rich dynam-ics and they allow to research the formation ofregular and chaotic spatio-temporal structures re-sulting from synchronization of oscillations. How-ever, the discrete time systems cannot demonstratephase and frequency phenomena which we observein real nature. Real oscillators demonstrate phasesynchronization phenomenon for regular [Huy-gens, 1673] and chaotic [Rosenblum et al., 1996;Fujigaki et al., 1996] motions. As a result theexistence of different phase structures is possible,when oscillations in subsystems have almost equalamplitudes and different phases and frequencies.These phenomena have been considered in a great

1895

1896 A. Shabunin et al.

number of works on phase oscillators. Most of thestudies was devoted to the global mean-field cou-pling systems [Matthews & Strogatz, 1990; Daido,1996; Tass, 1997; Yeung & Strogatz, 1999, 2000].It was demonstrated that very simple periodic os-cillators can demonstrate complex macroscopic be-havior: periodic, quasi-periodic and even chaoticthrough quasi-periodic and period-doubling routes[Matthews et al., 1990]. The mean-field approachallowing to consider the behavior of the system as awhole, does not take into account local connectionsbetween elements, which can lead to the formationof local spatial structures. Locally coupled limit-cycle oscillators was intensively investigated in theworks by Daido [1997], Bressloff et al. [1997] andBressloff and Coombes [1998]. Phase regularities innearest-neighbor coupled oscillators have also beenconsidered for circle maps [Pando, 2000].

It is known that chains of the simplest limit-cycle oscillators with periodic boundary conditionsexhibit traveling wave regimes when oscillations inneighboring sites differ for constant phase shifts.Daido considered a more complex oscillators chain(see [Daido, 1997]) and demonstrated that takinginto account the second harmonics in the spec-trum of oscillations can lead to novel behavior.Namely, he has found that temporal periodic os-cillators can demonstrate spatially chaotic behav-ior. Hence, transition to more realistic models leadto new dynamics which cannot be realized in thesimplest phase oscillators arrays. A number ofworks [Matias et al., 1997a; Matias et al., 1997b;Marino et al., 2000; Hu et al., 2000] have demon-strated that traveling wave regimes are possible forrings of chaotic oscillators (in the works [Matiaset al., 1997a; Marino et al., 2000] they were called as“rotating waves”). Nevertheless, evolution of trav-eling waves occurring in complex systems withchange in parameters has not been described in de-tail until now. What traveling wave regimes arepossible in real chaotic oscillator chains? How docomplicating temporal dynamics influence spatialstructures? How do destroying of spatial structuresconnected with synchronization between nearest-neighbor oscillators. In our work we focus onthese questions. We have chosen a chain of period-doubling self-oscillators with diffusion symmetriccoupling. Similar systems have rich dynamics.They are characterized by developed phase mul-tistability, when many regular and chaotic attrac-tors coexist in phase space [Astakhov et al., 1989;Anishchenko et al., 1995]. They also demonstrate

the possibility of the phenomena of full [Fujisaka& Yamada, 1983] and partial [Inoue et al., 1998;Belykh & Belykh, 2000] synchronization of chaoswhen all elements in the lattice or a part of suchelements oscillate identically. A chaotic attrac-tor related to the regime of synchronization is lo-cated inside the symmetric subspace x1 = x2 =. . . xn, where n is the number of all or a part ofoscillators in the array. Destroying the synchro-nization is accompanied by the bubbling and rid-dled basins phenomena [Ashvin et al., 1994; Laiet al., 1996]. Bifurcational mechanisms of thesynchronization loss and the multistability forma-tion are found for several period-doubling bifur-cations of every periodic orbit in the cascade ofbifurcations [Astakhov et al., 1999]. These bifur-cations lead both to the appearance of new stableattractors outside the symmetric subspace and tothe appearance of regions of local transversal in-stability for the synchronous attractor [Astakhovet al., 1997a; Astakhov et al., 1998].

In Sec. 2 we consider the evolution of travelingwaves with complicated dynamics during period-doubling bifurcations. We build regions of sta-bility for the waves with different spatial periodsand demonstrate that short wavelength waves dis-appear with increasing coupling. Section 3 is de-voted to a more detailed description of behaviorof one family of regimes originating from a trav-eling wave with determined periodicity. We con-sider typical bifurcational transitions and build re-gions of different stable regimes on the parameterplanes. In Sec. 4 we consider synchronization ofoscillations. We demonstrate that the completesynchronization breaks after transition to chaos.After these oscillators are nearly synchronized, toappreciate the quantity measure of their synchro-nization we use the averaged coherence function ap-proach [Anishchenko et al., 2000]. The proposedmeasure makes clear physical sense: it shows whatpart of the full power relates to coherent oscilla-tions. Using this characteristic we demonstrate thatthe traveling wave regimes are destroyed graduallywith decreasing coupling and this process is accom-panied by decreasing mutual coherence functionsand therefore by decreasing the levels of synchro-nization. The complete destruction of the aver-age space-periodicity is connected with the loss ofthe coherence on the main frequencies in the spec-trum. The conclusion contains the summary of theobtained results.

Synchronization and Hierarchy of Multistability Formation 1897

2. The Evolution of Traveling Waveswith Parameters Changing

As a model we used a chain of 30 Chua’sself-oscillators coupled via a resistor:

xi = α(yi − xi − f(xi))

yi = xi − yi + zi + γ(yi−1 + yi+1 − 2yi) (1)

zi = −βyi ,

(where

f(x) =

bx+ a− b if x > 1

ax if |x| ≤ 1

bx− a+ b if x < −1,

i = 1, 2, . . . , N ; N = 30) with periodic boundaryconditions:

x1 = xN , y1 = yN , z1 = zN .

All oscillators are identical. The behavior of a sin-gle oscillator is widely described in the literature(see e.g. [Komuro et al., 1991]). It is characterizedby period-doubling bifurcations cascade and bista-bility, when two symmetric attractors formed neartwo nontrivial equilibria P1 and P2 coexist in thephase space. With parameter α increasing these at-tractors merge and as a result double scroll chaoticattractor appears. The system of two coupled oscil-lators behave in a more complex way [Anishchenkoet al., 1995]. It demonstrates both period-doublingbifurcations cascade and tori breaking routes tochaos. The system is characterized by the developedmultistability and the complete synchronization ofchaos phenomenon. The in-phase complete syn-chronization of chaos exists at sufficiently large cou-pling. With decreasing coupling the regime of syn-chronization breaks (see [Anishchenko et al., 2000]).This process is accompanied by the bubbling of at-tractor and riddled basins phenomena. Arrays andlattices of Chua’s oscillators with different types ofcoupling have been investigated in a series of works(see e.g. [Perez-Munuzuri et al., 1993; Alexeevet al., 1995; Matias et al., 1997a; Marino et al.,2000]).

We investigated the system (1) by means ofcomputer simulations. The parameter α and thecoefficient of coupling γ were controlling parame-ters. Other parameters were fixed at values: a =−8/7, b = −5/7, β = −22.

2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

i

0

0,51

1,52

2,5

x i

(a) N=1

2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

i

0

0,51

1,52

2,5

x i

(b) N=2

2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

i

0

0,51

1,52

2,5

x i

(c) N=3

2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

i

0

0,51

1,52

2,5

x i

(d) N=5

Fig. 1. Spatial structures of simplest traveling waves withdifferent wavelengths.

Until α < 8.78 there are no oscillations in thesystem. At α ≥ 8.78 nearly harmonic period-onelimit-cycle oscillation regimes with different spatialperiods can be found depending on the value of thecoupling and on the initial values. They are char-acterized by equal amplitude and shifted phases ofoscillations in every site in the chain. These regimesare traveling waves rotating along the ring with con-stant phase velocity. For visualization of the spa-tial structure of the traveling waves we used thePoincare section. We fixed the value of the xi vari-able of every oscillator while the variable y1 crossesthrough the zero level from positive to negative val-ues. The chosen simplest waves with different wave-lengths are presented in the Fig. 1. The abscissaaxis in Fig. 1 marks the location of oscillator in

1898 A. Shabunin et al.

the chain, the ordinate axis denotes the value ofxi at the Poincare section. If all oscillators movein-phase we can see a straight horizontal line. Ifthere are some phase shifts between oscillations wesee different periodic curves. Because of the space-periodicity of the chain the summary phase shiftalong the entire chain must be proportional to 2πn.Figure 1 presents states with n = 1 [Fig. 1(a)],n = 2 [Fig. 1(b)], n = 3 [Fig. 1(c)] and n = 5[Fig. 1(d)]. They correspond to phase shifts be-tween oscillations of neighboring oscillators of π/15,2π/15, π/5, and π/3, respectively. It is clear thatfor chains with finite numbers of elements only fi-nite number of spatially periodic phase waves are

possible because the length of the chain must bedivisible by the spatial period. In our case otherpossible states remain with n = 6 and n = 15. Thelatter case is related to the complete antiphase syn-chronization. Traveling waves with these spatial pe-riods (Λ = 5 and Λ = 2 oscillators) were not foundin the considered chain, maybe because of very nar-row regions of stability.

Bifurcations of the states represented in Fig. 1which take place with the increasing of parameter αlead to complicating temporal dynamics in every os-cillator. These bifurcations are the period-doublingand tori birth bifurcations of regular attractors, andbound-merging bifurcations of chaotic attractors. A

5 10 15 20 25 30

i

0

1

2

xi

(a) α=1.14, γ=0.02

0 1 2x

1

0

1

2

x15

0 1 2x

1

-0,4

-0,20

0,2

0,4

y1

5 10 15 20 25 30

i

0

1

2

xi

(b) α=11.48, γ=0.02

5 10 15 20 25 30

i

0

1

2

xi

(c) α=11.54, γ=0.1

5 10 15 20 25 30

i

0

1

2

xi

(d) α=11.4, γ=0.2

0 1 2x

1

-0,4

-0,20

0,2

0,4

y1

0 1 2x

1

-0,4

-0,20

0,2

0,4

y1

0 1 2x

1

-0,4

-0,20

0,2

0,4

y1

0 1 2x

1

0

1

2

x15

0 1 2x

1

0

1

2

x15

0 1 2x

1

0

1

2

x15

Fig. 2. Developing of the regular traveling waves with the parameter α increasing.

Synchronization and Hierarchy of Multistability Formation 1899

more detailed structure of the plane of the regimeswill be described in Sec. 3. Figure 2 presents thecomplications of regular traveling waves with pa-rameter α increasing. It describes the evolutionof regimes based on the initial state with spatialperiod Λ = N/2 [Fig. 1(b)]: the traveling wavewith a period-two cycle 2CN/2 temporal behavior[Fig. 2(a)], with a period-four cycle 4CN/2 behav-ior [Fig. 2(b)], a two-band torus 2TN/2 [Fig. 2(c)]and a one-band torus 1TN/2 [Fig. 2(d)] behavior. In

this figure we have built the spatial structures andprojections of phase portraits to the planes x1 − y1

and x1 − x16. The latter projection demonstratesthat the mentioned regular regimes have exact spa-tial periodicity xi = xi+15. We see that increase ofthe temporal period does not lead to changing ofthe spatial period and temporal quasi-periodicitydoes not lead to spatial quasi-periodicity. Thedevelopment of temporal regimes take place ona background of the original spatial periodic

Fig. 3. Developing of the chaotic traveling waves with the parameter α increasing.

1900 A. Shabunin et al.

structure that is destroyed only after transition tochaos. The evolution of chaotic regimes is pre-sented in Fig. 3. This figure presents the cases of afour-band attractor [Fig. 3(a)], a two-band attrac-tor [Fig. 3(b)] and one-band attractor Figs. 3(c) and3(d). The chaotic regimes do not remain exactlyspatio-periodic. It can be seen from the “mutual”projection x1−x16 which is not a line more, but hasfinite “thickness”. For many-band chaotic attrac-tors it is very thin, i.e. the regimes are almost ex-actly space-periodic [Fig. 3(a)]. With the develop-ment of chaos the projection becomes more “thick”that means that the regime loses spatio-periodicity.In the regime of developed one-bound chaotic at-tractor the mutual projection looks like a square[Fig. 3(c)]. The mutual projection of the phase por-traits and the spatial diagrams demonstrate thatchaotic oscillations are not space-periodic more.However, as it will be demonstrated later, they re-main space-periodic on an average.

Stability of traveling waves depends on theirwavelength and on the coupling strength. Increas-ing the coupling coefficient γ leads to loss of sta-bility for short wavelength modes and as a re-sult of transition to more long wavelength ones.Figure 4 presents regions of stability for the fam-ilies of regimes originating from the traveling wavesdescribed in Fig. 1. The upper boundaries (sym-bols) mark destroying of the spatial structure due to

0 0,2 0,4

γ

8,5

9

9,5

10

10,5

11

11,5

12

12,5

13

α

1

2

3

4

Fig. 4. Regions of stability for the families of waves withdifferent wavelengths.

developing chaos. The right boundaries (lines)mark the loss of stability in a sudden way. Asa result of it the system transits to regimes withlonger wavelengths. The line “1” bounds a family ofregimes with spatial period N [see Fig. 1(a)]. Thisline is almost horizontal. From the right side of thisline only spatially-homogeneous regime is possible.The lines “2”, “3”, “4” are boundaries of familiesof regimes with spatial periods N/2, N/3, N/5, re-spectively. From the left-hand sides of these lineswaves with these spatial structures are observed.On crossing these lines only stable waves withlonger wavelengths can exist. Namely, from the leftside of line “4” the families of regimes with wave-lengthsN/5, N/3, N/2, N and spatio-homogeneousregime coexist. From the left side of the line“3” — with wavelengths N/3, N/2, N and spatio-homogeneous, from the left side of the line “2” —with wavelengths N/2, N and spatio-homogeneousregimes coexist. The spatio-homogeneous regimesexist at any coupling. We see that stability of thetraveling waves depend on coupling. The increasingof coupling leads to loss of stability for regimes withshorter spatial periods.

Unfortunately, the large dimension of the con-sidered system does not lead to the possibility tocarry out a detailed bifurcational analysis. Tak-ing into account the location of bifurcational lines1 − 4 and bifurcations in the coupled oscillatorswith smaller dimensions [Anishchenko et al., 1995;Astakhov et al., 1997b] we can propose a hypoth-esis that the mentioned spatially periodic regimesoriginated as a result of bifurcations of the sameequilibrium P1,2. In our opinion the bifurcationalmechanism can be the following. If we have asingle oscillator, the periodic period-one cycle ap-pears through the Hopf bifurcations of the equilib-ria P1 or P2. After this, a stable period-one cy-cle appears. In the system of two coupled oscilla-tors [Astakhov et al., 1997b] the equilibrium under-goes two consistent Hopf bifurcations. As a resultof the first bifurcation the stable in-phase (or an-tiphase, depending on the type of coupling) period-one cycle appears from the stable equilibrium pointand the point becomes saddle. After the secondHopf bifurcation the saddle equilibrium becomesunstable along another direction and the saddleantiphase (or in-phase) period-one cycle appears.Then, with increasing of the parameter α the saddlecycle can become stable through the pitchfork (forexample) bifurcation. If we have many-oscillatorssystem the equilibrium point can undergo several

Synchronization and Hierarchy of Multistability Formation 1901

Hopf bifurcations. The first Hopf bifurcation leadsto appearance of period-one spatio-homogeneousstable regime. After this the equilibrium becomesa saddle. The cycles that satisfy the spatio-periodicity conditions appears as a result of the fol-lowing Hopf bifurcations. They are saddle. Theregimes with longer spatial periods appear earlierthan the short wavelength ones. Then, with theparameter α increasing these saddle cycles becomestable. In the framework of the considered hypoth-esis the lines 1–4 are bifurcational lines with a bi-furcational condition for a multiplier of the corre-spondent cycle: µ = +1.

3. Typical Bifurcational Transitionsand Structure of the ParametersPlane for a Family ofSpatio-Periodic Regimes

In this section we carry out a more detailed anal-ysis of traveling waves formed on the base of theperiod-one cycle regime with Λ = N/2 [Fig. 1(b)].The common structure of the regimes on the param-eters plane γ −α is presented in Fig. 5. The regionof stability of this family of regimes is bounded bylines “1”, “5” and “6”. The line “1” bounds this re-gion from the right. On crossing the line the regimeswith wavelength Λ = N/2 loses their stability in asudden way and the system transits to waves withlarger wavelengths. Over this line and before theline “2” the stable period-one cycle is observed. Onthe line “2” the system transits to quasi-periodic be-havior. This is the line of the torus 2TN/2 [Fig. 2(d)]birth. For large coupling the system evolves at thebase of this torus and demonstrates transition tochaos through the torus breaking (dashed line “4”in Fig. 5). Over this line a one-band chaotic at-tractor exists. For small coupling the transition tochaos occurs through the period-doubling bifurca-tions and quasi-periodic behavior originates fromcycles with double periods. This region of smallercoupling is bounded by the line “3”. Over this linethe period-two cycle is observed [Fig. 2(a)] whichtransits to the two-torus 2TN/2 on the line “5”[Fig. 2(c)]. Then on the base of this torus thereare two possibilities: at larger coupling the torusbreaks with the appearance of two-band chaotic at-tractor, and for smaller coupling (on the line “6”)it transits to a period-four cycle [Fig. 2(b)] whichthen undergoes tori birth bifurcation. The pictureof regimes and bifurcational lines in Fig. 5 is ratherrobust. We restrict ourselves to only several first

0 0,05 0,1 0,15 0,2 0,25 0,3 0,35 0,4 0,45 0,5

γ

9

9,5

10

10,5

11

11,5

12

α

1C

1T2C

4C 2T 2 A 1AN

�

1

23

45

6

abc

Fig. 5. Diagram of regular and chaotic regimes originatingfrom the wave with spatial period N/2.

period-doubling and tori birth bifurcations, up toperiod eight. The analysis holds, and comparingwith the analysis of typical bifurcations in systemsof two coupled via resistor [Anishchenko et al., 1995]and via capacity [Astakhov et al., 1997b] oscillatorspresent typical scenarium for evolutions of regimeson the base of the space-periodic period-one cycle.The route to chaos goes through several period-doubling and tori birth bifurcations. The numberof bifurcations decreases with increasing coupling.At large coupling we have a chain of two bifurca-tions: 1C → 1T → 1A (the arrow a in Fig. 5), atsmaller coupling there is a chain of five bifurcations:1C → 1T → 2C → 2T → 2A → 1A (the arrow bin Fig. 5), at even smaller coupling 1C → 1T →2C → 2T → 4C → 4T → 4A → 2A → 1A (thearrow c in Fig. 5). The described dependence ofthe bifurcation chains on coupling suggest the com-mon bifurcational mechanisms for the changing oftemporal behavior of the family of spatio-periodicregimes: At finite coupling we have a finite chain ofperiod-doubling and tori birth bifurcations whichlead to chaos through the final tori breaking bifur-cation. The length of the chain increases with de-creasing coupling and at the limit of zero couplingtends to infinity.

The structures of the phase-space and typicalbifurcations are similar for spatio-periodic regimeswith other wavelengths.

1902 A. Shabunin et al.

4. Synchronization of Chaos in theChain of Oscillators

The considered complex spatial structures of os-cillations in the chain are results of synchroniza-tion. Usually the term “complete synchronization”is used for the behavior when all oscillations in thechain are equal in every moment of time (xi(t) =xk(t) for all i and k) namely, when the attractor be-longs to the symmetric subspace x1 = x2 = . . . xN[Heagy et al., 1994]. The dimension of the sub-space is equal to the dimension of the single oscil-lator. The case when oscillations are equal up totime-delay: xi(t) = xi+1(t + τ) was called as “lagsynchronization” [Rosenblum et al., 1997]. Fromanother point of view the case when all oscillatorscan be divided into several groups in which everyone of the oscillations are equal was called partialsynchronization [Belykh & Belykh, 2000]. There-fore, the case of considered spatio-periodic struc-tures can be called as both lag-synchronization andas partial synchronization. There is a more gen-eral approach to the chaotic synchronization def-inition which can be applicable to different casesof mutual behavior. This is the “generalized syn-chronization” [Rulkov et al., 1995]. In the frame-work of this approach we shall call oscillations asfully synchronized if there is deterministic connec-tion between states of any oscillators in the chain:xi(t) = f(xk(t−τ)). This is the case of full synchro-nization. When this equality is fulfilled not exactly,or it takes place from time to time (for example inthe case of bubbling phenomenon) we have partiallysynchronized oscillations. In this case it is necessaryto have quantity measure of this synchronization,which will demonstrate changing of synchronizationlevel from 0 for unsynchronized oscillators till 1 forfully synchronized ones. This measure must makeclear the physical sense, and must be easily calcu-lated by standard algorithms using time-series fromcoupled oscillators and it must be universal, namelycan be applicable to different cases of synchroniza-tion. We have proposed such synchronization mea-sure in the work [Anishchenko et al., 2000] whereit has been used for investigation of complete syn-chronization loss and bubbling phenomenon in thesystem of two coupled chaotic self-oscillators. Thismeasure is constructed on the base of the coherencefunction:

σ(ω) =

∣∣∣∣∣⟨

χx1(ω)χ∗x2(ω)

|χx1(ω)‖χx2(ω)|

⟩∣∣∣∣∣ (2)

where χxi is spectral density on the time-series ofthe ith oscillator, ω is a frequency, < · · · > denotesaveraging on an ensemble of realizations. The co-herence function is equal to 1 for the frequencieswhere oscillations are coherent and equal to 0 wherethey are independent. To evaluate the level of syn-chronization on all frequencies it is necessary to av-erage the coherence functions with weight function.The weight function must take into account whatpart of the entire power belongs to the correspond-ing spectral terms. The complete formula for cal-culating the synchronization level has the form:

S =

∫ ∞0

σ(ω)(〈|χx1(ω)|2〉+ 〈|χx2(ω)|2〉)dω∫ ∞0

(〈|χx1(ω)|2〉+ 〈|χx2(ω)|2〉)dω(3)

Using the formula (3) we can interpret the value Sas a part of the power which relates to coherent mo-tions i.e. the motions on the frequencies where thephases are locked. The proposed measure is sim-ilar to the “order parameter” which was used forthe description of measure of collective synchroniza-tions in ensembles of phase oscillators [Kuramoto &Nishikava, 1987; Matthews & Strogatz, 1990], but itis applicable for more complex regular and chaoticoscillators. Phase coherence was used as a chaoticsynchronization measure in the work [Mormannet al., 2000] where authors used instantaneous phaseapproach. In our opinion the coherence functionapproach is more universal because it is applicablealso for the cases when instantaneous phase cannotbe defined.

Let us consider the family of traveling waveswith wavelength N/2 [Fig. 1(b)]. The boundaryof the family of regimes with this space-periodicityconsists of three segments. The first one is the line“1” [Fig. 5]. On this line the regimes lose theirstability in a sudden way and the system transitsto regimes with longer wavelengths. From the leftside of this line and until lines “6” and “7” there arethe exactly space-periodic and nearly space-periodicregimes with this wavelength. The exact space-periodicity is destroyed at once after transition tochaos. In the chaotic region there is no exact space-periodicity, but it is observed “in the average”.Figure 6 demonstrates averaged spatio-temporal di-agrams with decreasing coupling. In this figure weobserve gradual destruction of the averaged space-periodic structure. The corresponding temporal be-havior is in a one-band chaotic attractor. For large

Synchronization and Hierarchy of Multistability Formation 1903

5 10 15 20 25 30i

00,5

11,5

22,5

3

<x i >

(a) α=11.78 γ=0.15

5 10 15 20 25 30i

00,5

11,5

22,5

3

<xi>

(b) α=11.78 γ=0.045

5 10 15 20 25 30i

00,5

11,5

22,5

3

<xi>

(c) α=11.78 γ=0.035

5 10 15 20 25 30i

00,5

11,5

22,5

3

<xi>

(d) α=11.78 γ=0.03

5 10 15 20 25 30i

00,5

11,5

22,5

3

<xi>

(e) α=11.78 γ=0.01

Fig. 6. The averaged spatial structure of the wave destroyedwith developing temporal chaos.

coupling the averaged spatial diagram looks like theoriginal traveling regime [compare Figs. 1(a) and5(a)]. Then, at γ < 0.045 it begins to change itsform. At γ = 0.035 the form of the spatial picturebecomes more flat, though it preserves the origi-nal structures with two maximums [Fig. 5(c)]. Forsmaller coupling it completely “forgets” the previ-ous structure. Due to the gradual nature in pro-cessing the destruction of space-periodicity, the line“6” is not a bifurcational line, but it characterizes adetermined step in this process. Namely, the pointsof this line (they are marked by “◦”) denotes theminimum coupling behind which the spatial pic-ture becomes qualitatively different. The gradualdestroying of averaged space-periodicity is observedonly for developed temporal chaos. If we choose theparameter α correspondent to two-band (or many-band) chaotic attractor the averaged space-periodicstructure is preserved until zero coupling. From the

top the region of the averaged space-periodicity isbounded by line “7”. Increasing of the parame-ter α leads to developing of temporal chaos and todestroying of space-periodic structure. For middlecoupling values it is destroyed gradually like in thecase of decreasing coupling [see Figs. 6(a)–6(c)]. Forstrong coupling the averaging periodic structure ispreserved until the transition to the double-scrollregime [Fig. 6(d)].

Let us look on the process of destroying spa-tial periodic structure from the point of view ofmutual synchronization of oscillators. In the reg-ular region, until transition to chaos we have thecase of full synchronization. These regimes arecharacterized by exact equalities xi(t) = xi+N/2(t)and xi(t) = xi+1(t + τi) for every oscillator (seeFig. 2). The synchronization level S is exactly equalto 1. With the increasing of parameter α when

(a)

(b)

Fig. 7. Dependence of the level of chaotic synchroniza-tion on (a) developing temporal chaos and (b) decreasingcoupling.

1904 A. Shabunin et al.

oscillations become chaotic the equality xi(t) =xi+N/2(t) is not fulfilled exactly. This leads to“thickness” of the mutual projection of the phaseportrait (Fig. 3). The synchronization level be-comes smaller than 1. After the transition to chaosit is near 0.99 (at α = 11.57), then with furtherchaotization it becomes smaller: 0.77 at α = 11.79[Figs. 4(a)–4(e), 0.56 at α = 12.0 and 0.13 atα = 12.25 (for double-scroll regime). The depen-dence of the synchronization level S on the param-eter α is built in Fig. 7(a). Figure 7(b) demon-strates decreasing of the synchronization level withdecreasing coupling. The gray color marks the re-gions where the averaged periodic structure is de-stroyed. It relates to the level of synchronizationless than approximately 0.8.

The synchronization level depends on the dis-tance between interacting oscillators in the chain.We have built this dependence for the cases of two-band chaotic attractor, one-band attractor and fora developed temporal chaos with destroyed spatial

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

N

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Fig. 8. Dependence of the level of synchronization on thedistance between oscillators in the chain.

structure (Fig. 8). For two-band attractor (markedby “◦”) we see almost full synchronization (S ≈ 1)which is gradually decreased with increasing of dis-tance. The developed temporal chaos with almost

0 0,5 1 1,5 2

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y1

(a) α=11.57 γ=0.05

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(b) α=11.61 γ=0.05

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(c) α=11.79 γ=0.05

Fig. 9. Changing in the power spectrum of oscillations and of the module of the coherence function with developing oftemporal chaos.

Synchronization and Hierarchy of Multistability Formation 1905

0 0,5 1 1,5f

�

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α=11.78 γ=0.15

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�

(c)

α=11.78 γ=0.02

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�

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α=11.78 γ=0.002

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�

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�0 0,5 1 1,5

f�

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�

Fig. 10. Losing of the coherence between dynamics of the neighbor oscillators with the decrease in coupling.

periodic spatial structure is characterized by de-creased level of synchronization (S ≈ 0.8) whichdecreases rapidly with distance. When the spa-tial structure is fully destroyed the level of synchro-nization very quickly falls to almost zero level (inour case it is a value of several percents, the pre-cision of the experiment). At γ = 0.02 (“*”) thedistance at which the level of synchronization fallsfrom S ≈ 0.4 till zero is five oscillators. At smallercoupling (γ = 0.002) the level of synchronizationfalls to zero at distance of one oscillator.

Figure 9 demonstrates rebuilding of the powerspectrum and the coherence function with chaoti-zation of oscillations. The coherence functions isbuilt for oscillations of nearest-neighbor oscillators.Figure 9(a) corresponds to a four-band attractor.The coherence function is equal to 1 for peaks onharmonics 2π/T and first sub-harmonics 2π/2T ofspectra and has smaller values for remaining fre-quencies. Figures 9(b) and 9(c) correspond to two-and one-band attractors. It is seen that the coher-ence is equal to 1 for main peaks and it remainsvery close to 1 for first sub-harmonics [Fig. 9(b)].The coherence for all frequencies except main har-monics is smaller than 1 [Fig. 9(c)]. With furtherchaotization the level of coherence for other spec-tral components become lower but the coherence

on the main frequency remain near 1 until tempo-ral chaos is developed when the coherence on themain frequency is also decreased. Figure 10 demon-strates the dependence of the coherence functionon the coupling value. The correspondent temporalregime was chosen as developed one-band chaoticattractor. Figures 10(a) and 10(b) correspond tospace-periodic regimes. The coherence function isless than one for all frequencies except main peaks.Figures 10(c) and 10(d) relate to regimes with de-stroyed spatial structure. This case is characterizedby value of coherence less than one on the mainfrequencies too. Comparing these figures with thespatio-temporal dynamics we see that spatial peri-odicity is connected with coherence on main peaks.Until the spectra of oscillations contain frequencieson which motions are fully coherent the chain pre-serves its averaged periodic spatial structure. If co-herent function decreases for every frequency thespatial periodic structure is destroyed.

5. Conclusion

In the work we considered developing of the dynam-ics on the base of spatio-periodic phase structuresin the chain of coupled period-doubling oscillators

1906 A. Shabunin et al.

with periodic boundary conditions. These struc-tures are traveling waves propagating along thechain with constant phase velocity. We have foundthat exact space-periodicity is preserved throughperiod-doubling and tori birth bifurcations untiltransition to chaotic temporal behavior. Chaotictemporal behavior induces spatial chaos which re-mains spatio-periodic “on an average”. The aver-aged spatio-periodic structure exists in the wide in-terval of the parameters and it is destroyed bothwith developing of chaos (at small coupling) andwith decreasing of coupling (in the regime of one-band chaotic attractor). The presence of the aver-aged spatio-periodic structure is connected with thecoherence of oscillations on main peaks in the mu-tual spectrum. It exists until the coherence functionremains equal to 1 on base frequencies. We havebuilt regions of stability for waves with differentwavelengths and found that increasing of couplingleads to instability of short waves. In the work wedescribe typical bifurcations and present the mapof main regimes originating from a simple travelingwave with changing parameters.

Evidently, considered particular spatio-periodicregimes and correspondent phase diagrams takeplace only for chains with the chosen length. How-ever common regularities are found to exist also forarrays with other numbers of elements. We testedour results on chains with other lengths and ob-tained results are qualitatively similar to those de-scribed in this work (in our investigations we chosearrays up to 1024 elements length).

Acknowledgments

This material is based on work supported bythe Naval Research Laboratory under ContractNo. N68171-00-M-5430. The authors also thankthe Civilian Research & Development Foundation(Grant REC-006) and Russian Foundation of BasicResearch (Grant 00-02-17512) for partial financialsupport.

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