Controlling spatiotemporal chaos in chains of dissipative Kapitza pendula

R. Chacón1 and L. Marcheggiani21Departamento de Física Aplicada, Escuela de Ingenierías Industriales, Universidad de Extremadura, Apartado Postal 382,

E-06071 Badajoz, Spain2Dipartimento di Architettura, Costruzioni e Structure, Università Politecnica delle Marche, via Brecce Bianche, 60131 Ancona, Italy

�Received 21 May 2009; revised manuscript received 11 March 2010; published 1 July 2010�

The control of chaos �suppression and enhancement� of a damped pendulum subjected to two perpendicularperiodic excitations of its pivot �one chaos inducing and the other chaos controlling� is investigated. Analytical�Melnikov analysis� and numerical �Lyapunov exponents� results show that the initial phase difference betweenthe two excitations plays a fundamental role in the control scenario. We demonstrate the effectiveness of themethod in suppressing spatiotemporal chaos of chains of identical chaotic coupled pendula where homoge-neous regularization is obtained under localized control on a minimal number of pendula. Additionally, wedemonstrate the robustness of the control scenario against changes in the coupling function. In particular,synchronization-induced homogeneous regularization of chaotic chains can be highly enhanced by consideringtime-varying couplings instead of stationary couplings.

DOI: 10.1103/PhysRevE.82.016201 PACS number�s�: 05.45.Gg, 05.45.Xt, 74.81.Fa

I. INTRODUCTION

During the last two decades, the problem of controllingthe chaotic dynamics of generic systems has been the subjectof intense research in very many situations appearing in di-verse scientific and technological areas, including biology�1�, laser systems �2�, fluid mixing �3�, secure informationprocessing �4�, and plasma systems �5�, to cite just a few.This has been reflected in an abundant literature �6–9�, in-cluding experimental control �10–15�. Among the differentcontrol methods which have been proposed, the introductionof so-called open loop small perturbations �16–20� has beenshown to be quite reliable and readily applicable experimen-tally �15,20�. It is especially advantageous in situations inwhich there is no possibility of measuring the system’s vari-ables, such as ultrafast processes at the molecular or atomiclevel.

In particular, since the early 1990s there has been thedevelopment of a theoretical approach to chaos control indissipative nonautonomous systems that is analytically de-monstrable by Melnikov analysis �MA� techniques, andwhich involves adding periodic chaos-controlling �CC� exci-tations �21�. This MA-based approach has been shown to beeffective in taming chaos in coupled arrays of damped, peri-odically forced, nonlinear oscillators �22�. Up to now, twoclasses of cases have been considered with respect to the twoperiodic excitations involved �one chaos-inducing, CI, andthe other CC�: �i� the two perturbations are external excita-tions �forcings�, and �ii� one excitation is external while theother is a parametric excitation. The physically relevant casein which both excitations are parametric has, however, to thebest of our knowledge yet to be studied in detail, in spite ofits great theoretical and technological interest. Indeed, thegeneral equation of a dissipative driven pendulum whosepivot is subjected to an arbitrary plane motion is

ml2� + mgl sin � = − �� − mly0 sin � − mlx0 cos � , �1�

where � is the angle of the pendulum, ��d� /dt, l is thelength of the �massless� pendulum arm, m is the mass of the

pendulum bob, � is the viscous damping coefficient, g is theacceleration due to gravity, and �x0 ,y0� are the Cartesian co-ordinates of the pivot. Here, we assume a periodic ellipticmotion of the pivot which is described by the equations

x0�t� = − Ax cos��t + �� ,

y0�t� = − Ay cos��t� , �2�

where Ax and Ay are the amplitudes of the horizontal andvertical displacements, respectively. The relative amplitude��Ax /Ay and the phase difference � change both the eccen-tricity and the inclination of the ellipses. For example, theperiodic motion of the pivot follows a straight line of slopeAy /Ax for �=0 while its trajectories are upright ellipses for�= �� /2 �see Fig. 1�. A numerical bifurcation analysis forthe particular case where the driving of the pivot of the pen-dulum follows a narrow upright ellipse was considered inRef. �23�. The fact that only the elliptic component of an

Ε � 0.1 , Φ � Π � 10 Ε � 0.3 , Φ � Π � 10 Ε � 0.9 , Φ � Π � 10

Ε � 0.7 , Φ � 0 Ε � 0.7 , Φ � Π � 10 Ε � 0.7 , Φ � Π � 2

FIG. 1. Trajectories of the pendulum pivot �Eqs. �2� and �3�� inthe Cartesian plane �x ,y� for Ay =1 and diverse values of the rela-tive amplitude � and phase difference �.

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arbitrarily shaped periodic excitation has an effect on a ro-tating pendulum for large excitation frequencies �24� pro-vides a physical motivation for the consideration of the el-liptic excitation �Note that such an excitation is typical if thependulum base is floating on water waves �23��. Horton et al.�23� demonstrated that the region in the parameter plane ofamplitude and frequency of excitation where rotations arepossible increases with the ellipticity. This suggests the pos-sibility of controlling the chaotic dynamics of a classicalKapitza pendulum ��=0� by considering an elliptic excita-tion ���0� instead of a purely vertical excitation ��=0�.Previous work on chaos control of a dissipative pendulumsubjected to two external excitations �25� �i� demonstratedthat a CC external excitation can suppress the chaotic behav-ior induced by a CI external excitation over certain regionsof parameter space. This reliable control scenario has beenalso demonstrated for the case of a dissipative pendulumsubjected to a CI external excitation by means of the appli-cation of an additional CC parametric excitation �26� �ii�. Itshould be stressed, however, that the effectiveness of CCparametric excitations in controlling chaos of general nonau-tonomous systems is far from being obvious: diverse coun-terexamples are known �27,28� showing that this is not nec-essarily the case. In particular, the application of MAtechniques becomes illegitimate in those cases where the ex-citations cannot longer be considered as perturbative terms.However, we show below that, even in such an unfavorablecircumstance, MA-based predictions provide useful qualita-tive information regarding regularization regions in param-eter space for the case of a dissipative pendulum subjected toan elliptic excitation.

The rest of the paper is organized as follows. Section IIprovides the MA-based analytical predictions for the dissipa-tive generalized Kapitza pendulum �Eqs. �1� and �2��, andthen compares numerical results �Lyapunov exponents� withthe analytical predictions, as well as characterizing the regu-larization routes in the control parameter plane. Section IIIstudies the effectiveness of the control method under local-ized application of the CC excitations in chains of coupledchaotic Kapitza pendula. We consider the application of theCC excitations on a minimal number of pendula, and studythe robustness of the control scenario under changes in thesize of the chains and the coupling function. Finally, Sec. IVpresents some concluding remarks.

II. CONTROL SCENARIO

After the scaling

0 � �g/l, � 0t, � �/0,� � Ax/Ay ,

� � �/�ml20�,� � Ay�2/g , �3�

Equations �1� and �2� can be recast into the form

� + sin � = − �� − � cos��sin � − �� cos� + ��cos � ,

�4�

where all variables and parameters are dimensionless

���d� /d�. Here, the term describing the vertical

�horizontal� oscillations of the pivot, −� cos��sin ��−�� cos�+��cos ��, is to be regarded as the CI �CC�excitation. When the CC excitation is absent ��=0�, we as-sume that the pendulum has a chaotic attractor for a given setof the remaining parameters. The chaotic dynamics and re-lated phenomena of vertically forced pendula has been a sub-ject of study for nearly 30 years �29–34�. It is also assumedthat the complete system �Eq. �4�� satisfies the MA require-ments, i.e., the dissipation and excitation terms are small-amplitude perturbations of the underlying conservative pen-

dulum �+sin �=0 �see �35,36� for general background�.Straightforward application of MA to Eq. �4� yields theMelnikov function �MF�

M��0� = − D + A sin�0� B cos�0 + �� ,

D = 8� ,

A = 2��2 csch��/2� ,

B = 2���2 sech��/2� , �5�

where the positive �negative� sign refers to the top �bottom�homoclinic orbit of the conservative pendulum,

�0�� = � �4 arctan e − ��, �0�� = � 2 sech . �6�

Let us assume that, in the absence of any CC excitation��=0�, the associated MF

M0��0� = − D + A sin�0� �7�

changes sign at some t0, i.e., D�A. Clearly, the MF associ-ated with the top homoclinic orbit, M+�t0� �Eq. �4��, can berecast in the form

M+�0� = − D + �A + B sin ��sin �0� − B cos � cos �0�

� − D + ��A + B sin ��2 + B2 cos2 ��1/2. �8�

If one now lets the CC excitation act on the pendulum suchthat

B2 + 2AB sin � + A2 − D2 � 0, �9�

this relationship represents a sufficient condition for M+�0�to be negative �or null� for all 0. The equals sign in Eq. �9�yields the boundary of the region in the �−� parameterplane in which homoclinic chaos is suppressed,

� = �+ � �− sin � � �sin2 � − �1 − D2/A2��R , �10�

with

R � coth��/2� , �11�

where the sign +�−� before the square root corresponds to theupper �lower� branch of the boundary. Similarly, for the MFassociated with the bottom homoclinic orbit, M−�0��Eq. �5��, one obtains

� = �− � �sin � � �sin2 � − �1 − D2/A2��R . �12�

The lower branch of the boundary functions �Eqs. �10� and�12�� has a minimum at the optimal suppressory phase dif-

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ferences �=�optsup,+�3� /2 and �=�opt

sup,−�� /2, respectively,which is �min= �1−D /A�R. Also, the area enclosed by thesetwo boundary functions, Eqs. �10� and �11�, is given by

Asup = 4DR/A , �13�

which provides a simple quantifier of the suppressory effec-tiveness of the CC excitation. One notes that, since Asup de-pends upon the ratio D /A which in turn provides a measureof the strength of the initial chaotic state, Eq. �13� indicatesthat the suppressory effectiveness of the CC excitation ex-hibits sensitivity to the particular initial chaotic state. Figure2 shows an example of the regions in the �−� parameterplane in which homoclinic bifurcations are frustrated �Eqs.�10� and �12��.

Consider now the enhancing effect of the CC excitation.Clearly, enhancement of the initial chaos can be achieved bymoving the pendulum from the homoclinic tangency condi-tion �D=A� even more than in the initial situation with noCC excitation. This means that constraining M0

��0� to be inphase with −B cos�0+�� is a sufficient condition forM+�0� to change sign at some 0. This condition providesthe optimal enhancing phase difference, �=�opt

enh,+�� /2, inthe sense that M+�0� presents its highest maximum at �opt

enh,+,i.e., one obtains the maximum gap from the homoclinic tan-gency condition. Similarly, one has �=�opt

enh,−�3� /2 as theoptimal enhancing phase difference for M−�0�. Thus, onewould expect that these enhancing values �opt

enh,� wouldin fact be the observed enhancing values if and only if thependulum separatrix were entirely formed by just one of thetwo homoclinic orbits in each case. Actually, since�opt

sup,+=�optenh,−, �opt

sup,−=�optenh,+, and the optimal suppressory

phase differences are obtained by requiring M0��0� to be in

phase opposition with B cos�0+��, one concludes thatan intermediate superposition of the involved phases willyield the true enhancing values: �opt

enh= �0,��.We now compare the theoretical predictions obtained

from MA with the Lyapunov exponent �LE� calculations forthe dissipative generalized Kapitza pendulum �Eq. �4��.

Computer simulations of Eq. �4� showed that MA-based pre-dictions provide useful qualitative information regardingcontrol scenario, even when the excitation amplitudes do notreasonably satisfy the MA requirements �35,36�. In this re-gard, it is worth recalling that even in the case of smallamplitudes one cannot expect too good a quantitative agree-ment between these two kinds of approaches, because MA isa perturbative method generally related to transient chaos,while LE provides information solely concerning steady re-sponses. We computed LEs using a version of the algorithmintroduced in �37�, with integration typically up to 104 drivecycles for each fixed set of parameters.

Figure 3 bottom shows an illustrative examplecorresponding to an unavoidable situation �see Fig. 3 top�which is clearly outside the perturbative requirements��=0.1, �=0.87, =1.6�. In the absence of the CC excita-tion ��=0�, the dissipative Kapitza pendulum exhibits a cha-otic strange attractor characterized by a maximal LE�+��=0�=0.061. The maximal LE was calculated for eachpoint on a 100�100 grid in the parameter planes −��Fig. 3 top� and �−� �Fig. 3 bottom�, respectively. FromFig. 3 bottom, one sees that complete regularization��+���0��0� mainly appears in certain regions inside is-lands which symmetrically contain the respective theoreti-cally predicted areas �Fig. 2�. The fact that regularization isnot achieved inside the theoretically predicted areas is a di-rect consequence of the necessarily large excitation ampli-tude used in the simulations �see Fig. 3 top�. Remarkably,two key features of the MA-based predictions are found toremain in the LE calculations. First, the diagram of Fig. 3bottom is periodic along the �-axis, with fundamental periodequal to �, just as the regularization islands of Fig. 2.Second, the widest ranges of the suppressory phasedifference are centered at the theoretically predicted optimalvalues �opt

sup,�= �� /2,3� /2�. Regarding enhancement ofchaos, we typically found that the LE calculations fit thetheoretical predictions for sufficiently small relative ampli-tudes, as in the case �=0.08 �see Fig. 4 top� where maximumenhancement occurs at �=�opt

enh= �0,��. It is noteworthy thatthe dual enhancer/suppressor role of the phase differenceholds for relative amplitudes which are clearly beyond theperturbative regime, as in the case �=1.25 �see Fig. 4 top�,although the vast majority of suppressor values of � aresymmetrically shifted to lower and higher values from theircorresponding theoretically predicted values for such largevalues of �. Figure 4 bottom shows an example of drasticenhancement of chaos at �=�opt

enh=0 as � increases from 0��+��=0�=0.061� to 0.15 ��+��=0.15�=0.2�. Also, the far-ther the phase difference is from �=�opt

enh=0, the higher theminimum relative amplitude needed to find chaos, as ex-pected from MA-based predictions.

Figure 5 shows illustrative examples of the regularizationroutes as � and � are changed. Typically, the Kapitza pen-dulum goes from a strange chaotic attractor at �=0��� to aperiod-1 attractor at �=� /2�3� /2� as the phase differenceincreases for a sufficiently small fixed value of � �see Fig. 5top�. The overall evolution of the initial chaotic state is char-acterized by the angular velocity undergoing an inverseperiod-doubling route as � is increased from 0,�, which ispreceded by an inverse interior crisis. The regularization

FIG. 2. �Color online� Boundary functions �cf. Eqs. �10� and�12�� encircling the regularization regions where homoclinic bifur-cations are frustrated in the �−� parameter plane for �=0.1,�=0.87, and =1.6. The quantities plotted are dimensionless.

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route exhibits approximate mirror symmetry with respect tothe optimal suppressory values �opt

sup,�= �� /2,3� /2�. Also,for fixed �=�opt

sup,�= �� /2,3� /2�, the Kapitza pendulumgoes from the strange chaotic attractor existing at �=0 to aperiod-1 attractor as � is increased via an inverse period-doubling route, which is preceded by an inverse interior cri-sis �see Fig. 5 bottom�.

III. LOCALIZED CONTROL OF SPATIOTEMPORALCHAOS

Previous studies have shown that chaos in coupled arraysof damped, periodically driven, nonlinear oscillators can be

suppressed by random shortcuts �38�, parametric disorder�39�, global disordered driving forces �40�, impurities �41�,and localized resonant forces �22�. We consider in the fol-lowing the application of the above control scenario to achain of N chaotic coupled Kapitza pendula described by thedimensionless equations of motion

�n + sin �n = − ��n − � cos��sin �n + ���n+1 + �n−1 − 2�n�

+ �����n+1 − �n�3 + ��n−1 − �n�3� ,

�i + sin �i = − ��i − � cos��sin �i − �� cos� + ��cos �i,

�14�

where n=2, . . . ,N−1, i=1,N, while � and �� are the linearand nonlinear coupling constants, respectively �42�, i.e., theCC excitation is solely applied to a minimal number of pen-dula. Thus, with fixed values of �, �, and corresponding toa given initial chaotic state of the uncoupled pendula, there

1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.60.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

frequency, ω

ampl

itude

,γ

0.0 0.5 1.0 1.5 2.00.0

0.5

1.0

1.5

phase difference, φ/π

rela

tive

ampl

itude

,ε

1.25

0.08

0.25

0.75

0.25/π

1/π2π/π

0.5/π

FIG. 3. �Color online� Top: maximal LE �+ versus frequency and amplitude � for a purely vertical excitation ��=0�, �=0.1, anda grid of 100�100 points in the parameter plane −�. Red �darkgray� and green �light gray� squares indicate that the respectivemaximal LE is �+��=0��0 and �+��=0��0, respectively. Bottom:Maximal LE �+ versus phase difference � and relative amplitude �for �=0.1, �=0.87, =1.6, and a grid of 100�100 points in theparameter plane �−�. Red �dark gray� and green �light gray�squares indicate that the respective maximal LE is �+���0��0and �+���0��0, respectively. The solid black lines indicate theparticular values of � and � considered in Fig. 4 top and bottom,respectively. The quantities plotted are dimensionless.

0.0 0.5 1.0 1.5 2.0−0.1

0.0

0.1

0.2

0.3

phase difference, φ/π

Max

imal

LE,λ

+

ε=0.08 ε=0.25 ε=0.75 ε=1.25

0.0 0.5 1.0 1.5

0.0

0.1

0.2

relative amplitude, ε

Max

imal

LE,λ

+

φ=0φ=0.25φ=0.5φ=1

FIG. 4. �Color online� Maximal LE �+ versus �top� phase dif-ference � for four values of the relative amplitude: �=0.08�squares�, 0.25 �circles�, 0.75 �triangles�, and 1.25 �stars�, and�bottom� relative amplitude � for four values of the phase differ-ence: �=0 �squares�, 0.25 �circles�, 0.5 �triangles�, and 1 �stars� �b�.Parameters are the same as in Fig. 2. Solid lines are solely plotted toguide the eyes. The quantities plotted are dimensionless.

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are only four independent parameters characterizing the con-trol scenario: �, �, �, and ��. To visualize the global spa-tiotemporal dynamics of the chains, we calculate the averagevelocity

��jT� �1

Nn=1

N

�n�jT� , �15�

where j is an integer multiple of the excitation periodT�2� /, while the degree of synchronization is character-ized by the correlation function

C�jT� �2

N�N − 1��il� cos�i�jT� − �l�jT�� , �16�

where the summation is over all pairs of pendula. Note thatC�� is 1�0� for the perfectly synchronized �desynchronized�state.

A. Linear coupling

Consider first the simple case of a purely linear couplingfunction ���=0�. Figure 6 shows the average velocity �red�gray�� and the correlation function �blue �black�� at=380T , . . . ,400T vs the coupling constant for chains of 5,7, and 9 pendula, respectively. We find that the individualpendula go from chaos to a period-1 attractor as the couplingincreases from 0, while the whole chain goes from a desyn-chronized state to a perfectly synchronized state. The evolu-tion of the desynchronized states exhibits great complexity inthe “weak” coupling �WC� regime, as can readily be appre-ciated in Fig. 6 for the case N=9. Indeed, we find windowswhere the chains present chaotic desynchronized states inter-spersed with windows containing regular states correspond-ing to periodic attractors, with periods larger than 1 and dif-ferent degrees of synchronization, as well as abrupttransitions between them via crisis phenomena. The final partof the WC regime is characterized by the average velocity

and the correlation function undergoing an inverse period-doubling route as the coupling is increased to reach the low-est coupling �=�min where the whole chain exhibits perfectsynchrony corresponding to a period-1 attractor. The depen-dence of the lowest coupling �min on the chain size is shownin Fig. 7. We typically find an exponential increase law of theform �min=a0+a1 exp�a2N�, with ai , i=0,1 ,2, being con-stants which depend upon the remaining parameters of thechain. Since exponential laws are generally associated withuniversal behavior, we expect the present law to remain validfor general dissipative, periodically driven chains where theindividually chaotic oscillators are subjected to linear cou-pling.

FIG. 5. Bifurcation diagrams of the angular velocity as a func-tion of �top� the phase difference for �=0.25 and �bottom� the rela-tive amplitude for �=� /2. The remaining parameters are the sameas in Fig. 2. The quantities plotted are dimensionless.

FIG. 6. �Color online� Bifurcation diagrams of average velocity�red �gray�� and correlation function �blue �black�� as a function ofthe linear coupling constant for chains of N=5,7, and 9 pendula,respectively. Parameters: �=0.1, �=0.87, =1.6, �=� /2, �=0.25, ��=0. The quantities plotted are dimensionless.

FIG. 7. �Color online� Lowest linear coupling constant �min �seethe text� as a function of the number of pendula N �dots� and ex-ponential fit �solid line�. Parameters are the same as in Fig. 6. Thequantities plotted are dimensionless.

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B. Nonlinear coupling

In many real systems, such as in neural networks for ex-ample, the couplings are typically nonlinear. Since the syn-chronization scenario is highly sensitive to coupling functionin the context of periodically forced coupled oscillators �43�,it is pertinent to compare the effects of linear and nonlinearcoupling functions on the above control scenario. In particu-lar, numerical experiments indicated that the suppressory ef-fectiveness of a purely cubic coupling ��=0,���0� ishigher than that of a purely linear coupling ���=0,��0� �cf.Fig. 8 top and Fig. 6 top, respectively�, in the sense thathomogeneous regularization of the chain is achieved forlower values of the coupling strength. This suppressory ef-fectiveness of the cubic coupling also works in the unfavor-able case where the strength of a purely linear coupling is notenough to tame the chaotic chain �see Fig. 8 medium andFig. 6 top�, while in those cases where homogeneous regu-larization of the chain is achieved for a purely linear cou-pling, the addition of a cubic coupling typically reinforcesthis synchronization-induced effect �see Fig. 8 bottom�.

C. Time-varying coupling

Now we would like to discuss the robustness of the con-trol scenario against the presence of time-varying couplings.It has been shown that the synchronizability of a network canbe significantly improved by using couplings of time-varyingstrength �44�. This suggests a possible improvement of theabove regularization scenario by considering couplings oftime-increasing strength instead of stationary couplings. Nu-

merical experiments confirmed this hypothesis for the chainEq. �14� with the simple substitution

�� → ���1 + �� , �17�

where ��0 is the time rate of change of the nonlinearcoupling strength. Figure 9 shows the average velocity�red �gray�� and the correlation function �blue �black�� at=280T , . . . ,300T vs the initial nonlinear coupling constant�� for �=0, 0.1, and 0.01, respectively. Starting from a situ-ation where regularization is not achieved for a stationarycoupling ��=0� over the complete range ��� �0,5� �Fig. 9top�, we typically found homogeneous regularization of thechains over wide ranges of �� even for small values of ��Fig. 9 medium and bottom�. Clearly, the time-increasingstrength of the coupling highly enhances the speed of thesynchronization-induced regularization of the chaotic chains.

IV. CONCLUDING REMARKS

We have shown through the example of a dissipativeKapitza pendulum, in which chaos is induced by a paramet-ric excitation, that the localized application of a second para-metric excitation on a minimal number of such coupled pen-dula is a reliable method of taming the chaos of the wholechain. For sufficiently large values of a linear coupling con-stant, perfect synchronization of the chain associated with aperiod-1 attractor is obtained irrespective of the chain size.This control scenario was found to be robust against the ad-dition of a nonlinear �cubic� coupling function to the linear

FIG. 8. �Color online� Bifurcation diagrams of the average ve-locity �red �gray�� and correlation function �blue �black�� as a func-tion of the nonlinear coupling constant for three values of the linearcoupling constant �=0, 0.3, and 2, respectively. Parameters:�=0.1, �=0.87, =1.6, �=� /2, �=0.25, N=5. The quanti-ties plotted are dimensionless.

FIG. 9. �Color online� Bifurcation diagrams of the average ve-locity �red �gray�� and correlation function �blue �black�� as a func-tion of the initial nonlinear coupling constant for three values of thetime rate of change of the nonlinear coupling strength �=0, 0.1,and 0.01, respectively. Parameters: �=0.1, �=0.87, =1.6, �=� /2, �=0.25, �=0, N=7. The quantities plotted aredimensionless.

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one. Additionally, the synchronization-induced homogeneousregularization of chaotic chains subjected to localized chaos-controlling excitations can be highly enhanced by consider-ing time-varying couplings instead of stationary couplings.

Preliminary numerical studies have indicated that the con-trol scenario is robust against the presence of additive noiseand also occurs in two-dimensional arrays—a complete re-port will be provided elsewhere. In view of the robustness ofthe present method, we expect it to be quite readily testableby experiment �42� and find technological application to im-

portant systems such as superconducting Josephson arrays�45�.

ACKNOWLEDGMENTS

We gratefully acknowledge financial support from theMinisterio de Ciencia e Innovación �MCI, Spain� through“Acciones Integradas Hispano-Italianas” Grants No. HI2006-0157 and No. IT007777A9DL �R. Ch. and L. M.� andProject No. FIS2008-01383 �R. Ch.�.

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