Dec 1, 1998 - 1 Nonlinear Optics Group, Air Force Research Laboratory. 3550 Aberdeen ... Bifurcation diagram of the single isolated pendulum. The angular ...
EUROPHYSICS LETTERS
Europhys. Lett., 44 (5), pp. 559-564 (1998)
1 December 1998
Self-organization of coupled nonlinear oscillators through impurities A. Gavrielides1, T. Kottos2 3, V. Kovanis1 and G. P. Tsironis3 ;
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Nonlinear Optics Group, Air Force Research Laboratory 3550 Aberdeen Ave., SE Kirtland AFB, NM 87117-5776 USA 2 Department of Physics of Complex Systems The Weizmann Institute of Sciences, Rehovot 76100, Israel 3 Physics Department, University of Crete and Foundation for Research and TechnologyHellas - P.O. Box 2208, 71003 Heraklion, Crete, Greece
(received 6 April 1998; accepted in nal form 9 October 1998) PACS. 05.30?d { Quantum statistical mechanics. PACS. 05.40+j { Fluctuation phenomena, random processes, and Brownian motion. PACS. 73.20Dx { Electron states in low-dimensional structures (superlattices, quantum well structures and multilayers).
Abstract. { The e ects of impurities introduced into a lattice of strongly coupled oscillators are studied. Contrary to our expectations, a single impurity can control the chaotic behavior of a long array of chaotic oscillators and bring about complete synchronization. This e ect appears to be independent of the length of the chain, obtaining self-organization for chains as long as 512 lattice sites. By appropriately introducing more that one impurity, if needed, longer chains of oscillators can be tamed.
In this letter we study the e ect of impurities introduced at a site in the forced FrenkelKontorova model, or equivalently in the di usively coupled Josephson junctions model, in which the applied current at each junction is modulated by a common frequency [1]. In the general case, when the individual oscillators execute a chaotic motion, the chain of such N coupled oscillators exhibits a chaotic spatio-temporal pattern. The possibility of a ecting synchronized motion in such an array has been investigated recently by Braiman et al. [2] for the case of one- and two-dimensional arrays of strongly coupled pendula. They found that a chaotic array of forced damped pendula can exhibit complex but frequency-locked spatio-temporal patterns by introducing a certain amount of disorder in the lengths of the pendula. Thus introduction of disorder in an array of coupled chaotic pendula induces rich spatio-temporal patterns. We will investigate the same phenomenon but from a di erent viewpoint. Namely to what extent and for how large an array may we induce synchronization, given that all the pendula are identical and chaotic except for a single impurity introduced at a particular site of the array. The idea is to induce small synchronized domains that crystallize around nonchaotic oscillators and in particular impurities placed at strategic locations in the
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Fig. 1. { Bifurcation diagram of the single isolated pendulum. The angular velocity _ at each period of the driving frequency is plotted as a function of the length l. The inset shows an expanded version of the region about l 1:0 and the abscissa is de ned by x = 1 ? l.
chain. This physical concept is borrowed from the physics of disordered systems and has already been used in some previous nonlinear studies [3]. It is well known in this eld that defects can create localized excitations in the space around their positions [4]. Thus it is hoped that this study will provide a further explanation and will illuminate the results obtained in [2], by considering the simplest possible disordered array of oscillators, that which contains a single impurity. To demonstrate this, we will use the model examined in ref. [2]
ml2 + _ = ?mgl sin + + sin !t + 0
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where n = 1; 2 : : :N and will assume free boundary conditions, i.e. 0 = 1 , = +1 . The parameters used in the calculations are the gravitational acceleration g = 1, the mass of the pendulum m = 1, the dc torque 0 = 0:7155, the ac torque = 0:4; the angular frequency ! = 0:25, and the damping rate = 0:75. Thus, at each site there is an underdamped oscillator with a pendulum length l . The parameter k is the coupling between neighboring pendula and its value is 0:5 as in ref. [2], a value that corresponds to the strong-coupling regime. Each isolated pendulum is chaotic for values of l 1:0. For values larger than that, the pendulum executes a libration in which it oscillates about its equilibrium position without overturning. On the other hand, for pendulum lengths shorter than one, the pendulum executes a rotation where the combined torque rotates the pendulum over the top. It is instructive to examine the bifurcation diagram as a function of the length of a single isolated pendulum and identify the possible chaotic regimes. Figure 1 shows the bifurcation of a single pendulum using the pendulum length as bifurcation parameter. The velocity of the pendulum _ is plotted at every period of the drive, thus obtaining a stroboscopic Poincare plot. We nd that there are three distinct chaotic regimes, extremely narrow and located at l 1:0; 0:84, and 0:52. In addition, there is a chaotic band below l 0:35. We can see in N
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Fig. 2. { Gray scale evolution of _n (jT ) for N = 128, and ln = 0:8 and limp = l64 = 1:0. The pattern is plotted for 50 periods of the drive.
the inset of g. 1 that the chaotic regime around l 1:0 extends roughly from 0:998 to 1:002. This allows us to select with con dence a given pendulum in the chain to be chaotic or not. The simplest con guration we examine rst is that of a single chaotic pendulum in a sea of identical nonchaotic pendula. Our numerical calculations indicate that in the absence of chaotic impurity the solution for the chain is a uniform periodic solution with _ (t) = (t) (2) independent of the site. The inclusion of the chaotic impurity is to simply distort a small region around it in a symmetric manner in which to a large extent the rest of the lattice does not participate, however, maintaining frequency synchronization. In g. 2 we show a sample calculation of a chain of 128 oscillators having l = 0:8 with an impurity of limp = 1:0 placed in the middle of the chain at site n = 64. This is a gray scale plot of the velocity as a function of site number. In the vertical scale, time in integral multiples of the forcing signal period is depicted. The darker shading indicates higher velocity. It is evident that synchronization is maintained. Around the defect the velocity di erence is very small and symmetric. It broadens slightly and increases in value as the length di erence increases. Thus the velocity of the impurity in this case is very close to that of the rest of the array with the highest velocity mismatch a couple sites away. It is apparent that the length of the array does not modify the e ects of the impurity which are totally con ned to its closest neighbors. Of course, the most interesting area of investigation is what happens when a single impurity is placed in a chain of identical chaotic pendula. We select the length of the identical pendula to be equal to 1:0 and then insert an impurity at various sites, preferably the center of the chain, and observe its e ects. We surprisingly nd that for a large parameter range of the pendulum length, position and coupling strength, simple spatio-temporal patterns with period one (P1) emerge. This synchronization was obtained also for quite large chains, even though most of the calculations were performed with chains of oscillators as large as 128 pendula. It appeared that short-length impurities, in general, produced periodic patterns more often than impurities with large pendulum lengths for k = 0:5. This is quite evident in g. 3a) where a spatio-temporal pattern is produced by an limp = 0:8 impurity introduced at site 64 in a n
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Fig. 3. { a) Spatio-temporal pattern produced by an limp = 0:8 impurity at site 64 in an array of 128 oscillators with ln = 1:0. b) Chaotic pattern produced by an limp = 1:2 impurity at site 64 in an array of 128 oscillators with ln = 1:0. c) Spatio-temporal pattern produced by an limp = 0:8 impurity placed in the middle of an array of 512 oscillators with ln = 1:0.
chain of 128 oscillators. A careful examination shows that after the transients have died out approximately after 25 periods, the majority of the oscillators cluster into a P1 pattern except at the edges where it appears that the last three oscillators at each side belong to a P4 pattern. In contrast, in g. 3b) we show a chaotic pattern obtained by simply changing the length of the impurity to limp = 1:2. As can be seen the pattern remains chaotic. Finally the e ect of an impurity with limp = 0:8 was evaluated as a function of the length of the array of oscillators. It was found that the impurity always induced synchronization for quite large chains. In g. 3c) we show a gray scale pattern produced by the impurity placed in the middle of a chain of 512 oscillators. The crystallization of synchronization around the impurity and its growth towards the edges as time increases can be clearly observed. It is evident that the pattern grows in time linearly with respect to the length of the array. In view of these results we tend to favor the following explanation. We know that a collection of N uncoupled chaotic oscillators will have N positive Lyapunov exponents. We further expect that for very weak coupling each oscillator will retain its individuality, with the coupling
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Fig. 4. { Bifurcation diagram of (jT ) as a function of limp for k = 0:5, and N = 32 (a). Bifurcation diagram of (jT ) as a function of limp for k = 5:0, and N = 32 (b).
simply acting as an external source of noise. Thus the attractor of the individual oscillator will be unchanged except for some noisy manifestations of its coupling to the rest of the chain. However, for very strong coupling the dynamics of the system can change drastically, the whole system being essentially similar to that of a single oscillator with a single positive Lyapunov exponent [5]. This was investigated in ref. [5] for coupled maps and driven Dung oscillators and this particular regime of a single Lyapunov exponent is denoted as regime 3. We believe that it is in this regime where the collection of the oscillators are very strongly coupled and where the dynamics of the whole system are essentially similar to that of the single oscillator, and that the insertion of a single nonchaotic oscillator could e ectively modify the system's attractor and thus produce spatio-temporal organization. If we repeat the same calculation but use instead limp = 0:52 for the length of the impurity we still obtain a P1 pattern formation, if one excludes edge e ects which tend to disappear after quite long transients. We note here that, apart from the chaotic regime around l = 1:0 that exists for the isolated pendulum, there are also other chaotic regimes as discussed previously. In particular, if we use for the length of the impurity limp = 0:52, then the isolated pendulum exhibits chaotic dynamics. Nevertheless, there is still clear self-organization. This seems to support our previous hypothesis and render our explanation plausible. To obtain a more global bifurcation diagram of the e ects of the impurity as a function of pendulum length at xed coupling strength, we use a measure (jT ) to obtain a convenient representation of the bifurcation structure. This is de ned as a global property namely the average of the velocity of the individual oscillators computed at each successive period of the drive period
(jT ) = N1
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The results are shown in g. 4(a) in which we plot (jT ) as a function of limp for k = 0:5, and N = 32: For each length of the impurity the calculation was carried out for a sucient number of periods to eliminate transients and the next 16 values of (jT ) were used as a representation of the attractor. Thus P16 attractors or attractors of higher periodicity are not recognized as such but rather as chaotic attractors. Nevertheless, g. 4(a) shows the existence of a P1 attractor coexisting with a P5 attractor for impurity lengths up to limp = 0:6. We were able to nd more than one P5 attractor, however, for clarity only one is included in the bifurcation diagram. The P1 attractor is stable from very low values below 0:2 and as high as
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0:83: Notice that this spans a large window in which the isolated impurity is in itself chaotic. Figure 4(b) shows the bifurcation diagram as a function of the impurity length for the same conditions as before but now the coupling constant has been increased to 5:0. In this case the P1 behavior is predominant with a small chaotic region around limp 1:0 which develops through a very clear period-doubling sequence. On the right-hand side there is a small portion of what seems to be two P2 attractors which quickly bifurcate as the length of the pendulum is decreased. It appears that all the higher-order coexisting attractors are associated with edge e ects and that there could be a multiplicity of them. Indeed as the number of pendula increases, calculations show that edge e ects involve only as few as a couple of the extreme pendula. It is noteworthy to point out that the pattern is robust to small spatial disorder. We have performed calculations in a chaotic chain of oscillators with lengths uniformly drawn from the interval [0.998{1.002], i.e. when the individual oscillators are still chaotic. We found that the presence of the impurity still synchronizes the lattice of chaotic oscillators. In conclusion we have demonstrated that a chain of chaotic pendula can be frequency-locked into a spatio-temporal pattern by introducing an appropriate impurity at a site in the lattice. In most cases a single impurity can tame chaos. We found that a single nonchaotic impurity with an appropriate pendulum length can control and self-organize the dynamics of a chain as long as 512, the limit to which our calculations were performed. Increasing the coupling constant increases the domain of the lengths of the impurity pendulum for which self-organization can be observed. Excluding edge e ects, we found that the pattern has always a repetition period equal to the period of the driving signal. In addition we found numerically that the spacing of the induced pattern in the self-organized oscillator regime as can be seen in g. 3a) and c) depends critically on the nearest-neighbor oscillator coupling. The spacing seems to increase monotonically when the coupling is also increased. Our results suggest that strongly coupled systems have an attractor whose dynamics are similar to the attractor of the single pendulum, and that the introduction of a single impurity can change the dynamics drastically. *** We acknowledge partial support from ENE grant 95?115-E of the General Secretariat of Research and Technology of Greece. AG and VK would also like to acknowledge partial support from AFOSR. The authors would like to thank K. Georgiou for technical assistance. REFERENCES [1] Floria L. M. and Mazo J. J., Adv. Phys., 45 (1996) 505; Ustinov A. V., Cirillo M. and Malomed B. A., Phys. Rev. B, 47 (1993) 8357; Wiesenfeld K., Colet P. and Strogatz S., Phys. Rev. Lett., 76 (1996) 404. [2] Braiman Y., Lindner J. F. and Ditto W. L., Nature, 378 (1995) 465. [3] Molina M. I. and Tsironis G. P., Int. J. Mod. Phys. B, 9 (1995) 1899. [4] Economou E. N., Green's Functions in Quantum Physics, Springer Ser. Solid State Phys., Vol. 7 (Springer-Verlag, Berlin) 1979. [5] Lai Y. C. and Winslow R. L., Physica D, 74 (1994) 353.
