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PHYSICAL REVIEW E DECEMBER 1997VOLUME 56, NUMBER 6

BRIEF REPORTS

Brief Reports are accounts of completed research which do not warrant regular articles or the priority handling given toCommunications; however, the same standards of scientific quality apply. (Addenda are included in Brief Reports.) A Brief Reporno longer than four printed pages and must be accompanied by an abstract. The same publication schedule as for regular afollowed, and page proofs are sent to authors.

Synchronization of strange nonchaotic attractors

Ramakrishna RamaswamySchool of Physical Sciences, Jawaharlal Nehru University, New Delhi 110 067, India

~Received 29 July 1997; revised manuscript received 21 August 1997!

Strange nonchaotic attractors~SNAs!, which are realized in many quasiperiodically driven nonlinear sys-tems, are strange~geometrically fractal! but nonchaotic~the largest nontrivial Lyapunov exponent is negative!.Two such identical independent systems can be synchronized by in-phase driving: Because of the negativeLyapunov exponent, the systems converge to a common dynamics, which, because of the strangeness of theunderlying attractor, is aperiodic. This feature, which is robust to external noise, can be used for applicationssuch as secure communication. A possible implementation is discussed and its performance is evaluated. Theuse of SNAs rather than chaotic attractors can offer some advantages in experiments involving synchronizationwith aperiodic dynamics.@S1063-651X~97!10612-2#

PACS number~s!: 05.45.1b

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Pecora and Carroll@1# showed that identical~or nearlyidentical! nonlinear systems can be made to synchronizcoupled by a common drive signal. If one considers the ovall system as separated into drive and response subsysthen a necessary and sufficient condition for synchronizato occur is that the Lyapunov exponents corresponding toresponse subsystem are all negative. This property is roand is easy to realize in the laboratory@1–3#, even when thedynamics of the drive is chaotic and unstable. An applicatof chaotic synchronization that has been extensivelyplored is the possibility of secure communications: A nuber of different schemes based on a variety of coding pciples have been proposed@4–7#.

The property of synchronization of nonlinear systemsextremely general. One situation where this is most eaachieved is between quasiperiodically driven systems inregime wherein the dynamics lie on strange nonchaotictractors~SNAs! @8,9#. The purpose of this Brief Report is tsuggest that such systems possess advantages thatthem ideal for applications in communications that use ariodic signals.

SNAs, which are found in quasiperiodically driven sytems, are geometrically strange, namely, they are fractal,the largest nontrivial Lyapunov exponent is negative ahence the dynamics are not chaotic. They can be crethrough a variety of mechanisms@10# and exist over a rangeof parameter values~i.e., they are not exceptional or nongneric!. SNAs have been observed in several experimesystems@11,12# and have been verified through the usepower spectral methods and attractor dimension estimaAlthough the largest nontrivial Lyapunov exponent is negtive, the dynamics areaperiodicsince the underlying attractor is strange: This makes it difficult to deduce the Lyapun

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exponents, or indeed thenonchaotic dynamics, by attractorreconstruction using standard methods.

Synchronization of two such systems is trivial becausethe negative Lyapunov exponents. Regardless of wheresystems are started, they eventually converge to the sdynamics so long as thephaseof the quasiperiodic driving ismatched. There is no requirement of coupling the syste~other than the coupling implicit in the matched phase;below!.

As an example of this behavior, consider the followinsystem introduced by Zhou, Moss, and Bulsara@13#, whichdescribes a driven-damped superconducting quantum inference device:

x1kx52~x1b sin 2px!1q1 sin v1t1q2 sin v2t,~1!

where the ratio of frequencies is taken to be irrationv1 /v25(A511)/2. This system~and related variants! hasbeen extensively studied in both numerical and analog silations and is thus a typical example of a system that canexperimentally realized. An identical copy of this systewith phase differencef has the equation of motion

y1ky52~y1b sin 2py!1q1 sin v1~ t1f!

1q2 sin v2~ t1f!. ~2!

The two systems can be synchronized regardless of the invalues ofx,x,y,y, so long as there is no phase lagf50 andthe parameters~herek,b and q1 ,q2! are such that the dynamics are on a SNA. Explicitly, it is observed thux(t)2y(t)u→0 rapidly; results for a typical orbit are showin Fig. 1~a!.

Rewriting the above system in autonomous form

7294 © 1997 The American Physical Society

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x15x2 ,

x252kx22~x11b sin 2px1!1q1 sin v1x31q2 sin v2x3 ,

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makes it evident that phase matching corresponds to reping y3 in Eq. ~3! by x3 and thereby coupling the two systemThis then conforms to the general framework of synchrozation in the manner of Pecora and Carroll@1# with the xsystem the drive and they effectively the response.

In other parameter ranges, the system in Eq.~1! can bechaotic. In such a case, both the drive and the responsepositive Lyapunov exponents and synchronization cannotcur; see Fig. 1~b!. When there is a phase mismatch, nameif fÞ0 in Eq. ~2!, again synchronization does not occ@Fig. 1~c!#, even when the parameters correspond to Sdynamics.

Secure communications using aperiodic dynamicsbeen implemented in several ways@4–7# and the techniqueof synchronization with SNAs rather than chaotic attractcan be employed in several of them. It should be mentionhowever, that some of the simpler schemes have been shto be susceptible to unmasking@14# by inference of the un-derlying attractor.

FIG. 1. Time series of the signals from the two systemsx1(t)~solid line! and y1(t) ~dashed line!. The parameters are setk5b52, q152.768, andv152.25. ~a! Whenq250.88 andf50,the dynamics are on a SNA and the two systems synchronize~b!Whenq250.38 andf50, the dynamics are on a chaotic attractThe Lyapunov exponents are all positive and synchronization ispossible.~c! Whenq250.88 andfÞ0, the dynamics are on a SNAbut synchronization does not occur.

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The most direct method of secure communication throuthe use of chaotic synchronization uses a chaotic signamask information@4#. The alternate strategy suggested heis to use the signal from a strange nonchaotic system inanalogous manner, by transmitting the low-amplituinformation-bearing signalm(t) that is added to~and maskedby! the output from the first systemx(t), namely,x8(t)5x(t)1m(t). It is also necessary to simultaneoustransmit a means of phase locking, say, a train ofd-function

FIG. 2. Robustness of synchronization with respect to noisealters the parameters of the response system, as described itext. The values of the parameters arek5b52, q152.768,q250.88, andv152.25 and the noise strength iss51022 for ~a!m[b @see Eq.~4!#, ~b! m[q2 , and ~c! m[v2 . Reduction of thenoise strength improves the synchronization in the last case~d!wheres51024 andm[v2 .

FIG. 3. Demonstration of the viability of the secure communcation scheme. ~a! The signal being communicated im(t)50.1 sinv2t ~dotted line!, which is added to the output fromthe SNA, i.e.,x8(t) ~solid line!. The parameterq2 of the responsesystem differs from that of the drive by 10%. Other SNA parameters are as in Fig. 1~a! and the recovered signal is the dashed cur~b! The signal being communicated ism(t)50.1 sinv2t sinv1t~dotted line!, which is added to the output from the SNA~solidline!. The frequencyv2 of the response system has fluctuationwith s51023. Other SNA parameters are as in Fig. 1~a!. The re-covered signal is the dashed curve. Thed-function spikes inx8(t)are used by the response system for phase matching.

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7296 56BRIEF REPORTS

pulses. The signalm(t) can be recovered by allowing thsystems to synchronize and subtracting the output of theond system, i.e.,x8(t)2y(t).

Since the two systems evolve independently, the effecadditive noise is minimal: noise added tox8(t) will be un-changed upon subtraction. On the other hand, it is necesto consider the mismatch between the transmitter systemthe response in some detail. One way to explore the effecsuch mismatch is by introducing fluctuations in the paraeters of the response

m5m0@11sj~ t !#, ~4!

wheres is the noise amplitude,j(t) is ad-correlated randomvariable with zero mean, andm0 is the value of the parametein the transmitter system. For the response system in Eq.~2!,we considerm[q2 , b, andv2 . Results are shown in Fig.for the case of noise amplitudes51022 for the three param-eters indicated above. The plot ofx vs y shows that thedegree of synchronization in the presence of noise is fagood, except for the case ofm[v2 , namely, when the quasiperiodic driving frequency is subject to fluctuations. Ideed, variation of the parametersq2 and b by up to 10%does not significantly alter the synchronization exceptshort bursts in time. The drive frequency is much more ssitive to fluctuations and only by reducing the noise amptude to 1024 is it possible to greatly improve the synchronzation in this case@Fig. 2~d!#.

The viability of the above scheme is demonstrated usthe SNA of Eqs.~1! and~2! and the results are shown in Fig3, wherein the signal to be communicated is a sinusoform. In the absence of noise, the recovery of the signaexact~and is therefore not shown!; with noise added follow-ing Eq. ~4! in the driving frequency or in the other parameters, the recovery of the signal is of good quality. Indeeven when the parameters of the two systems do not ma

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signal recovery is possible: In Fig. 3~a! the parametersq2 ofthex andy systems differ by 10%. The occasional errors dto loss of synchronization that are apparent in Fig. 3 dopersist over long times.

Note that it is important for this means of application thinterception ofx8(t) can have no potential value in the asence of knowledge of the underlying dynamical systsince the dynamics is intrinsically aperiodic. One can ustandard methods to reconstruct the dynamics@15#, but theextraction of reliable values for~small! negative Lyapunovexponents from experimental time-series data for SNAsproved to be difficult@11,12#. Thus it may be more problematic to reliably reconstruct the underlying attractor, in cotrast to the example of the Lorenz system, which was csidered by Perez and Cerdeira@14#.

Related schemes that use chaotic attractors, for examthe modulation-detection procedure described by CuomoOppenheim@4#, can be similarly adapted to the caseSNAs. A somewhat different implementation of secure comunications using SNAs that transmits digital informatiby switching parameter values has also been proposedcently @16#.

The synchronizing property arises directly from the usea common in-phase driving: The negative Lyapunov exnents alone do not guarantee that thex and y signals willcoincide.~In the extreme case when both systems are ingrable, in the absence of a common driving term, there wbe no synchronization.! Other applications that use the sychronization of chaotic systems@17# can also be effectedusing strange nonchaotic systems. In general, as a coquence of the negative Lyapunov exponents, the stabilityrobustness using SNAs is greater than that with comparchaotic attractors. This may make quasiperiodically drivsystems particularly suitable for applications that involvesynchronization of large numbers of nonlinear dynamisystems.

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