- Email: [email protected]
Communication with chaos over band-limited channels
Available online at www.sciencedirect.com
PERGAMON
Acta Astronautica 53 (2003) 465-475
COMMUNICATION
W ITHCHAOS OVER BAND-LIMITED CHANNELS
Elbert E. N. Macau and CleversonM . P. Marinhe Labt$at&io de Integra#o e Testes-LIT Institute National de PesquisasEspaciais - IlfPE ’ Sio Josk dos Gzmpos- SP - Brazil E-mail: elbert6iIit.ium.b~ - Fax: 55-12-3941-1884 Recently,manyworks havepointedout the fact that the complexityinherentin the dynamicsof a chaoticsystemscan be advantageously usedto implementdigital communicationsystem[1,8-141. Digital infiion is encodedin kge-scale fm of the wavefmmby using small perhubations to co&o1 the symbolic dynamics[l]. Chaotic oscillationscover a wide spectraldomainand can efficiently mssk sn i&ormstion signalscrambledby the chaoticarcoder[l-2]. However,the wide q&rum poses intrinsic difficulties iu the decodingprocessif the codified chaotic sigtutl is transmittedover m l cmnumicaticn channelswith lim ited bandwidth[12,13]. We addressthis problemnumerically,and we proposean approachto improvechaoticeonununicaticuover bandlim ited andnoisy channels..Our msultssuggestthst chaoticconununication conceptis a convenient approachto be exploitedto efficiently transmit largeamountof datewith a good level of security, which makesthis conceptas a good option to be used for the next generationcommunication satellites. 0 2003 Elsevier Science Ltd. All rights reserved.
1. IntroductioJJ Chaos in physical systemswas thought to bc a form of noise that interfere with desired patterns of behavior of systems and so was unwanted. Now it is known that chaos is not a noise, but a deterministic broadband non-periodical signal that arises as a behavior of nonlinear systems that are extremely sensitive to changesin initial conditions [2]. Despite this mmarkable sensitivity, it has recently been demonstratedthat chaos can be comrollad. In fact. by using just small perturbation it is possible to drive a chaotic system among desired statesand even stabilize a chaotic evolution on periodic orbits. Hayes, Gmbogi, and Ott [l] conceived the idea of use the control of chaos to causethe symbolic dynamics of a chaotic system to follow a prescribed symbol sequence.By doing so, they showed that it was possible to encodeany desired messagein a trajectory of a chaotic oscillator. This idea, that was subsequentlydemonstratedexperimentally, wss the start of a shift paradigm in communication, in which chaos is used as a fimdamentalbuilding block for constructing of remsrkablc simple, efficient and cost-effective communication systems[ 141. In this work, we present a chaotic based commuuication strategy that not only codify the information on the chaotic invariant set by applying small perturbations, but also take advantageof the inherent complexity of the chaotic dynamics to make it difficulty for an eavesdrop to decode the communication. In fact, chaotic oscillations cover a wide spectral domain and can efficiently mask an information sigual scrambled by the chaotic cncodcr. However, the wide spectmm poses intrinsic difliculties in the decoding processif the wdiiicd chaotic signal is transmitted over real communication channels with lim ited bandwidth. We addressthis problem numerically and we propose approachesto improve the use of the chaotic communication paradigm over band-limited channels. 0094-5765/03/$- seefront matter 0 2003 Elsevier ScienceLtd. All rights reserved. doi: lO.l016/SOO94-5765(03)00150-4
466
E.E.N. Macau, C.M.F! Marinho /Acta Astroliautica
53 (2003) 465-475
To apply our ideas,we consider a Chua’s circuit operating in the double scroll chaotic regime. As discussedlatter, Chua’s circuit is a remarkably simple and robust electricalcircuit made of only four linear elementsand a nonlinearelement(the so-called Chuadiode):[5,1I]. More recentlythe Chua diode hasbeenfabricatedas a microelectronic IC chip. At first, we explainhow the conceptsof control of chaose chaoticsynchronization can be tied togetherto allow the reliable and efficiently transmkion of a messageover a ccummmicationchannel.After that, we take in count the band limit characteristicsof a real communicationchannelaswell as the effect of the presenceof noiseon the channel.
2. Theoretical background
Typically, a chaoticattractorhas embeddedon it a dense.setof unstableperiodic orbits (21. Using small perturb&ions,so sqklI that thkoriginal dynam@ of the systemis not
considerablechanged,it is possibl?ioot.$ control a trajecto~ ti its evolution th~~gb the chaotic invariant set as will as to’stabili!zethe systemOna d&died W&Able periodic orbit [15,16].~~,theso~~controlofchtsos[17]tachniqu~casbeus~~controla~ trajectoryso that a messagecartbe convenientlycodified on it. A chaotic signal is characterizedby an aperiodic and apparentlyraudom waveform that producesa.se@ience of maximumsahdminimums [2J. We can -iate the digital symbol 1 eachtime the wave ove$assa previously speci@value,while thksy%nlkl0 is associated to signal e+rsions that u&pass ano@r prior established, value. By doing so, we can view ‘ixcli&ic systemas a bii sequencegenerat& I&thermom~ this generatorcan be controlled by using small perturbationsin a tiamework of ari&ipl&ne4~ control of chaos strategy [ 15-161.Thus, the chaotic dynamicscan be controlledto produce desireddigital sequence. As the control of chaos philosophy does not dnunatically change the system dynamics,but just allow us to take advantageof the inherentcomplexity of the chaotic dynamics to control it by using small perturbations[17] (and so with expensesof low energy),typically not all the sequencescan be produced.Thus, thereare sequencesthat the systemdynamicsisxrnableto produce.Thesesequenceare namedby forbid&n seqwnces. For a specific system,we call iti grammar the set of rules that specify the allowable sequencesth+t its dynamicscan produce For a given system,then sre various proposed methodsfor the determinationof its grammar[l]. In a commtication scenario,the souicc of informationmust be allowed to produce any desired sequence.To cod& this sequencein a chaotic wavefbrm, some conversion algorithm must be used on the original some stream.‘l%is algotithm is usually ea8ily implementedby using a conversiontable. Let us now qply thy ideas to the Chua’s circuit that appearson figure 1. The mathematicalequationsthat describeits dynamicsarethe following: c,it, = Gbq - vc,)-&c, ) C,+c2= G(vc,- vc,)+ iL LiL = -I+,
inwhich C,, C,, G=lIR
e L aregivenconstants.
(1)
E.E.N. Macau, C.M.P. Marinho /Acta Astronautica
53 (2003) 465-475
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00
Figure 1: (a) Chua’scircuit; (b) Nonlinearnegative-xe&ancechamcteristicof the Chua’sdiode. The nonlinear resistor (Chua’s diode) presents a 3-segmentodd-symmetric voltagecurrent characteristic that cau be describedby the following equation:
Inthisworkweusethefollowingwmmlizeparametervalues: C, =1/9, C, =l, L=1/7, G=0.7, Gb =-OS, Go =-0.8 and BP = 1. For a Poinear6 surface of section, we take the iarhces iL =fGF, JQ)SF, whereF=B,(G,-G,,)/(G+G,), so that these halfplanes intersect the attractor with edges at the unstable fixed points at the center of the attractor lobes. As described in figure 2 and 3, these planes allow us to associatethe digital symbols 0 or 1 to the system dynamic, which depends on each of the two planes the trajectory crosses.
468
E.E.N. Macau, C.M.P. Marinho /Acta Astronautica
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Figure2: Doublescrollchaoticoscillatorstate-space trajectoryprojectedon thein-vet plane &owingthewafirccofsection. The system’s grammar is determinedby allowing the system to evolve f?eely, following its intrinsic dynamics. Each time the system crossesthe Poincar6section, we register the x value and the correspondingsequenceof symbols. Thus, supposethat the system generatesthe sequencedf symbols 4@,... . We can associateto this sequencethe binary number O.&&..,
which correspondsto the real number r = 242-“. In this n-l schema,r representsthe symbolic state of the system. By keep foliowing the system dynamics and its intersectionwith the de&d PO&& section, we obtainedthe relation betweenx and its symbolic codifxation. This relation is called by codificution @action and appearsinfigure4.
E.E.N. Macau. C.M.F! Marinho /Ada
Astronautica
53 (2003) 445-475
Figure 3: Double-scrollchaoticattractorandtbe used Poincar6section
8
0.26 0.2 0.16 0.1 0.06
8 Figure 4: Binary codingfunctionr(x) for the doublescroll system. From the analyze of the symbolic sequence,we can realize that the grammar for this system is very simple [ 1,141: any sequenceof bits can happen with the exception of two conaemtive maximums. Thus, we can codify any sequenceof bits generatedby the source if a symbol 1 is “artificially” introduced a&r each sequenceof 1 and a 0 after each string of OS.Asa consequence,k oscillations of a given polarity representsk - 1 bits of information. Let us now introduce the control strategy that allow us to impose to the system dynamics to follow a desired string of bits. Say the system state point passes through branch 0 of the surface of section at x = x, , and next crosses the surface of section at
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410
x = X, , on eitherbranch0 or 1. Becausewe havepreviously detetmined.thefunction r(x) , use the stored V&ES to find the symbolicstate r(x, ) . We then convertthe number P-(X,)to its correspondingbinary sequencetruncatedat some chosenlength A!, and store this finite-lengthsymbol sequencein a coderegister;As the systemstatetrajectoriesmoves towardsits next encounterwith the :surface section 4 *of .- .._ _“.~,. x = x, , we shift the sequencein the coderegisterleft, discardingthe mostk&fictit .bi$ and insert the first desiredinformation code bit in the now empty’:l~~&$&@f :sl&,of.the ‘coderegister.We then concertthis new symbolsequenceto its c&q&ridmg s$x&rotic-state$. Now, when the systemstate point crossesthe surfaceof @&nat 3 .= T&,we & a searchalgorithmto find the nearest value of the coordinatex that &&$onds to the de&redsymbolic state r; . Let us call this value as XL. By construction,{I&) - r(zL 4 S 2-N. Now let & = x, - &. Becausewe have chosenthe branchesof the surfaceof sectionat constantvaluesof the inductor current it, the deviationdk correspondsto a small deviationin the voltagesve, and vc, . we w
3. Channel equalization to allow chaotic synchronization
Standarddesignof a systemof two synchronizedchaotic oscillatorsusually assumesa perfect channelwith infinite bandwidth and no phasedistortions[S-7]. However, even an ordinarytransmissionwire behavesas a low-passfilter which attenuatesat leastslightly the transmitterdriver signal. As we are working with nonlinear chaoticsystems,it is possible that eventhis small attenuationcan imp&ssynchronization. We considerthis problem in the contextof the Chua’scircuit. Let us considerthe Chua’scircuit equations:
vc $$=BP r=- tG G
z=L i BpG
(3).
b=$
a=- G c,
Theseequationscan be written in the following adimensionalform: fW,u) = aO, - x-g(x)), q = (u-y+z),
(4)
F, = -&I. Nonlinearity is taken in the form g(x) = bx + ;(a - b>Qx+ ‘I- Ix - 11) (5)
and parametersa = 10.00, /? = 14.87, a = -1.27, b = -0.68. If no information is being transmitted,the driver signalreducesto u, = x, and for the responsesystem
E.E.N. Macau, C.M.P. Marinho /Acta Astronautica 53 (2003) 465-475
qj = 40 Qgo,
471
(6)
where c(t) is an impulse responsefbnction of the channel, an the sigrnl Q derioteslinear convolution. The Chua’s chaotic double scroll attractor for the system is shown in Figure 5. The power spectrum of the x-component of the driving system is shown in Figure 6. In Figure 7 we show the maximum value of the module error (m,(t)-m(t)) between the transmittedand the receiver messageas a fimction of the cutoff frequency of the channel. -7
.-.
3
.
-2
..‘I
.-
..7--1.
.. x
-
I
. ..a
3
Figure5: Doublescrollattractorin the (x,y)-plane.
Figure6: Powerspectraldensityof thex componentof the doublescroll systemBy comparing figures 6 and 7, we can see that the maximum value of the module error accumulatequickly and become significant larger as the cutoff fkquency decreases.This effect happensbecausea significative part of the chaotic content is eliminated.
472
&EN,
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53 (2003) 465-475
0 0
0 Ooo
Figure7: maximum value of the module error as a function of the cutoff frequency. Two difkent approacheshave beenproposedto cope with channeldistortions in order to preservesynchronizationof chaotic oscillators. Both of these methods have an advantagethat they allow for exact synchronizationwithout a&cting the structure of chaotic attractors.The first scheme[8,9] involves learning the channelresponse8mction c(t) andbuilding an inversefllter, c”(t) (seeFigure 8). This methodrequiresthat channel distortions are invertible, and the inverse filter is stable,which is not necessarilytrue in general.It also is very sensitiveto noise in the channel.Indeedin the presenceof noise the recovered signal will have a form u’= c-‘(t) Q (c(t) 60u(t) + r,(t)), where q(t) denotes random noise.The consequenceof this approachis the noise outside the passbandof the filter c(t) will be amplified by the inversetilter c-*(l).
m =w
;----‘----‘----‘-‘---------“‘-“----,
Figure8: Channelequalization basedon theinversefilter The secondmethod due to Carrol [lo] employs augmentingthe chaotic signal filtered be the channel,by a signal of the decoderpassedthrougha complimentaryband-stopfilter Ud(C) = c(r) Q u,(t) + (1-c(t)) @X&)
(7) .
E.E.N. Macau. C.M.P. Marinho /Acta Astronautica
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In this case (seeFigure 9) the filter responsefunction also must be learned at the receiveing end. As comparedto the previous method, there is no filter inversion and associatedwith it problems of filter instability and noise amplification. As Carroll [9] has shown, for mild channel distortions exact chaos synchronization can be achieved with this approach. However, the conditional Lyapunov exponent characterizing stability of the synchronized state is less than in the no-filter case. Furthermore, the last term in (7) is the one which provides a positive feedback in the responsesystem and so may causereduction of stability.
Figure9: Carroll methodfor channelequalization A difherent approachto achieve our goal is to preparea rather narrow-band chaotic signal which would not be significantly affected by the chamrel distortions. If, in addition, the information message is also narrow-band, it can be recovered at the decoder without distortion. In Figure 10 we show a method that performs like that. The x-component of the encoder is divided i&o a filtered part xf =#Sn, andacomplimentaryp&t.u,(t).The information signal ‘is added to x,’ and the sum u, = xf + m(t) is sent to the decoder.By doing so, the the signal which is injected into the nonlinear element of the system is simply m(t) +x,(t), as if there were no filter in the faadback loop. At the decoder, the x,, component is similarly split into the filtered part xi = 4 8 X, and its complimentary part -* 4 = x, -xi. The information signal is recovered as a difference m,(t) = u,, -xi, which means that the transmitted signal is passedthrough a channelwithout distortion, and so the originalmessagecanbemcovered. This method was applied to the Chua’s circuit. We used a low pass fourth order elliptical filter in the feedback loop, and the cutoff fiquency was adjusted to optimize the system performance. In Figure 10 we compare the maximum value of the module error for the situations with the presence and without the presenceof the filter. The message is repn?sentedas m(t) = O.Lwn(c.at), with w = 50 x ld rad/s.
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Lt.07 .
Figure10- (a) maximum value of the moduleerror (m, -m) asa fimctionof the channelcutofffrequency.. (b) maximum value of the module error (m, - m) (m, - m) asa fiulctionof theLPScutofftrequcncy. 4. Bibliography
[l] S. Hayes, C. Grebogi and E. Gtt, “Communicationwith Chaos”.Phys. Rev. Lett., vol. 70, pp. 3031-3034,1993. [2] E. Ott, Chaos in dynamical systems, Cambridgeuniversity Press,N.Y., 1994. [3] P. CvitanoviC,G. GunamtneandI. Procaccia,Phys.Rev. A 38,1503 (1988) [4] D. P. Lathmp and E. J. Kostelich, Phys. Rev. A 40,4028 (1989); [5] H. Fujisaka and T. Yamada, “Stability theory of synchronizui motion in coupledoscillator systems.”Progr. Theor. Phys..,vol. 69, pp. 32-47,1983. [q P. Ashwin, J. Buescu and I. Stewart, “From at&actor to chaotic saddle: a tale of transverseinstability.”Nonlineariry, vol. 9, pp. 703-737,1996. [7] A. V. Gppenheim, K. M. Cuomo, R J. Baron and A. E. Fredman, “Channel for Communicationwith Chaotic Signals.”in Chaotic, Fractal and Nonlinear Signal Processing, R A. Katz, Ed., AIP Press,1996,pp. 289-301. EkpGation
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[8] Carroll, T. L., “Synchronizing chaotic systems using filtered signals.”Whys.Rev. E,
vol.
50, pp. 2580-2587,1994. [9] Carroll, T. L., “Communication with use of filtered, synchronized, chaotic signals.” IEEE Transactions on Circuits and SystemsI: Fundamental Theory and Applications, vol. 42, pp. 105-l 10, (1995). [lo] Carroll, T. L. and Pecora, L. M., “The effect of filtering on communication using synchronized chaotic ciqits.”
in: 1996 IEEE International S’posium
on Circuits and
Systems. Circuits and $ustems Connecting the World, ISCAS 96, New York, NY, USA: IEEE, 1996, pp. 1741”77,vol. 3. [l l] L. 0. Chua, T. Yand, G. Q. Zhong and C. W. Wu, “Synchronization of Chua’s circuits with time-varying channels and parameters.”IEEE Trans. Circuits and @stems 4 vol. 43, pp. 862-868, 1996. [12] K. M. Cuomo and A.,V. Qppenheim, “Circuit implementation of synchronized chaos with applications to communications.”Phys. Rev. Lett., vol. 71, pp. 6568,1993. [13] U. Parlitz, L. Kokarev, T. Stojanovski e H. Preckel, “Enconding messages using chaotic synchronization.”Phys. Rev. E, vol. 53, pp. 4351-4361,1996. [14] S. Hayes, C. Grebogi, E. Gtt, and A. Mark, “Experimental Control of Chaos for Communication”. Phys. Rev. Let& vol. 73, pp. 1781-1784,1994. [15] E. E. N. Macau and C. Grebogi, “Driving trajectories in chaotic systems”, Int. J. Bifurcat. Chaos, vol 12, pp. 1423-1442(2001). [16] E. E. N. Macau and I. L. Caldas, ‘Driving trajectories in chaotic scattering”, Phys. Rev E, vol. 65, art no. -027202 Part 2 (2002). [17] E. Ott, C. Grebogi, and J. A. Yorke, “Controlling Chaos”, Phys. Rev. L&t., ~0164, pp. 1196-l 199 (1990). [ 181 A. R. Volkovskii and N. F. Rulkov, “Synchronous chaotic response of a nonlinear oscillator system as principle for the detection of the information component of chaos.” Tech. Phys. Lett., vol. 19, pp. 97-99, 1993.
PERGAMON
Acta Astronautica 53 (2003) 465-475
COMMUNICATION
W ITHCHAOS OVER BAND-LIMITED CHANNELS
Elbert E. N. Macau and CleversonM . P. Marinhe Labt$at&io de Integra#o e Testes-LIT Institute National de PesquisasEspaciais - IlfPE ’ Sio Josk dos Gzmpos- SP - Brazil E-mail: elbert6iIit.ium.b~ - Fax: 55-12-3941-1884 Recently,manyworks havepointedout the fact that the complexityinherentin the dynamicsof a chaoticsystemscan be advantageously usedto implementdigital communicationsystem[1,8-141. Digital infiion is encodedin kge-scale fm of the wavefmmby using small perhubations to co&o1 the symbolic dynamics[l]. Chaotic oscillationscover a wide spectraldomainand can efficiently mssk sn i&ormstion signalscrambledby the chaoticarcoder[l-2]. However,the wide q&rum poses intrinsic difficulties iu the decodingprocessif the codified chaotic sigtutl is transmittedover m l cmnumicaticn channelswith lim ited bandwidth[12,13]. We addressthis problemnumerically,and we proposean approachto improvechaoticeonununicaticuover bandlim ited andnoisy channels..Our msultssuggestthst chaoticconununication conceptis a convenient approachto be exploitedto efficiently transmit largeamountof datewith a good level of security, which makesthis conceptas a good option to be used for the next generationcommunication satellites. 0 2003 Elsevier Science Ltd. All rights reserved.
1. IntroductioJJ Chaos in physical systemswas thought to bc a form of noise that interfere with desired patterns of behavior of systems and so was unwanted. Now it is known that chaos is not a noise, but a deterministic broadband non-periodical signal that arises as a behavior of nonlinear systems that are extremely sensitive to changesin initial conditions [2]. Despite this mmarkable sensitivity, it has recently been demonstratedthat chaos can be comrollad. In fact. by using just small perturbation it is possible to drive a chaotic system among desired statesand even stabilize a chaotic evolution on periodic orbits. Hayes, Gmbogi, and Ott [l] conceived the idea of use the control of chaos to causethe symbolic dynamics of a chaotic system to follow a prescribed symbol sequence.By doing so, they showed that it was possible to encodeany desired messagein a trajectory of a chaotic oscillator. This idea, that was subsequentlydemonstratedexperimentally, wss the start of a shift paradigm in communication, in which chaos is used as a fimdamentalbuilding block for constructing of remsrkablc simple, efficient and cost-effective communication systems[ 141. In this work, we present a chaotic based commuuication strategy that not only codify the information on the chaotic invariant set by applying small perturbations, but also take advantageof the inherent complexity of the chaotic dynamics to make it difficulty for an eavesdrop to decode the communication. In fact, chaotic oscillations cover a wide spectral domain and can efficiently mask an information sigual scrambled by the chaotic cncodcr. However, the wide spectmm poses intrinsic difliculties in the decoding processif the wdiiicd chaotic signal is transmitted over real communication channels with lim ited bandwidth. We addressthis problem numerically and we propose approachesto improve the use of the chaotic communication paradigm over band-limited channels. 0094-5765/03/$- seefront matter 0 2003 Elsevier ScienceLtd. All rights reserved. doi: lO.l016/SOO94-5765(03)00150-4
466
E.E.N. Macau, C.M.F! Marinho /Acta Astroliautica
53 (2003) 465-475
To apply our ideas,we consider a Chua’s circuit operating in the double scroll chaotic regime. As discussedlatter, Chua’s circuit is a remarkably simple and robust electricalcircuit made of only four linear elementsand a nonlinearelement(the so-called Chuadiode):[5,1I]. More recentlythe Chua diode hasbeenfabricatedas a microelectronic IC chip. At first, we explainhow the conceptsof control of chaose chaoticsynchronization can be tied togetherto allow the reliable and efficiently transmkion of a messageover a ccummmicationchannel.After that, we take in count the band limit characteristicsof a real communicationchannelaswell as the effect of the presenceof noiseon the channel.
2. Theoretical background
Typically, a chaoticattractorhas embeddedon it a dense.setof unstableperiodic orbits (21. Using small perturb&ions,so sqklI that thkoriginal dynam@ of the systemis not
considerablechanged,it is possibl?ioot.$ control a trajecto~ ti its evolution th~~gb the chaotic invariant set as will as to’stabili!zethe systemOna d&died W&Able periodic orbit [15,16].~~,theso~~controlofchtsos[17]tachniqu~casbeus~~controla~ trajectoryso that a messagecartbe convenientlycodified on it. A chaotic signal is characterizedby an aperiodic and apparentlyraudom waveform that producesa.se@ience of maximumsahdminimums [2J. We can -iate the digital symbol 1 eachtime the wave ove$assa previously speci@value,while thksy%nlkl0 is associated to signal e+rsions that u&pass ano@r prior established, value. By doing so, we can view ‘ixcli&ic systemas a bii sequencegenerat& I&thermom~ this generatorcan be controlled by using small perturbationsin a tiamework of ari&ipl&ne4~ control of chaos strategy [ 15-161.Thus, the chaotic dynamicscan be controlledto produce desireddigital sequence. As the control of chaos philosophy does not dnunatically change the system dynamics,but just allow us to take advantageof the inherentcomplexity of the chaotic dynamics to control it by using small perturbations[17] (and so with expensesof low energy),typically not all the sequencescan be produced.Thus, thereare sequencesthat the systemdynamicsisxrnableto produce.Thesesequenceare namedby forbid&n seqwnces. For a specific system,we call iti grammar the set of rules that specify the allowable sequencesth+t its dynamicscan produce For a given system,then sre various proposed methodsfor the determinationof its grammar[l]. In a commtication scenario,the souicc of informationmust be allowed to produce any desired sequence.To cod& this sequencein a chaotic wavefbrm, some conversion algorithm must be used on the original some stream.‘l%is algotithm is usually ea8ily implementedby using a conversiontable. Let us now qply thy ideas to the Chua’s circuit that appearson figure 1. The mathematicalequationsthat describeits dynamicsarethe following: c,it, = Gbq - vc,)-&c, ) C,+c2= G(vc,- vc,)+ iL LiL = -I+,
inwhich C,, C,, G=lIR
e L aregivenconstants.
(1)
E.E.N. Macau, C.M.P. Marinho /Acta Astronautica
53 (2003) 465-475
467
00
Figure 1: (a) Chua’scircuit; (b) Nonlinearnegative-xe&ancechamcteristicof the Chua’sdiode. The nonlinear resistor (Chua’s diode) presents a 3-segmentodd-symmetric voltagecurrent characteristic that cau be describedby the following equation:
Inthisworkweusethefollowingwmmlizeparametervalues: C, =1/9, C, =l, L=1/7, G=0.7, Gb =-OS, Go =-0.8 and BP = 1. For a Poinear6 surface of section, we take the iarhces iL =fGF, JQ)SF, whereF=B,(G,-G,,)/(G+G,), so that these halfplanes intersect the attractor with edges at the unstable fixed points at the center of the attractor lobes. As described in figure 2 and 3, these planes allow us to associatethe digital symbols 0 or 1 to the system dynamic, which depends on each of the two planes the trajectory crosses.
468
E.E.N. Macau, C.M.P. Marinho /Acta Astronautica
53 (2003) 465-475
Figure2: Doublescrollchaoticoscillatorstate-space trajectoryprojectedon thein-vet plane &owingthewafirccofsection. The system’s grammar is determinedby allowing the system to evolve f?eely, following its intrinsic dynamics. Each time the system crossesthe Poincar6section, we register the x value and the correspondingsequenceof symbols. Thus, supposethat the system generatesthe sequencedf symbols 4@,... . We can associateto this sequencethe binary number O.&&..,
which correspondsto the real number r = 242-“. In this n-l schema,r representsthe symbolic state of the system. By keep foliowing the system dynamics and its intersectionwith the de&d PO&& section, we obtainedthe relation betweenx and its symbolic codifxation. This relation is called by codificution @action and appearsinfigure4.
E.E.N. Macau. C.M.F! Marinho /Ada
Astronautica
53 (2003) 445-475
Figure 3: Double-scrollchaoticattractorandtbe used Poincar6section
8
0.26 0.2 0.16 0.1 0.06
8 Figure 4: Binary codingfunctionr(x) for the doublescroll system. From the analyze of the symbolic sequence,we can realize that the grammar for this system is very simple [ 1,141: any sequenceof bits can happen with the exception of two conaemtive maximums. Thus, we can codify any sequenceof bits generatedby the source if a symbol 1 is “artificially” introduced a&r each sequenceof 1 and a 0 after each string of OS.Asa consequence,k oscillations of a given polarity representsk - 1 bits of information. Let us now introduce the control strategy that allow us to impose to the system dynamics to follow a desired string of bits. Say the system state point passes through branch 0 of the surface of section at x = x, , and next crosses the surface of section at
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410
x = X, , on eitherbranch0 or 1. Becausewe havepreviously detetmined.thefunction r(x) , use the stored V&ES to find the symbolicstate r(x, ) . We then convertthe number P-(X,)to its correspondingbinary sequencetruncatedat some chosenlength A!, and store this finite-lengthsymbol sequencein a coderegister;As the systemstatetrajectoriesmoves towardsits next encounterwith the :surface section 4 *of .- .._ _“.~,. x = x, , we shift the sequencein the coderegisterleft, discardingthe mostk&fictit .bi$ and insert the first desiredinformation code bit in the now empty’:l~~&$&@f :sl&,of.the ‘coderegister.We then concertthis new symbolsequenceto its c&q&ridmg s$x&rotic-state$. Now, when the systemstate point crossesthe surfaceof @&nat 3 .= T&,we & a searchalgorithmto find the nearest value of the coordinatex that &&$onds to the de&redsymbolic state r; . Let us call this value as XL. By construction,{I&) - r(zL 4 S 2-N. Now let & = x, - &. Becausewe have chosenthe branchesof the surfaceof sectionat constantvaluesof the inductor current it, the deviationdk correspondsto a small deviationin the voltagesve, and vc, . we w
3. Channel equalization to allow chaotic synchronization
Standarddesignof a systemof two synchronizedchaotic oscillatorsusually assumesa perfect channelwith infinite bandwidth and no phasedistortions[S-7]. However, even an ordinarytransmissionwire behavesas a low-passfilter which attenuatesat leastslightly the transmitterdriver signal. As we are working with nonlinear chaoticsystems,it is possible that eventhis small attenuationcan imp&ssynchronization. We considerthis problem in the contextof the Chua’scircuit. Let us considerthe Chua’scircuit equations:
vc $$=BP r=- tG G
z=L i BpG
(3).
b=$
a=- G c,
Theseequationscan be written in the following adimensionalform: fW,u) = aO, - x-g(x)), q = (u-y+z),
(4)
F, = -&I. Nonlinearity is taken in the form g(x) = bx + ;(a - b>Qx+ ‘I- Ix - 11) (5)
and parametersa = 10.00, /? = 14.87, a = -1.27, b = -0.68. If no information is being transmitted,the driver signalreducesto u, = x, and for the responsesystem
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qj = 40 Qgo,
471
(6)
where c(t) is an impulse responsefbnction of the channel, an the sigrnl Q derioteslinear convolution. The Chua’s chaotic double scroll attractor for the system is shown in Figure 5. The power spectrum of the x-component of the driving system is shown in Figure 6. In Figure 7 we show the maximum value of the module error (m,(t)-m(t)) between the transmittedand the receiver messageas a fimction of the cutoff frequency of the channel. -7
.-.
3
.
-2
..‘I
.-
..7--1.
.. x
-
I
. ..a
3
Figure5: Doublescrollattractorin the (x,y)-plane.
Figure6: Powerspectraldensityof thex componentof the doublescroll systemBy comparing figures 6 and 7, we can see that the maximum value of the module error accumulatequickly and become significant larger as the cutoff fkquency decreases.This effect happensbecausea significative part of the chaotic content is eliminated.
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&EN,
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0 0
0 Ooo
Figure7: maximum value of the module error as a function of the cutoff frequency. Two difkent approacheshave beenproposedto cope with channeldistortions in order to preservesynchronizationof chaotic oscillators. Both of these methods have an advantagethat they allow for exact synchronizationwithout a&cting the structure of chaotic attractors.The first scheme[8,9] involves learning the channelresponse8mction c(t) andbuilding an inversefllter, c”(t) (seeFigure 8). This methodrequiresthat channel distortions are invertible, and the inverse filter is stable,which is not necessarilytrue in general.It also is very sensitiveto noise in the channel.Indeedin the presenceof noise the recovered signal will have a form u’= c-‘(t) Q (c(t) 60u(t) + r,(t)), where q(t) denotes random noise.The consequenceof this approachis the noise outside the passbandof the filter c(t) will be amplified by the inversetilter c-*(l).
m =w
;----‘----‘----‘-‘---------“‘-“----,
Figure8: Channelequalization basedon theinversefilter The secondmethod due to Carrol [lo] employs augmentingthe chaotic signal filtered be the channel,by a signal of the decoderpassedthrougha complimentaryband-stopfilter Ud(C) = c(r) Q u,(t) + (1-c(t)) @X&)
(7) .
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In this case (seeFigure 9) the filter responsefunction also must be learned at the receiveing end. As comparedto the previous method, there is no filter inversion and associatedwith it problems of filter instability and noise amplification. As Carroll [9] has shown, for mild channel distortions exact chaos synchronization can be achieved with this approach. However, the conditional Lyapunov exponent characterizing stability of the synchronized state is less than in the no-filter case. Furthermore, the last term in (7) is the one which provides a positive feedback in the responsesystem and so may causereduction of stability.
Figure9: Carroll methodfor channelequalization A difherent approachto achieve our goal is to preparea rather narrow-band chaotic signal which would not be significantly affected by the chamrel distortions. If, in addition, the information message is also narrow-band, it can be recovered at the decoder without distortion. In Figure 10 we show a method that performs like that. The x-component of the encoder is divided i&o a filtered part xf =#Sn, andacomplimentaryp&t.u,(t).The information signal ‘is added to x,’ and the sum u, = xf + m(t) is sent to the decoder.By doing so, the the signal which is injected into the nonlinear element of the system is simply m(t) +x,(t), as if there were no filter in the faadback loop. At the decoder, the x,, component is similarly split into the filtered part xi = 4 8 X, and its complimentary part -* 4 = x, -xi. The information signal is recovered as a difference m,(t) = u,, -xi, which means that the transmitted signal is passedthrough a channelwithout distortion, and so the originalmessagecanbemcovered. This method was applied to the Chua’s circuit. We used a low pass fourth order elliptical filter in the feedback loop, and the cutoff fiquency was adjusted to optimize the system performance. In Figure 10 we compare the maximum value of the module error for the situations with the presence and without the presenceof the filter. The message is repn?sentedas m(t) = O.Lwn(c.at), with w = 50 x ld rad/s.
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Lt.07 .
Figure10- (a) maximum value of the moduleerror (m, -m) asa fimctionof the channelcutofffrequency.. (b) maximum value of the module error (m, - m) (m, - m) asa fiulctionof theLPScutofftrequcncy. 4. Bibliography
[l] S. Hayes, C. Grebogi and E. Gtt, “Communicationwith Chaos”.Phys. Rev. Lett., vol. 70, pp. 3031-3034,1993. [2] E. Ott, Chaos in dynamical systems, Cambridgeuniversity Press,N.Y., 1994. [3] P. CvitanoviC,G. GunamtneandI. Procaccia,Phys.Rev. A 38,1503 (1988) [4] D. P. Lathmp and E. J. Kostelich, Phys. Rev. A 40,4028 (1989); [5] H. Fujisaka and T. Yamada, “Stability theory of synchronizui motion in coupledoscillator systems.”Progr. Theor. Phys..,vol. 69, pp. 32-47,1983. [q P. Ashwin, J. Buescu and I. Stewart, “From at&actor to chaotic saddle: a tale of transverseinstability.”Nonlineariry, vol. 9, pp. 703-737,1996. [7] A. V. Gppenheim, K. M. Cuomo, R J. Baron and A. E. Fredman, “Channel for Communicationwith Chaotic Signals.”in Chaotic, Fractal and Nonlinear Signal Processing, R A. Katz, Ed., AIP Press,1996,pp. 289-301. EkpGation
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[8] Carroll, T. L., “Synchronizing chaotic systems using filtered signals.”Whys.Rev. E,
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50, pp. 2580-2587,1994. [9] Carroll, T. L., “Communication with use of filtered, synchronized, chaotic signals.” IEEE Transactions on Circuits and SystemsI: Fundamental Theory and Applications, vol. 42, pp. 105-l 10, (1995). [lo] Carroll, T. L. and Pecora, L. M., “The effect of filtering on communication using synchronized chaotic ciqits.”
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