Russia

PHYSICAL REVIEW E FEBRUARY 1998VOLUME 57, NUMBER 2

Global reconstruction in application to multichannel communication

Vadim S. Anishchenko and Alexey N. PavlovNonlinear Dynamics Laboratory, Department of Physics, Saratov State University, Astrakhanskaya Street 83, 410026 Saratov,

~Received 14 August 1997!

We propose a method for restoration of time-dependent control parameters of dynamical system using thetechnique of global reconstruction. The technique presented is applied to multichannel confidential communi-cation by means of parameter modulation.@S1063-651X~98!11001-2#

PACS number~s!: 05.45.1b

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One of the methods for mathematical modeling of expemental data is the global reconstruction of dynamical stems~DS! @1,2#. The algorithm for global reconstruction hatwo stages. At the first stage the embedding dimensionN ischosen and the phase portrait of DS of one-dimensionaalization is restored@3#. At the second stage the general forof the mathematical model

dxW

dt5FW ~xW ,mW 0!, xWPRN, mW 0PRM, ~1!

is stated@the form of nonlinear vector functionFW (xW ,mW 0) isindicateda priori# and evolution equations are concretizei.e., the value of parameter vectormW 0 is found.

In practice we deal with scalar time series producedunknown DSs, so that it is impossible to indicate evenproximately the form of right-hand parts of Eq.~1!. Usuallynonlinear functions on the right-hand side of Eq.~1! are de-fined as a polynomial expansion and the expansion cocients are fitted numerically.

The task of global reconstruction on a scalar time serierelated to the class of ill-posed inverse problems becathere is no possibility to indicate uniquely thea priori formof nonlinearitiesF j (xW ,mW 0). Therefore, an arbitrary choice othe right-hand side of Eq.~1! is a serious shortcoming of thmethod of global reconstruction.

The situation changes dramatically if the explicit formnonlinear functionsF j (xW ,mW 0) is known and only the parameters have to be found. In this case the problem of recstruction is set forth and may be solved with the requiaccuracy. This may lead to interesting applications of recstruction problem. In particular, we further show that tmethod for reconstruction may be used successfully for cfidential communication.

Recently, the methods for secure communication baseexploiting broadband chaotic oscillations of some dynamchaos oscillator as a carrying~or masking! signal were sug-gested@4#. To extract information signal from the chaotbackground the phenomenon of chaotic synchronization@5#is usually used. Another possibility to protect transmittinformation may be realized by means of controlling cha@6#.

An important problem of simultaneous transfer of seveinformation signals in a single carrier has been studied in@7#.For this purpose several parameters of DS were modul

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by information signals and then extracted by means ofauto-synchronizationphenomena@7#.

In the present report we suggest an alternative methoddemodulation of encoded messages using the techniqureconstruction of DSs from a one-dimensional realizatiThe idea of the method is rather simple. Let a dynamichaotic oscillator, whose mathematical model is known@i.e.,the dynamics of the oscillator is described by the knononlinear system~1! with N phase variables andM constantparameters# be a transmitter. To transmitM messages simultaneously let us perform modulation of parametem i* 5m i

01m i(t), wherem i0, i 51,...,M , are the constant val

ues of control parameters of oscillator~1! andm i(t) are in-formation signals performing a relatively slow paramemodulation. The signal transmitted via the communicatchannel is now described by the system

dxW

dt5FW „xW ,mW 01mW ~ t !…. ~2!

Let the recipient know for sure the form of nonlinear vectofunction FW (xW ,mW 0). In other words, the recipient of information knows the mathematical model of oscillator~1! used forinformation transmission. The concrete values of the paraeter vectormW 0 may be unknown. The recipient of the infomation is able to uniquely solve the problem of reconstrtion of system~2! from one-dimensional realization, whiccan be measured at the input of the receiver. The solutiothe reconstruction problem means the extraction of modtion signalsm i(t).

In order to get the unique reconstruction ofm i(t), system~1! should be transformed into the form

dx1

dt5x2 ,

dx2

dt5x3 , ...,

dxn

dt5 f ~xW ,mW 0! ~3!

by means of variable substitution. If transformations frosystem~1! to system~3! do not exist, then one needs to knothe temporal dependences of more than one phase variabis indeed possible to realize this transformation for mawell-known models such as Lorenz and Ro¨ssler systems andChua’s circuit~see, e.g.,@2#!. Here we suppose that syste~1! permits transformation into the form~3!.

In radiophysical tasks of information transmission tmodulating signals are low frequency in comparison tocarrier. Using this circumstance, we study relatively slo

2455 © 1998 The American Physical Society

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2456 57BRIEF REPORTS

parameter modulation. Let us introduce a small time intert0 during which the values ofm i* 5m i

01m i(t) practically donot change. At the time scale of ordert0 the nonautonomousproperty of Eq.~2! may not be taken into account. As showbelow, the reconstruction technique works successfullyrelatively small values oft0 , i.e., for very short time seriesthe latter giving an opportunity to define the current paraeter valuesm i* for each t0 , i.e., to restore the modulatiosignals.

Consider the concrete examples. We useAnishchenko-Astakhov oscillator as a source of chaoticcillations @8#:

dx

dt5mx1y2xz,

dy

dt52x,

dz

dt52gz10.5g~x1uxu!x.

~4!

Bearing in mind that the signal from the oscillator beireceived by the recipient is the one-dimensional realizay(t), we transform Eq.~4! by means of variable substitutionY5y, Z52x, andX52mx2y1xz into the form

dY

dt5Z,

dZ

dt5X,

dX

dt5 f ~X,Y,Z!, ~5!

f ~X,Y,Z!5X~X1Y!

Z1~mg21!Z2g~X1Y!

10.5g~ uZu2Z!Z2. ~6!

The recipient of the information may define the valuesZ,X,dX/dt in system~5! at discrete time momentsiDt em-ploying numerical differentiation of the time seriesy( iDt)5Y( iDt), where Dt is sampling step, i 51,...,K;K5@ t0 /Dt#, @ t0/Dt# being the integer part of the quantitt0/Dt. Therefore, the problem of defining the current valuof m and g is reduced to the solution ofK algebraic equa-tions ~6! with two unknown quantitiesm,g by means of aleast-squares algorithm.

Consider the simplest case. Let the value ofm be fixed(m51.5) and the functionf contain only one unknownquantity g. We choose the stepwise temporal dependeobtained by the scanning of Einstein’s portrait@Fig. 1~a!# asa law of modulation. The range of parameter variationgP@0.15,0.25#) was split into 256 subranges, each corsponding to the shade of the black-and-white image. Tothe used method of demodulation for its robustness we ada normally distributed random value with variance 1024 tog(t). Figures 1~b! and 1~c! show the signal in the communcation channel and the reconstructed information signal,spectively.

To prove the possibility of simultaneous transmissionseveral messages under the noise influence, two paramwere modulated simultaneously as follows:m was modu-lated by a chaotic signal from the Ro¨ssler system, where thtime renormalization was performed to provide a slower pcess of parameter variation compared to oscillations in~4!; g was modulated by a harmonic signal. The resultstained are presented in Fig. 2 and testify to the reliabilitythe method in the case of simultaneous transmission ofindependent messages in the presence of noise.

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The following conclusions can be derived from the resupresented. First, the proposed technique does not suptheoretical limitations to the number of parameters bemodulated simultaneously, i.e., it allows one to use thegime of simultaneous transmission ofM information signals.However, a small value of sampling stepDt is necessary forthe computation of parameter values with good enoughcuracy~we usedDt50.025! as well as a rather high precsion of conversion of the carrier into digital form by aanalog-to-digital converter~ADC!. ~Of course, the higher theADC resolution, the smaller the error of the parameter vacomputation.! Second, if while making transformations fromEq. ~1! to Eq.~3! one needs to compute the derivatives of tparameters, the requirement of a slow modulation becoessential@as in the case ofm(t) in Eq. ~4!#. In this case thecondition of a slow parameter modulation allows one to nglect dm/dt in comparison todx/dt. However, for the pa-rameterg(t) the requirement of a slow modulation is not sessential. Whenm is fixed Eq.~6! is linear with respect tog.

FIG. 1. ~a! Initial portrait scanned with resolution 5003464,~b!the signal from the chaotic oscillator under modulation, and~c!restored portrait obtained by global reconstruction technique.

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In this case the value ofg may be estimated using the knowedge of phase variables and their derivatives only at a simoment in time. Since the solution of such an equation dnot require much computation, one may realize demodtion of the information signal in real time in fact.

FIG. 2. ~a! and ~b! Information signals performing the modulation of the parametersm andg, respectively,~c! signal in the chan-nel of communication, and~d! and ~e! restored signals. To test thmethod from the viewpoint of its stability with respect to noise wadded to both signals~a! and~b! noise with varianceD51024. Wealso added noiseD51025 into the first equation of system~4!. t ismeasured in seconds.

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To realize the proposed method for signal demodulatin practice, the recipient of the information must possesspecialized processor or computer and an ADC. We suppthat such a method for confidential communication may fiits effective application in the exchange of informationshort enough distances using a cable network, so that nwould not be created while broadcasting.

In the present paper we did not compare the descrimethod for demodulation with the ones suggested in@7#,which may be a topic of separate research. We supposein some situations the technique for global reconstructoffers the possibility of using a larger frequency rangemodulation signals than the method of autosynchronizati

The results obtained present an alternative applicationthe method for dynamical model reconstruction. In this pathe method for confidential communication is illustrated fthe modified oscillator with inertial nonlinearity as an eample. Similar results were obtained for the Lorenz aRossler systems, which is why we may state with confidenthat the workability of the method does not depend onchoice of the source of chaotic oscillations, whose maematical model can be transformed to the form~3!.

The authors express their sincere gratitude to N. B. Json, A. G. Balanov, A. N. Silchenko, and A. B. Neiman ftheir interest to the work, fruitful discussions, and helpmanuscript preparation. The work was partially supportedthe State Committee on Higher education~Grant No. 95-0-8.3-66!.

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