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SYNCHRONIZATION OF DRIVE-RESPONSE SYSTEMS ON CONDITION OF A LARGE MISMATCH OF PARAMETERS.
Andrey I. Panas
Institute of RadioEngineering and Electronics of the Russian Academy of Sciences, Mokhovaya St., 11, 103907 Moscow, Russia Email: chaos@mail.cplire.ru
Abstract - The method for synchronization of driveresponse systems on condition of a large mismatch of parameters is considered on example of the system based on modified Chua's circuit. The approach to information transmission using such method is proposed. The results of experiments are presented.

The aim of the report is to consider the possibility of synchronization drive-response system on condition a large mismatch of parameters (more than 1-2%). 2. SYNCHRONIZATION A generalized structure of the drive-response system is presented in Fig. 1.

1. INTRODUCTION Communication systems are a promising potential application of the dynamic chaos. Recently, a growing interest to this problem was observed, associated with the appearance of the papers [1-6] where different approaches to communications using chaotic signals were demonstrated. Experiments in low-frequency and RF bands confirmed the implementability of these approaches were performed [6-8]. However, application of the obtained results in practice proves to be difficult. At present, one of the main problems is the quality of the information signal retrieved in the receiver, i.e., SNR of analog and BER of digital signals. Operation of the most of known approaches is based on the chaotic synchronous response in the receiver [9]. Consequently, high-quality communication is possible only in the case of highquality chaotic synchronous response. Otherwise, desynchronization noise is observed in the output signal, which prevents us to retrieve the information [8]. In real conditions, this noise occurs due to various perturbing factors and is the main reason for the decrease of SNR and BER. By recognition of this fact, we change the direction of our investigations and now look for the ways to increase the robustness of the already known approaches [10] and for new approaches that would be stable to the perturbing factors. Parameters mismatch between drive (transmitter) and response (receiver) systems is one of above factors. The necessity of elements choice with accuracy 0.5-1 % (for qualitative information transmission) restricts a practical implementation of communication schemes based on chaotic synchronous response.

Figure 1: Drive-response system structure. Chaotic oscillator is the basis of the drive system. The response system has the same structure as the oscillator but with the disconnected feedback loop. The subtractor, whose output signal is the chaotic response R, is placed in the point of the feedback loop break. In the experiments, the communication channel between the drive and response systems was represented by a wire line. We used Chua's circuit as chaotic oscillator. Such circuit properties as a structure simplicity, a variety of chaotic modes, a decomposition possibility allowed us to use the Chua's circuit in different approaches for communications [1,3,6-8,10]. On the other hand, drive-response system based on Chua's circuit is extremely sensitive to parameters mismatch [6-8]. This circumstance make it attractive from point of view of the problem considered in the report.

Figure 2: Subsystems of the chaotic oscillator. The oscillator obtained by decomposition of the Chua's circuit was a main element in experiments on transmission of speech and music signals in low-

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frequency and RF bands [6-8]. Such oscillator consists of two subsystems (Fig. 2). The first subsystem (RLC) is a bandpass filter and the second subsystem (RCNR) is a low-frequency firstorder filter loaded at a nonlinear resistance NR with three-segment piecewise-linear voltage-current characteristics [11]. The use of the considered oscillator in the scheme in Fig. 1, when parameters of the drive and response systems are the same, leads to the next system of differential equations: C1· C1/dt) = (UC2 - UC1)/R1 - IN(UC1) (dU C2· C2/dt) = (UC1 - UC2)/R2 + IL (dU L·dIL/dt) = - UC2 ( C1· C1/dt) = (UC2 - U'C1)/R1 - IN(U'C1) (dU' C2· C2/dt) = (U'C1 - U'C2)/R2 + I'L (dU' L·dI'L/dt) = - U'C2, (

(1)

where (UC1, UC2, IL) and (U'C1, U'C2, I'L) - are voltages and currents of the drive and response systems, respectively. In this case, a response R is determined R=U
C2

As compared with Fig. 1, we can note on the following changes: 1. Negative feedback loop realized with the help of the amplifier (gain (<0)) is inserted into the response system. 2. The signal from the communication channel is amplified with the gain 1-. 3. The input of the response system is a sum of amplified signals from the communication channel (item 2) and feedback loop (item1). Thus, as against Fig. 1, system structure in Fig. 3 has additional block qualified to help to synchronize the input chaotic signal. This block is situated at the response system input. In this case, chaotic response R is determined as a difference between input and output signals of modified response system. If to take into account (1), such drive-response system will be described the following differential equations: C1· C1/dt) = (UC2 - UC1)/R1 - IN(UC1) (dU C2· C2/dt) = (UC1 - UC2)/R2 + IL (dU L·dIL/dt) = - UC2 ( (3) C1· C1/dt)=[(UC2· ) + U'C2· - U'C1)/R1 - IN(U'C1) (dU' (1 C2· C2/dt) = (U'C1 - U'C2)/R2 + I'L (dU' L·dI'L/dt) = - U'C2 ( In [8], the approach for possible appearance determination of the "on-off" intermittency was offered and testified. This approach is based on the eigenvalues analysis of a matrix of coefficients for equations describing the response system. In the case (1), the following eigenvalues were obtained: - for the phase space region corresponding to middle segments (Ga).of the equation (1) Ю1 = 0.3504, Ю2,3 = - 0.125 ± i· 0.9922, - for the phase space parts corresponding to outer segments (Gb) Ю1 = - 0.7007, Ю2,3 = - 0.125 ± i· 0.9922. The presence of a positive real part at one of the eigenvalues testifies that synchronization attractor is not stable to perturbing factors. This unstability is the reason of the "on-off" intermittency appearance. If parameters (1) and (3) are identical and = -2, application of the above approach to the system (3) leads to the following results: Ga: Ю1 = 0.1631, Ю2,3 = - 0.316 ± i· , 1.44 Gb: Ю1 = - 0.3187, Ю2,3 = - 0.316 ± i· . 1.44 We can see the system (3) is also not stable absolutely. However, decreasing of the factors a1 with a positive real part allows us to hope on weakening of the "onoff" intermittency effect. From a point of view of chaotic response quality, the weakening of the "on-off" intermittency leads to increase of SNR at the response system output (in average 20 dB). Such property

- U'

C2

(2).

As shown in [8], the effect of the perturbing factors, such as detuning parameter of the drive and response systems, external noise in the communication channel and other, leads to appearance of the "on-off" intermittency accompanied by response desynchronization. The reasons of the "on-off" intermittency appearance are discussed in [8]. Here is important to note that the system trajectory is stable with respect to small perturbations when goes through the phase space part corresponding to outer segments (Gb), and is not stable when it is in the phase region corresponding to middle segments (Ga). "On-off" intermittency is a undesirable phenomenon and can limit possibilities of considered oscillators application in practice. So far methods of the "on-off" intermittency decrease were an exact tuning of the drive-response system elements and maximum weakening of the effect of perturbing factors in the communication channel as far as possible (with accuracy 0.5-1 %). Here, other opportunity of "on-off" intermittency decrease is considered. The system structure realizing this opportunity is presented in Fig. 3.

Figure 3: Modified structure of drive-response system.

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makes possible to receive a synchronous response when detuning parameter of the drive and response systems is even more 5-10%. Experimental results are presented in Fig. 4 and Fig. 5.*

Figure 4: (a) - input (top trace) VC2 and output (bottom trace) V'C2 signals of the response system, (b) - VC2 versus V'C2 (bottom), top line corresponds to test identical signals, (c - input signal (top) and chaotic response R (bottom) of the receiver, scales are the same. Fig. 4 demonstrates input and output signals as well as chaotic response of the receiver (response) system when one of the controlling parameters (R1) differs from a similar parameter of the drive system by 6% and = 0 (feedback loop is absent). This case corresponds to the response system desynchronization and cannot be practically used for information transmission because of a synchronous response absence. Fig. 5 presents the case when the negative feedback loop ( = -2) is introduced into the response system. As can see in Fig. 5, the qualitative synchronous response is realized in the receiver system despite of large parameter detuning. As it was expected from the matrix eigenvalues analysis, bursts don't disappear absolutely. They become small and haven't practically influence on synchronization quality. * Experiments were performed for the following parameters of the drive and response systems: L = 40 mH, C1 = 15 nF, C2 = 100 nF, R1 = 1.8 kOhm, R2 = 2 kOhm. The nonlinear element NR was realized by means of the two opamp KR1401UD2b (analog LM324) [6-7]. The similar opamp played the role of buffer amplifiers.

Figure 5: (a) - input (top trace) VC2 and output (bottom trace) V'C2 signals of the modified response system, (b) - VC2 versus V'C2 (bottom), top line corresponds to test identical signals, (c - input signal (top) and chaotic response R (bottom) of the modified receiver, scales are the same. 3. APPROACH TO INFORMATION TRANSMISSION Thus, the additional feedback loop converts the response system to a good synchronizator. On the other hand, both simulation and experiments shows that the synchronization is broken when detuning parameter exceeds 10% or an external signal with the amplitude more 5% with respect to the amplitude of the chaotic signal is inserted into the drive system. This effect can be used, however, for digital or impulse-modulated information signal transmission. The example of system structure for transmission of such signals is presented in Fig. 6.

Figure 6: The example of system structure for information transmission. It unites the system structures in Fig. 1 and Fig. 3. External signal S (for instance, output signal of a harmonic generator) is fed at the drive system input through an electronic key. The amplitude and frequency of generator oscillations are chosen so that the signal in the communication channel was not unmasked by external signal presence and response

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system output signal was desynchronized enough. The operation of the electronic key is controlled by the transmitted digital (binary) or two-level impulsemodulated signal. Here we bear in mind that transmitted signals are lowfrequency with respect to chaotic signals of the drive system. The key is opened and passes through itself harmonic oscillations when information signal of one level is present at the key input. And vice versa, information signal of other level closes the key. Fig. 7 demonstrates this situation and presents signals in the communication channel and at the response system output (chaotic response) for binary information signals. As is seen in Fig., we can determine time intervals for output signal when signals of one or other level are present at the drive system input. It means that if to exploit corresponding signal processing of the output signal, we can recover the transmitted digital information. On the other hand, the signal form in the communication channel don't allows to undesirable observers to recognize the information.

This report is supported in part by a grant from the Russian Foundation for Fundamental Research (No. 97-01-00800). REFERENCES [1] Kocarev, L., Halle, K. S., Eckert, K., Chua, L. O. & Parlitz, U., "Experimental demonstration of secure communication via chaotic synchronization," Int. J. Bifurcation and Chaos, Vol. 2, No. 3, pp. 709-713, 1992. [2] Belsky, Yu. L. & Dmitriev, A. S., "Communication by means of dynamic chaos," Radiotekhnika I Elektronika, Vol. 38, No. 7, pp. 1310-1315, 1993, (Russian). [3] Dedieu, H., Kennedy, M. P. & Hasler, M., "Chaos Shift Keying: Modulation and Demodulation of a Chaotic Character Using Self-Synchronizing Chua' Circuits," IEEE Trans Circuits Syst., Vol. s 40, No. 10, pp. 634-642, 1993. [4] Volkovskii, A. R. & Rul' kov, N. V., "Synchronous chaotic response of a nonlinear oscillating system as the detection principle of chaos informational component", Pis' v GTF, Vol. 19, No. 3, pp. 71ma 75, 1993, (Russian). [5] Bohme, F., Feldman, U., Schwartz, W. & Bauer, A., "Information transmission by chaotizing," Proc. of Workshop NDES' Krakov, Poland, July 94, 1994, pp. 163-168, 1994. [6] Dmitriev, A. S., Panas, A. I., Starkov, S. O., "Experiments on transmission of speech and musical signals using dynamical chaos," Preprint IRE RAS N12(600), Moscow, Russia, pp. 1-42, 1994, (Russian). [7] Dmitriev, A. S., Panas, A. I. & Starkov, S. O., "Experiments on speech and music signals transmission using chaos," Int. J. Bifurcation and Chaos 5(4), pp. 1249-1254, 1995. [8]Dmitriev, A., Panas, A., Starkov, S. & Kuzmin, L., "Experiments on RF band communications using chaos", Int. J. Bifurcation and Chaos, Vol. 7, No. 11, 1997. [9]Pecora, L. M. & Carroll T. L., "Synchronization in chaotic systems", Phys. Rev. Letters, Vol. 64, pp. 821-824, 1990. [10]Dmitriev, A., Maximov, N., Panas, A. & Starkov, S., "Robustness of chaotic communications systems with nonlinear information mixing", Proc. of Workshop NDES' , Moscow, Russia, pp. 20997 216, 1997. [11]Madan, R., Chua' Circuit: A Paradigm for Chaos s (Singapore, World Scientific), 1993.

Figure 7: (a) - input (top trace) and output (bottom) signals of the drive system (transmitter), (b) - input signal (top) of the transmitter and output signal (bottom) of the receiver (response system). CONCLUSIONS Thus, it is shown that synchronization of the driveresponse system is possible even on condition of a large mismatch of system parameters. This property can be used for digital or impulse-modulated information signal transmission. ACKNOWLEDGMENTS

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