Experimental Realization of Binary Signals Transmission Based on Synchronized Lorenz circuits

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1 Experimental Realization of Binary Signals Transmission Based on Synchronized Lorenz circuits Cornelio PosadasCastillo. Faculty of Engineering Mechanic and Electrical (F.I.M.E.), Nuevo León Autonomous University (U.A.N.L.), Pedro de Alba s/n Cd. Universitaria San Nicolas de los Garza Nuevo León, MÉXICO. César CruzHernández and Ricardo NuñezPérez. Electronics and Telecommunications Department, Scienti c Research and Advanced Studies of Ensenada (CICESE), Km. 0, Carretera TijuanaEnsenada, Ensenada, B. C., 0, MÉXICO. January 0, 00 Abstract In this paper, an experimental study of practical realization to transmit binary signals using chaos is presented. In particular, we use a Generalized Hamiltonian systems approach to synchronize two unidirectionally coupled Lorenz circuits. Experimental results are in close accordance with the simulations. KeyWords: Synchronization, chaotic systems, Lorenz system, passivity based observers, private communication. Introduction Acknowledgments Since Pecora and Carroll reported the discovery of synchronized chaos in [Pecora & Carroll, 990], the possibility of secure communications using chaos This work was supported by the CONACYT, México under Research Grant No. A. and by F.I.M.E. of the U.A.N.L. Correspondence to: César CruzHernández, CICESE, Electronics & Telecom. Dept., P.O. Box 9, San Diego, CA 99, USA, Phone: , Fax: , ccruz@cicese.mx has received considerable attention. Synchronization of chaotic systems has in recent years become an area of active research. Di erent approaches are proposed and being pursued (see, e.g., [Chua et al., 99; Kocarev et al., 99; Ogorzalek, 99; Wu & Chua, 99; Ding & Ott, 99; Feldmann, et al., 99; Nijmeijer & Mareels, 99; Special Issue, 99, 000; Cruz & Nijmeijer, 000; SiraRamírez & Cruz, 000; 00; Pikovsky et al., 00] and the references therein). Data encryption using chaotic dynamics was reported in the early 990 s as a new approach for signal encoding that di ers from the conventional methods

2 that use numerical algorithms as the encryption key. Thereby, the synchronization of chaotic systems plays an important role in chaotic communications. In particular, several techniques, such as chaotic masking [Cuomo et al., 99], chaotic switching [Parlitz et al., 99; Cuomo et al., 99; Dedieu et al., 99] and chaotic parameter modulation [Yang & Chua, 99] have been developed. Our objective in this paper consists in describing an experimental study of practical realization to transmit binary signals based on synchronized chaotic systems. In particular, we use a Generalized Hamiltonian systems approach developed in [Sira Ramírez & Cruz, 000; 00] to synchronize two unidirectionally coupled Lorenz circuits. We enumerate several advantages over the existing methods: ² it enables synchronization be achieved in a systematic way and clari es the issue of deciding on the nature of the output signal to be transmitted; ² it can be successfully applied to several wellknown chaotic oscillators; ² it does not require the computation of any Lyapunov exponent; ² it does not require initial conditions belonging to the same basin of attraction. This paper is organized as follow. In Section, we present the system description to transmit binary information by chaotic switching. In Section, we obtain the synchronization of two Lorenz systems through a Generalized Hamiltonian systems approach. In Section, we present the stability analysis related to the synchronization process. In Section 5, we show the physical system implementation and experimental results: 5. synchronization and 5. binary communication. Finally, in Section, we give some concluding remarks. System description In this section, we describe the communication system based on synchronized chaos and its operating principles. Figure shows a block diagram to transmit binary messages by chaotic switc We will use the Lorenz system as chaos generator. The state equations are given by [Lorenz, 9] _x = ¾ (x x ) ; () _x = rx x x x ; _x = x x bx ; with ¾ =0, r =,andb == thelorenzsystem exhibits chaotic behavior. m(t) p Maestro p Transmisor Esclavo x ( ) ξ ( t) t p Receptor Figure : Block diagram to transmit binary messages by chaotic switching. For communication purposes here we apply the Hamiltonian synchronization of Lorenz circuits to chaotic switching. In this technique, the message m (t) is supposed to be a binary message, and is used to modulate one or more parameters of the (switching) transmitter, i.e. m (t) controls a switch whose action changes the parameter values of the transmitter. Thus, according to the value of m (t) at any given time t, the transmitter has either the parameter set value p or the parameter set value p 0. At the receiver, m (t) is decoded by using the synchronization error to decide whether the received message corresponds to one parameter value, or the other (it can be interpreted as an one or zero). In the next section we describe the Generalized Hamiltonian systems approach to synchronize two unidirectionally coupled Lorenz circuits. Synchronization is thus between the transmitter and the dynamics with the receiver being givenbyanobserver. e ( t ) Synchronization of Lorenz system Consider the following ndimensional autonomous system _x = f (x) ; x R n ; () which represents a circuit exhibiting a chaotic behavior. Following the approach provided in [Sira Ramírez & Cruz, 00], many physical systems described by Eq. () can be written in the following Generalized Hamiltonian Canonical form, _x = J (x) S (x) F(x); () where H (x) denotes a smooth energy function which is globally positive de nite in R n. The column gradient vector of H, denoted by =, isassumedto exist everywhere. We frequently use quadratic energy function H (x) ==x T M x with M being a, _

3 constant, symmetric positive de nite matrix. In such case, = = M x. Thesquarematrices,J (x) and S (x) satisfy, for all x R n, the following properties, which clearly depict the energy managing structure of the system, J (x)j T (x) =0and S (x) =S T (x). The vector eld J (x) = exhibits the conservative part of the system and it is also referred to as the workless part, or workless forces of the system; and S (x) depicting the working or nonconservative part of the system. For certain systems, S (x) is negative de nite or negative semide nite. In such cases,the vector eld is addressed to as the dissipative part of the system. If, on the other hand, S (x) is positive de nite, positive semide nite, or inde nite, it clearly represents, respectively, the global, semiglobal and local destabilizing part of the system. In the last case, we can always (although nonuniquely) descompose such an inde nite symmetric matrix into the sum of a symmetric negative semide nite matrix R (x) and a symmetric positive semide nite matrix N (x). And where F(x) represents a locally destabilizing vector eld. We consider a special class of Generalized Hamiltonian systems with destabilizing vector eld and linear output map, y, givenby _x = J (y) (I S) F(y); x Rn ; y = C ; y Rm ; () where S is a constant symmetric matrix, not necessarily of de nite sign. The matrix I is a constant skew symmetric matrix. The vector variable y is referred to as the system output. The matrix C is a constant matrix. We denote the estimate of the state vector x by», and consider the Hamiltonian energy function H(») to be the particularization of H in terms of». Similarly, we denote by the estimated output, computed in terms of the estimated state». The gradient vector (»)= is, naturally, of the form M». A dynamic nonlinear state observer for the system () is readily obtained as _» = J (y) (I S) F(y)K(y ); = C ; (5) where K is a constant vector, known as the observer gain. The state estimation error, de ned as e = x» and the output estimation error, de ned as e y = y, are governed by _e = J (I S ; e Rn ; e y = ; e y R m ; () where the vector, =@e actually stands, with some abuse of notation, for the gradient vector of the modi ed energy function, (e)=@e = = = = M(x») = Me. Below, we set, when needed, I S = W. We say that the receiver system (5) synchronizes with the transmitter system (), if e (t)! 0 as t!. Lorenz system. To synchronize the Lorenz circuit using the Generalized Hamiltonian systems approach we the Eq. (). The set of state equations describing Lorenz system in Hamiltonian Canonical form with a destabilizing eld (master) isgivenby _x _x 5 = 0 ¾ 0 ¾ 0 x 5 () _x 0 x 0 ¾ ¾ 0 ¾ rx 5 ; 0 taking as the Hamiltonian energy function the scalar function H (x) = ¾ x x x : () The destabilizing vector eld evidently calls for y = x = ¾ 0 0 = to be used as the output of the master (). The matrices C, S and I, aregiven by C = ¾ 0 0 ; S = ¾ ¾ 0 ¾ 0 5 ; I = 0 ¾ 0 ¾ : Thepairofmatrices(C; S) already constitutes a pair of detectable, but nonobservable, matrices. Even though the addition of the matrix I to S does not improve the lack of observability, the pair (C; W) = (C; S I) remains, nevertheless, detectable. In this case, the dissipative structure of the system is fully damped due to the negative de niteness of the matrix S. Then, there is no need for an output estimation error injection for complementing, or enhancing, the system s natural dissipative structure. The slave is designed as _» _» 5 = 0 ¾ 0 ¾ 0 x 5 _» 0 x 0

4 ¾ ¾ 0 ¾ rx 5 ; 0 (9) and the synchronization error is therefore governed by the globally asymptotically stable system _e _e 5 = 0 ¾ 0 ¾ 0 x 5 0 x 0 ¾ ¾ 0 ¾ 0 : If the negative real part of the observable eigenvalues, related to the constant dissipation structure matrix of the error dynamics, must be enhanced, one can still use the above observer but now including an output reconstruction error injection term. The resulting observer (slave) is given by _» _» _» 5 = 0 ¾ 0 ¾ 0 y 5 0 y 0 ¾ ¾ 0 ¾ ry 5 K K 5 e : 0 K (0) The asymptotically stable reconstruction error dynamics is then governed by _e 0 _e 5= ¾( K ) ¾K ¾( K ) ¾K y 5 _e () y 0 ¾(¾ K ) ¾( K ) ¾K ¾( K ) 0 5 : 0 b Onemaynowspecifythevaluesofk i, i =; ; in order to guarantee a faster asymptotic stability to zero of the state reconstruction error trajectories. Synchronization stability analysis In this section, we examine the stability of the synchronization error () between the Lorenz system () and observer (0). A necessary and su cient condition for global asymptotic stability to zero of the estimation error is given by the following theorem. Theorem (SiraRamírez and Cruz, 00) The state x of the nonlinear system () can be globally, exponentially, asymptotically estimeted, by the state» of the observer (0) if and only if there exists a constant matrix K such that the symmetric matrix [W KC][W KC] T = [S KC][S KC] T = S T KCC T K is negative de nite. 5 Physical system implementation and experimental results In this section, we describe the physical implementation of the synchronization and communication schemes. Figure shows the analog circuit implementation of the Lorenz equations () 0 0 R R 9 9 U R9 AD 0 K R AD 0 K R R 5 V 0 K R K R5 M R Figure : Analog circuit implementation of the Lorenz system. 5. Synchronization The experimental setup of Generalized Hamiltonian systems approach to synchronize the Lorenz circuits is shown in Figure (). The basic idea of this scheme is as follows: a voltage 5 Vdc:is injected into the Lorenz circuit (transmitting system) which modi es 00 K XR 00 K XR 50 K X 5 V M 5 V M 0. UF C U 0 K R0 5 V 0. UF C 0 K R0 0 K R U 0. UF C U 0 M R 5 V M R 5 V U M XR U5 M XR U x x x

The signal error e = x» ; is injected into the block of U. The choice of gains k =50and k = k =0was used. Figure shows ξ U U ξ 5.

5 K RK K RK K RK 5 V U9 U0. K R0 U U 00 K 00 K 00 K. K R. K R 5 V 5 V 5 V. K R 0 0 R R 9 9 R U R9 0 K R 0 K R 5 V 0 K R K R5 M R 00 K XR 00 K XR 50 K X 5 V M 5 V M 0. UF C 0 K R0 5 V. K 0. UF C 0 K R0. K R 0. UF C 0 K R. K R9 0 M R 5 V M R 5 V M XR M XR the voltage x. This voltage x is then used as driving signal on the Lorenz circuit (receiving system) across I:C: AD. The signal error e = x» ; is injected into the block of U. The choice of gains k =50and k = k =0was used. Figure shows ξ U U ξ 5. Binary communication The experimental setup to transmit binary signals by chaotic switching is shown in Figure 5. The basic idea of this scheme is as follows: We generate two chaotic attractors of the Lorenz circuit (Eq. ()), utilizing a square signal of Vdc:and Vdc:With these values, the Lorenz circuit exhibits two di erent but qualitatively similar chaotic attractors (to encoding \" or \0" ). The Figure shows the transmission of K x AD U U5 ξ R R R U 5 V R9 M R 00 K XR 0. UF C U 0 K 5 V R0 U x Transmitted signal 0 M R K 0 AD 5 V 0. UF C M XR U5 x M U AD U U ξ message m(t) p 9 0 K R 0 K R 00 K XR 0 K 5 V R0 K M R 0 AD 5 V 0. UF C M XR U x M U 0 K 5 V R 9 K Figure : Experimental setup of Generalized Hamiltonian systems approach to synchronize the Lorenz circuit. 0 K R R5 50 K X the quality of synchronization between the transmitter and receiver. The horizontal axis corresponds to x of the transmitter and the vertical axis to» of the receiver. Figure 5: Experimental setup to transmit binary signals by chaotic switching. an alternating sequence of and 0 (transmitted signal x ). The Figure, top of gure: shows binary Figure : Transmitted signal x. signal to be transmitted. Bottom of gure shows the original information is recovered by synchronization error detection at the receiver. Figure : Synchronization between the master and slave: x of the master versus» of the slave. Concluding remarks In this paper, we have developed an experimental study of practical realization to transmit binary sig 5

[5] Ding, M. y Ott, E. (99) Enhancing synchronism of chaotic systems, Phys. Rev. E 9(), 959 p. [] Feldmann, U. Communication by chaotic signals: the inverse system approach, Int. J.

6 [5] Ding, M. y Ott, E. (99) Enhancing synchronism of chaotic systems, Phys. Rev. E 9(), 959 p. [] Feldmann, U. Communication by chaotic signals: the inverse system approach, Int. J. Circuit Theory and Applications,, pp. 5559, 99. [] Kocarev, L., Halle, K.S., Eckert, K., Chua, L.O.y Parlitz, U. (99) Int. J. Bifurc. Chaos (), 09p. Figure : Top of gure: Binary signal to be transmitted. Bottom of gure: Synchronization error detection e = x». nals using synchronized chaos. Synchronization of two Lorenz circuits was obtained from the perspective of Generalized Hamiltonian systems including dissipation and destabilizing terms. The approach allows to give a simple design procedure for the receiver system and clari es the issue of deciding on the nature of the output signal to be transmitted. We have shown that the experimental results are in close accordance with the simulations. In a forthcoming article we will be concerned with a physical implementation of the method combining with conventional cryptographic schemes to enhance the security of lowdimensional chaosbased secure communication systems. References [] Chua, L.O., Kocarev, Lj., Eckert, K. & Itoh, M. [99] Experimental chaos synchronization in Chua s circuit, Int. J. Bifurc. Chaos. (), 050. [] Cruz, C. y Nijmeijer, H. (000) Synchronization through ltering, Int. J. Bifurc. Chaos 0(), 5. [] Cuomo, K.M., Oppenheim, A.V. y Strogratz, S.H. (99) Synchronization of Lorenzbased chaotic circuits with applications to communications, IEEE Trans. Circuits Syst. II 0(0), p. [] Dedieu, H., Kennedy, M. y Hasler, M. (99) Chaotic shift keying: Modulation and demodulation of a chaotic carrier using selfsynchronizing Chua s circuits, IEEE Trans. Circuits Syst. II 0(0), p. [] Lorenz, E.N. (9) Deterministic nonperiodic ow, J. Atmospheric Sci. 0, 0p. [9] Nijmeijer, H. y Mareels, I.M.Y. 99 An observer looks at synchronization, IEEE Trans. Circuits Syst. I (0), 90p. [0] Ogorzalek, M.J. (99) Taming chaospart I: Synchronization, IEEE Trans. Circuit Syst. I 0(0), 999p. [] Parlitz, U., Chua, L.O., Kocarev, Lj., Halle, K.S. y Shang, A. (99) Transmission of digital signals by chaotic synchronization, Int. J. Bifurc. Chaos (), 99p. [] Pecora, L.M. y Carroll, T.L. (990) Synchronization in chaotic systems, Phys. Rev. Lett. (), p. [] Pikovsky,A.,Rosenblum,M.yKurths,J.(00) Synchronization: A Universal Concept in Nonlinear Sciences, Cambridge University Press. [] SiraRamírez, H. y Cruz, C. (00) Synchronization of chaotic systems: A generalized Hamiltonian systems approach, Int. J. Bifurc. Chaos (5), 95. Y en Procs. of American Control Conference (ACC 000), al0 de junio del 000, Chicago, Illinois, USA., 9 p. [5] Special Issue (99) on Chaos synchronization and control: Theory and applications, IEEE Trans. Circuits Syst. I (0). [] Special Issue (000) on Control and Synchronization of chaos, Int. J. Bifurc. Chaos 0( ). [] Yang, T. y Chua, L.O. (99) Secure communication via chaotic parameter modulation, IEEE Trans. Circuits Syst. I (9), 9p. [] Wu, C.W. & Chua, L.O. [99] A simple way to synchronize chaotic systems with applications to secure communication systems, Int. J. of Bifurc. Chaos, (), 9.

Encrypter Information Software Using Chaotic Generators

Vol:, No:6, 009 Encrypter Information Software Using Chaotic Generators Cardoza-Avendaño L., López-Gutiérrez R.M., Inzunza-González E., Cruz-Hernández C., García-Guerrero E., Spirin V., and Serrano H.