INTERNATIONAL JOURNAL OF CIRCUIT THEORY AND APPLICATIONS Int. J. Circ. ¹heor. Appl., 27: 543}553 (1999) SYNCHRONIZATION OF HIGH-ORDER OSCILLATORS BY OBSERVER DESIGN WITH APPLICATION TO HYPERCHAOS-BASED CRYPTOGRAPHY GIUSEPPE GRASSI1,s AND SAVERIO MASCOLO2* 1 Dipartimento di Matematica, Universita% di Lecce, 73100 Lecce, Italy 2 Dipartimento di Elettrotecnica ed Elettronica, Politecnico di Bari, 70125 Bari, Italy SUMMARY The aim of this paper is to make a contribution in the context of both hyperchaos synchronization and cryptography. The approach illustrated herein, which exploits the concept of observer from system theory, is at "rst applied for synchronizing via a scalar signal two recent examples of high-order oscillators, characterized by several positive Lyapunov exponents. Successively, an example of hyperchaos-based cryptography is illustrated. The advantages of the proposed tool are discussed in detail. In particular, the utilization of both synchronization of complex dynamics and cryptography seems to make a contribution to the development of communication systems with higher security. Copyright ( 1999 John Wiley & Sons, Ltd. KEY WORDS: hyperchaotic circuits; applied cryptography; observer-based synchronization 1. INTRODUCTION The goal of synchronization is to design a coupling between two chaotic systems, called drive system and response system, so that their dynamics become identical after a transient time.1 Recently, the synchronization of hyperchaotic systems (i.e. systems with more than one positive Lyapunov exponent) has become a "eld of active research.2}6 For instance, in Reference 3 hyperchaos synchronization is achieved by exploiting linear and non-linear feedback functions, whereas in Reference 4 a linear combination of the original state variables is used to synchronize hyperchaos in RoK ssler's systems. The synchronization issue has generated a great interest in transmitting information from one location to another using chaotic signals.7}9 In particular, di!erent methods have been developed in order to hide the contents of a message using chaotic waveforms. However, the attacks illustrated in References [10}12] have shown that most of the proposed methods, based on low-dimensional chaotic systems, are not secure or have a low security. The aim of this paper is to make a contribution in the context of both hyperchaos synchronization and cryptography. The paper is organized as follows. By exploiting the concept of observer,5 in Section 2 the issue of hyperchaos synchronization is brie#y illustrated, whereas in Section 3 the technique is applied to synchronize two recent examples of high-order oscillators,13, 14 characterized by several positive Lyapunov exponents. The approach can be easily generalized for synchronizing via a scalar signal two wide classes of high-order oscillators with complex dynamics. In Section 4 an example of hyperchaos-based cryptography is illustrated, which combines conventional cryptographic methods and synchronization of hyperchaotic *Correspondence to: Saverio Mascolo, Dipartimento di Elettrotecnica ed Elettronica, Politecnico di Bari, Via Orabona 4, 70125 Bari, Italy. sand Department of Innovation Engineering, University of Lecce, 73100 Lecce, Italy CCC 0098}9886/99/060543}11$17.50 Copyright ( 1999 John Wiley & Sons, Ltd. Received August 1998 Revised February 1999 544 G. GRASSI AND S. MASCOLO systems. Finally, in Section 5 the advantages of the proposed tool are discussed in detail. It is worth noting that the increased complexity of the transmitted signal as well as the adoption of high-order oscillators enable to weaken the low-security objections against low-dimensional chaos-based schemes.15 2. HYPERCHAOS SYNCHRONIZATION BY OBSERVER DESIGN In this paper, the attention will be focused on the following class of hyperchaotic systems: x5 (t)"Ax(t)#b f (x(t))#c (1) y(t)"h(x(t)) (2) where x3Rn is the state, y3R is the scalar output, f : RnPR and h : RnPR are the non-linear vector "elds whereas A3Rn]n, b3Rn]1 and c3Rn]1. The synchronization issue is closely related to the observer problem in control theory.5 Namely, given the system description (1)}(2), the objective is to reconstruct the state x(t) starting from measurements of the output y(t). An approach in solving the observer problem is to use as a response system a copy of the drive one modi"ed with a term depending on the di!erence between the received signal and its prediction derived from the observer. Taking into account the results in Reference 5, if the scalar synchronizing signal (2) is chosen as y(t)"f (x(t))#kx(t) (3) with k"[k , k , 2 , k ]3R1]n and the response system is 1 2 n xQL (t)"Ax( (t)#b f (x( (t))#c#b (y(t)!yL (t)) (4) yL (t)"f (x( (t))#kx( (t) (5) where (5) is the prediction of the received signal derived from the observer, it can be easily shown that the following linear time-invariant error system is obtained: e5 (t)"Ae(t)!bke(t)"Ae(t)#bu(t) (6) where e(t)"(xL (t)!x(t)) represents the synchronization error whereas u(t)"!ke(t) plays the role of a state feedback. In order to make the response system (4) an observer for the state of the drive system (1), the error system (6) has to be globally asymptotically stabilized at the origin. By exploiting results from system theory,16 it can be stated that if the controllability matrix xb Ab A2b 2 An!1by of (6) is full rank, all the error system eigenvalues are controllable, i.e. they can be placed anywhere by proper state feedback u(t)"!ke(t). On the other hand, if the controllability matrix is not full rank, some eigenvalues are uncontrollable, i.e. they are not a!ected by the introduction of any state feedback. In this case, system (6) is globally asymptotically stabilized at the origin by suitable k provided that its uncontrollable eigenvalues have negative real parts.5 3. SYNCHRONIZATION OF HIGH-ORDER OSCILLATORS 3.1. Synchronization of seventh-order oscillators The main problem in designing and building chaotic oscillators for applications such as chaos-based communications is the reproducibility of the circuits.13 Because of the manufacturing spread of the element parameters, the analogue circuits commonly match to within an accuracy of about 1%. The most unreliable Copyright ( 1999 John Wiley & Sons, Ltd. Int. J. Circ. ¹heor. Appl., 27: 543}553 (1999) 545 SYNCHRONIZATION OF HIGH-ORDER OSCILLATORS devices of the chaotic circuits are the non-linear elements, such as diodes and transistors. These elements exhibit large manufacturing spread and high-temperature sensitivity of their parameters. In order to build highly reproducible and temperature stable circuits, a family of oscillators has been proposed in Reference 13, which contains no discrete diodes or transistors. The circuit considered in this section is a seventh-order oscillator. It includes an opamp, an RC phase shift circuit, two bu!ers, sixth-order LC resonance loops and a voltage comparator used as a non-linear device. By taking the parameter values reported in Reference 13, the circuit dynamics can be written in dimensionless form as xR xR 1 xR 2 xR 3 xR 4 xR xR 5 6 7 " 0)5 !1 0 !1 0 !1 1 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 !1 0 0 0 0 0 0 0 0 1 0 1 0 !1 0 !1 0 0 0 0 0 0 !1/0)7 0 0 0 !0)5 x 0 1 0 x 2 0 x 3 x # 0 4 0 x 5 0 x 6 200/7 x 7 H(x !1) 1 (7) where H is the Heaviside function describing the comparator behaviour, that is, H(u)"0 if u(0 and H(u)"1 if u*0. The hyperchaotic behaviour of the oscillator has been con"rmed by both numerical simulations and experimental results.13 In particular, system (7) is characterized by four positive Lyapunov exponents. The projection of the hyperchaotic attractor on the plane (x , x ) is reported in Figure 1. By 1 2 considering the scalar signal: 7 y(t)"H(x !1)# + k x 1 i i i/1 (8) and by applying the proposed method, the response system is xLQ 0)5 !1 0 !1 0 !1 !0)5 1 xLQ 1 0 0 0 0 0 0 2 xLQ 0 0 0 1 0 1 0 3 xLQ " 1 0 !1 0 0 0 0 4 xLQ 0 0 0 0 0 1 0 5 xLQ 1 0 !1 0 !1 0 0 6 xLQ 0 0 0 0 0 0 !1/0)7 7 xL 0 1 xL 0 2 0 xL 3 xL # 0 4 0 xL 5 0 xL 6 200/7 xL 7 0 0 0 H(xL !1)# 1 0 (y!yL ) 0 0 200/7 (9) where 7 yL (t)"H(xL !1)# + k xL 1 i i i/1 Copyright ( 1999 John Wiley & Sons, Ltd. (10) Int. J. Circ. ¹heor. Appl., 27: 543}553 (1999) 546 G. GRASSI AND S. MASCOLO Figure 2. Synchronization between the state variables x and 2 xL of systems (7) and (9), respectively 2 Figure 1. Projection of the hyperchaotic attractor of system (7) on the plane (x , x ) 1 2 is the prediction of (8) derived from (9). Since the controllability matrix of the error system A eR 1 eR 2 eR 3 eR " 4 eR 5 eR 6 eR 7 0)5 !1 0 !1 0 !1 !0)5 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 !1 0 0 0 0 ! 0 0 0 0 0 1 0 0 1 0 !1 0 !1 0 0 0 0 0 0 0 !1/0)7 0 0 0 200/7 [k k k k k k k ] 1 2 3 4 5 6 7 B e e e e e e e 1 2 3 4 5 6 7 is full rank, its eigenvalues can be moved anywhere. For instance, they can be placed in !1 for k" [!1)3125 0)4550 0)6300 !0)2450 !0)2100 0)3850 0)2125]. It follows that e(t)"(xL (t)!x(t))P0 as tPR for any initial condition e(0), that is, given the initial condition x(0), the dynamics of systems (7) and (9) are identical for each initial condition xL (0). This means that system (9) is a global observer5 of system (7). The synchronization between the selected state variables x and xL is shown in Figure 2. 2 2 3.2. Synchronization of eighth-order oscillators A novel two-terminal device exhibiting hysteretic current voltage characteristic has been recently proposed in Reference 17. It is a combined circuit including a negative impedance converter and the Schmitt trigger. The device has been called chaos-diode since the only external element needed to generate chaos is a second-order resonance loop connected in parallel to the diode.17 Further results have been presented in Reference 14, where hyperchaos is generated by means of extending the external loop from the second-order Copyright ( 1999 John Wiley & Sons, Ltd. Int. J. Circ. ¹heor. Appl., 27: 543}553 (1999) 547 SYNCHRONIZATION OF HIGH-ORDER OSCILLATORS Figure 3. Projection of the hyperchaotic attractor of system (11) on the plane (x , x ) 1 2 circuit to higher-order ones. The oscillator considered in this section is a eighth-order circuit. Simulation and experimental results have con"rmed the hyperchaotic behaviour of this oscillator, which is characterized by six positive Lyapunov exponents.14 The projection of the hyperchaotic attractor on the plane (x , x ) 1 2 is reported in Figure 3. By considering the parameters reported in Reference 14, the state equations are given by xR 1 xR 2 xR 3 xR 4 " xR 5 xR 6 xR 7 xR 8 0)6 !1 0 !1 0 !1 0 !1 1 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 1 0 !1 0 0 0 0 0 0 0 0 0 1 0 1 1 0 !1 0 !1 0 0 0 0 0 0 0 0 1 1 0 !1 0 !1 0 0 0 0 0 !1 x !4.2 1 0 x 2 0 x 3 0 x 4 # 0 x 5 0 x 6 0 x 7 0 x 8 s (x , s ) j 1 j~1 (11) where s "H(x !1#0)9s ), j"1, 2, 2, is the discrete state function that describes the hysteretic action j 1 j~1 of the Schmitt trigger belonging to the chaos-diode. This function depends on the variable x as well as on the 1 previous state s and alternates between 1 and 0, since H is the Heaviside function. By taking the j~1 synchronizing signal: 8 y(t)"s (x , s )# + k x j 1 j~1 i i i/1 Copyright ( 1999 John Wiley & Sons, Ltd. (12) Int. J. Circ. ¹heor. Appl., 27: 543}553 (1999) 548 G. GRASSI AND S. MASCOLO Figure 4. Synchronization between the variables x (t) and xL (t) of systems (11) and (13), respectively 1 1 and by applying the suggested technique the response system is xLQ 0.6 !1 0 !1 0 !1 0 !1 1 xLQ 1 0 0 0 0 0 0 0 2 xLQ 0 0 0 1 0 1 0 1 3 xLQ 1 0 !1 0 0 0 0 0 4 " xLQ 0 0 0 0 0 1 0 1 5 xLQ 1 0 !1 0 !1 0 0 0 6 xLQ 0 0 0 0 0 0 0 1 7 xLQ 1 0 !1 0 !1 0 !1 0 8 xL !4.2 !4.2 1 0 0 xL 2 0 0 xL 3 0 0 xL 4 # s (xL , s )# (y!yL ) j 1 j~1 0 0 xL 5 0 0 xL 6 0 0 xL 7 0 0 xL 8 (13) where 8 yL (t)"s (xL , s )# + k xL (14) j 1 j~1 i i i/1 is the prediction of (12) derived from (13). Since the controllability matrix of the error system between (11) and (13) is full rank, its eigenvalues can be moved anywhere. By placing them in !1 it results k"[!2)0476 0 0 !5)0000 !3)8095 1)9048 5)7143 !1)1905] and system (13) becomes a global observer of system (11). Figure 4 shows how the hyperchaotic waveform xL (t) tracks x (t). 1 1 4. APPLICATION TO HYPERCHAOS-BASED CRYPTOGRAPHY A block diagram illustrating the proposed scheme is reported in Figure 5. The idea is to combine synchronization of hyperchaotic systems and conventional cryptographic methods to design hyperchaosbased cryptosystems. The fourth-order oscillator considered herein contains an op-amp, two LC circuits and a diode.18 Its hyperchaotic behaviour has been con"rmed by both laboratory experiment and numerical simulation.18 The encrypter is based on the fourth-order oscillator and an encryption function, which is used to encrypt the message signal p(t) by means of the chaotic key K(t).8 The decrypter, based on the fourth-order Copyright ( 1999 John Wiley & Sons, Ltd. Int. J. Circ. ¹heor. Appl., 27: 543}553 (1999) 549 SYNCHRONIZATION OF HIGH-ORDER OSCILLATORS Figure 5. Hyperchaos-based cryptography: block diagram oscillator and a decryption function, enables the message signal to be retrieved when synchronization is achieved between the transmitting and receiving systems. The dynamics of the proposed encrypter is written in dimensionless form as xR 0)7 !1 !1 0 1 1 0 0 0 xR 2 " 3 0 0 !3 xR 3 0 0 3 0 xR 4 x 0 1 0 x 2 # 0 x 3 !1 x 4 0 0 30(x !1)H(x !1)# 4 4 0 e (t) en (15) !1 Here the encrypted signal e (t) is generated by a n-shift cipher [8]: %/ e (t)"f ( 2 f ( f (p(t), K(t)), K(t)), 2 , K(t)) %/ 1 1 1 where the non-linear function (16) G (17) f (x, K)" 1 (x#K)#2h (x#K), (x#K)!2h, !2h)(x#K))!h !h((x#K)(h h)(x#K))2h is recursively used for the encryption, with n"30, h"4, K(t)"x (t) and p(t)"0)5 sin t. By considering the 2 transmitted signal z(t)"y(t)#e (t) (18) %/ where 4 y(t)"30(x !1)H(x !1)# + k x (19) 4 4 i i i/1 is the output of the oscillator, the dynamics of the decrypter can be written as xLQ 0)7 !1 !1 0 1 xLQ 1 0 0 0 2 " xLQ 3 0 0 !3 3 xLQ 0 0 3 0 4 Copyright ( 1999 John Wiley & Sons, Ltd. xL 0 1 xL 0 2 # 0 xL 3 !1 xL 4 0 30(xL !1)H(xL !1)# 4 4 0 0 (z(t)!yL (t)) (20) !1 Int. J. Circ. ¹heor. Appl., 27: 543}553 (1999) 550 G. GRASSI AND S. MASCOLO Figure 6. Time waveform of the transmitted signal (18) where the prediction of the observer is 4 yL (t)"30(xL !1)H (xL !1)# + k xL (21) 4 4 i i i/1 From (18) notice that y(t) masks the encrypted signal e (t), which in turn hides the message signal p(t). Since %/ the error system between (20) and (15) is linear time-invariant and its controllability matrix is full rank, decrypter (20) becomes a global observer of encrypter (15) by choosing, for instance, the set of the eigenvalues as M!0)5G3)486j, !0)5G0)8222jN, which gives k"[!3)6937 0)2445 1.0727 !2)7000]. By considering the encrypted signal recovered by the decrypter: eL (t)"z(t)!yL (t) (22) %/ and by using the key KK (t)"xL (t) generated by the decrypter, the following message signal is retrieved: 2 pL (t)"f ( 2 f ( f (eL (t),!KK (t)),!KK (t)), 2 , !KK (t)) (23) 1 1 1 %/ where the decryption rule is the same as the encryption one.8 Since encrypter and decrypter are synchronized, it results xL (t)Px(t), KK (t)PK(t), eL (t)Pe (t), that is, pL (t)Pp(t). The validity of the proposed scheme is %/ %/ con"rmed by simulation results. In particular, the hyperchaotic transmitted signal (18) is reported in Figure 6, whereas the recovered message signal (23) is shown in Figure 7. This "gure clearly highlights that pL (t)Pp(t). 5. DISCUSSION The previous examples suggest the following considerations. (i) Although systems (7) and (11) exhibit complex dynamics and are characterized by several positive Lyapunov exponents, hyperchaos synchronization is achieved via a scalar signal. This is a remarkable feature, since a single channel is usually available for communication applications. Copyright ( 1999 John Wiley & Sons, Ltd. Int. J. Circ. ¹heor. Appl., 27: 543}553 (1999) SYNCHRONIZATION OF HIGH-ORDER OSCILLATORS 551 Figure 7. Time waveform of the recovered signal (23) (ii) The initial conditions of drive and response systems do not need to belong to the same basin of attraction. In some sense, the proposed approach overcomes the drawback related to the sensitive dependence on the initial condition of the chaotic systems to be synchronized.1 (iii) In order to verify synchronization, it is not necessary to compute any Lyapunov exponent. (iv) Since the error systems are controllable, all their modes can be arbitrarily assigned. As a consequence, any short synchronization time can be obtained. This is another remarkable feature for designing chaos-based communication scheme, since it is possible to avoid that part of the message be lost during the transient behaviour. (v) By considering 2Nth-order LC loops, the dynamic equations of the seventh order oscillator can be generalized to obtain the following (2N#1)th order oscillator:13 dx N i"a (x !z)#b + y e i dt i i i k k/i i dy i"! + b x k k k i dt k/1 dz c "!z#cH(x !1) 1 dt (24) where a "a, b "!1 and a "0, b "1 for i, k"2, 3, 2 , N whereas a, e , k , c and c are 1 1 i i,k i i constants, which depend on the circuit parameter values. In a similar way, by considering 2Nth-order LC loops connected in parallel to the chaos-diode, the dynamic equations of the eighth-order oscillator can be generalized to obtain the following 2Nth order oscillator:14 dx N i"a (x !cs )#m + y e i dt i i j i k k/i Copyright ( 1999 John Wiley & Sons, Ltd. Int. J. Circ. ¹heor. Appl., 27: 543}553 (1999) 552 G. GRASSI AND S. MASCOLO dy i i"! + m x k i dt k k k/1 (25) s "H(x !1#s (1!b)), J"1, 2 2 j 1 j~1 (vi) (vii) (viii) (ix) (x) (xi) where a "a, m "!1 and a "0, m "1 for i, k"2, 3, 2 , N whereas a, e , k , b and c depend on 1 1 i i,k i i the circuit parameter values. Note that, by increasing the value N, a large number of positive Lyapunov exponents can be obtained.13,14 Taking into account the results illustrated in Section 2, it is easy to show that the proposed technique can be applied for synchronizing the wide classes of hyperchaotic oscillators represented by (24) and (25). The application to cryptography proposed herein shares all the above-mentioned synchronization features. For the sake of simplicity, a fourth-order oscillator has been considered in Section 4. However, it is easy to show that this example can be easily generalized to the class of hyperchaotic systems described by (1). As a consequence, several cryptosystems can be designed by making use of di!erent hyperchaotic circuits, including the high-order oscillators described by (24) and (25). The adoption of hyperchaotic systems with several positive Lyapunov exponents enhances the level of security of the communication scheme.19}21 Namely, since low-dimensional chaotic systems exhibit very regular geometric structures when viewed in some suitable phase space,10 for these systems it is possible to reveal the hidden information by creating the geometric structure with a forecasting approach.11 On the contrary, when the chaotic dynamics are more complex (see (24) and (25)) the forecasting approach may fail, since it becomes di$cult to recreate the underlying geometric structure in the phase space.21 The level of security is further enhanced by using a complex transmitted signal.10,15 Namely, in Reference 10 it is suggested that two chaotic signals can be added together to create a carrier signal of su$cient complexity. In this way, it is not possible to predict the carrier dynamics based on a reconstruction of the geometric structure in the phase space.10 The proposed approach makes a contribution in this direction, since a transmitted signal of high complexity is used. In (18) the "rst addend is related to the non-linear element of the fourth-order oscillator, the second one is a linear combination of all the chaotic state variables whereas the third one is the encrypted signal. The forecasting approach developed in References 10 and 11 enables the behaviour of the chaotic carrier to be predicted in low-dimensional systems. Although this is hard to do for hyperchaotic systems, consider the transmitted signal (18) and suppose that the carrier y(t)"30(x ! 4 1)H(x !1)#+4 k x is reconstructed in some way by an intruder. By considering the results i/1 i i 4 available in literature, it seems that it is not possible for an intruder to reconstruct the key x (t) 2 starting from a completely di!erent signal y(t). This conjecture leads to conclude that, even if the encrypted signal e (t)"z(t)!y(t) is reconstructed, it is not possible for an intruder to obtain the %/ message signal p(t). Taking into account the considerations reported in Reference 15, it can be concluded that the proposed approach, by considering transmitted signals of increased complexity as well as hyperchaotic systems with complex dynamics, enable to weaken the low-security objections against low-dimensional chaos-based schemes. 6. CONCLUSION By exploiting the concept of observer, in this paper two recent examples of high-order oscillators have been synchronized using a scalar signal. The approach can be easily generalized for synchronizing two wide classes of oscillators characterized by several positive Lyapunov exponents. Furthermore, by combining cryptographic methods and synchronization of high-order oscillators, an example of hyperchaos-based Copyright ( 1999 John Wiley & Sons, Ltd. Int. J. Circ. ¹heor. Appl., 27: 543}553 (1999) SYNCHRONIZATION OF HIGH-ORDER OSCILLATORS 553 cryptography has been illustrated. The advantages of the suggested tool have been discussed in detail. The conclusion of the analysis is that the proposed synchronization technique has some interesting features, which are suitable for communication purpose. In particular, the approach seems to make a contribution to the development of communication schemes with higher security. REFERENCES 1. T. L. Carroll and L. M. Pecora, &Synchronizing chaotic circuits', IEEE ¹rans. CAS, CAS-38, 453}456 (1991). 2. A. Tamasevicius, G. Mykolaitis, A. Cenys and A. Namajunas, &Synchronization of 4D hyperchaotic oscillators', IEE Electron. ¸ett., 32, 1536}1537 (1996). 3. M. K. Ali and J. Q. Fang, &Synchronization of chaos and hyperchaos using linear and nonlinear feedback functions', Phys. Rev. E, 55, 5285}5290 (1997). 4. J. H. Peng, E. J. Ding, M. Ding, and W. Yang, &Synchronizing hyperchaos with a scalar transmitted signal', Phys. Rev. ¸ett., 76, 904}907 (1996). 5. G. Grassi and S. Mascolo, &Nonlinear observer design to synchronize hyperchaotic systems via a scalar signal', IEEE ¹rans. CAS, CAS-44, 1011}1014 (1997). 6. G. Grassi and S. Mascolo, &Synchronization of hyperchaotic oscillators using a scalar signal', IEE Electron. ¸ett., 34, 424}425 (1998). 7. C. W. Wu and L. O. Chua, &A simple way to synchronize chaotic systems with applications to secure communication systems', Int. J. Bifurcation Chaos, 3, 1619}1627 (1993). 8. T. Yang, C. W. Wu and L. O. Chua, &Cryptography based on chaotic systems', IEEE ¹rans. CAS, CAS-44, 469}472 (1997). 9. G. Kolumban, M. P. Kennedy and L. O. Chua, &The role of synchronization in digital communications using chaos-Part I: fundamentals of digital communications', IEEE ¹rans. CAS, CAS-44, 927}936 (1997). 10. K. M. Short, &Steps toward unmasking secure communications', Int. J. Bifurcation Chaos, 4, 959}977 (1994). 11. K. M. Short, &Unmasking a modulated chaotic communications scheme', Int. J. Bifurcation Chaos, 6, 367}375 (1996). 12. T. Yang, &Recovery of digital signals from chaotic switching', Int. J. Circuit ¹heory Appl., 23, 611}615 (1995). 13. G. Mykolaitis, A. Tamasevicius, A. Cenys and A. Namajunas, &A family of chaotic and hyperchaotic oscillators for synchronization experiments', Proc. 6th Int. =orkshop on Nonlinear Dynamics of Electronic Systems (NDES 98), Budapest, July 1998, pp. 253}256. 14. E. Lindberg, A. Tamasevicius, A. Cenys, G. Mykolaitis and A. Namajunas, &Hyperchaos via s-diode', Proc. 6th Int. =orkshop on Nonlinear Dynamics of Electronic Systems (NDES 98), Budapest, July 1998, pp. 125}128. 15. T. Yang and L. O. Chua, &Impulsive control and synchronization of nonlinear dynamical systems and application to secure communication', Int. J. Bifurcation Chaos, 7, 645}664 (1997). 16. W. L. Brogan, Modern Control ¹heory, Prentice-Hall, Englewood Cli!s, NJ, 1991. 17. A. Tamasevicius, G. Mykolaitis, A. Cenys and A. Namajunas, &Chaos diode', Proc. 6th Int. =orkshop on Nonlinear Dynamics of Electronic Systems (NDES 98), Budapest, July 1998, pp. 177}180. 18. A. Tamasevicius, A. Namajunas and A. Cenys, &Simple 4D chaotic oscillator', IEE Electron. ¸ett., 32, 957}958 (1996). 19. L. M. Pecora, T. L. Carroll, G. Johnson and D. Mar, &Volume-preserving and volume-expanding synchronized chaotic systems', Phys. Rev. E, 56, 5090}5100 (1997). 20. T. L. Carroll and L. M. Pecora, &Synchronizing hyperchaotic volume-preserving maps and circuits', IEEE ¹rans. CAS, CAS-45, 656}659 (1998). 21. C. Zhou and T. Chen, &Extracting information masked by chaos and contaminated with noise: some considerations on the security of communication approaches using chaos', Phys. ¸ett. A, 234, 429}435 (1997). Copyright ( 1999 John Wiley & Sons, Ltd. Int. J. Circ. ¹heor. Appl., 27: 543}553 (1999)

1/--страниц