- Email: [email protected]
Secure communication via multiple parameter modulation in a delayed chaotic system
Chaos, Solitons and Fractals 23 (2005) 1071–1076 www.elsevier.com/locate/chaos
Secure communication via multiple parameter modulation in a delayed chaotic system Er-Wei Bai a
a,*
, Karl E. Lonngren a, Ahmet Uc¸ar
b
Department of Electrical and Computer Engineering, University of Iowa, Iowa City, IA 52242, USA b Department of Electrical and Electronics Engineering, Firat University, 23119 Elazig, Turkey Accepted 8 June 2004
Abstract The problem of secure communication via parameter modulation in a chaotic system is considered. Information signals are used to modulate various parameters of a chaotic system. The resulting chaotic signal is later demodulated and the information signals are recovered using an adaptive demodulator. The convergence of the demodulator is established and simulation results validate the theoretical predictions. Ó 2004 Elsevier Ltd. All rights reserved.
1. Introduction Synchronization plays a significant role in modern communication system that has received significant attention recently [1–5]. It has been suggested that a chaotic signal could be used to carry information from one location to another in a secure manner. Several techniques have been proposed and analyzed recently including additive masking, chaotic switching, chaos shift keying, and chaotic frequency modulation [6–12]. In particular, parameters [6,7] can be modulated with the information. In such cases, either all of the parameters of the system are known with the additional restriction that they be identical or one of the parameters is modulated. This may impose some limitations to the applicability of the technique. The modulation of multiple parameters of a chaotic system is likely to increase the flexibility and security of the system. In the present paper, we show that it is possible to design a secure communication system using synchronization techniques of two chaotic systems based on modern non-linear control theory. We take an information signal and use it to modulate various parameters of a chaotic system. The resulting transmitted signal consists of the information hidden in the signal from the chaotic system. The recovery of the information at the receiver is achieved with an appropriate demodulator which is described in this paper. In Section 2, we develop the technique of creating a secure communication system along with the necessary mathematical foundation that is required in order to create the system. This system is developed with reference to a particular model that is known to admit chaotic solutions [13]. The model is based on a non-linear delay-differential equation
*
Corresponding author. Tel.: +1 319 335 5949; fax: +1 319 335 6028. E-mail addresses: [email protected] (E.-W. Bai), [email protected] (K.E. Lonngren), aucar1@firat.edu.tr (A. Uc¸ar). 0960-0779/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2004.06.072
1072
E.-W. Bai et al. / Chaos, Solitons and Fractals 23 (2005) 1071–1076
that has been analyzed elsewhere [14–16]. Our choice of this particular model does not impose any limitations on the technique. Using the chosen model, we can easily prove the feasibility mathematically and then present numerical solutions that demonstrate the veracity of the technique in this section. We generalize the results in Section 3. Section 4 contains the concluding comments concerning this method.
2. Secure communications system The secure communication system involves the development of a signal that contains the information that is to remain undetectable by others within a carrier signal. We can ensure the security of this information by inserting it into a chaotic signal that is transmitted to a prescribed receiver who would be able to detect and recover the information from the chaotic signal. In the present application, we propose a secure communication system as shown in Fig. 1. The technique takes the information and modulates certain parameters of a non-linear signal that is generated by a chaotic signal generator. The resulting signal is transmitted through the hostile environment to a receiver. The receiver consists of a chaotic signal generator of the type that was used in the chaotic transmitter and a non-linear controller. This will permit the demodulation of the received signal and the recovery of the information. As the signal is transmitted through the hostile environment, it is secure since it requires that an interloper possesses an identical chaotic signal generator and a non-linear controller in order to intercept the information. Consider a chaotic system that is described by the following first-order delay equation: n X x_ ðtÞ ¼ hj fj ðxðt sj ÞÞ; sj P 0 ð1Þ j¼1
with the initial function x(t) = W(t) for t 2 [maxjsj, 0]. In (1), x(t) is the transmitted signal and the hjÕs are unknown parameters. This equation has been studied and shown to admit chaotic solutions for certain values of the parameters [13]. To illustrate the idea, we consider a simple case of n = 2 and f1 ðxðt sÞÞ ¼ xðt sÞ; f2 ðxðt sÞÞ ¼ xðt sÞ3 ;
ð2Þ 3
x_ ðtÞ ¼ h1 xðt sÞ h2 ½xðt sÞ : Suppose that information signals are two constant signals taking values of d and e, respectively. We set h1 = d, and h2 = e. In other words, the information signals are modulated into the coefficients of the transmitter which produces a chaotic output x(t). In order to recover these two constant information signals based only on the received chaotic signal x(t), we have to develop an adaptive demodulator at the receiver that will actually detect these two constant information signals. This can be accomplished by developing a suitable non-linear demodulator in the receiver, similar to the one that we have previously developed for the synchronization purpose in [11] for the system described by (1). Let the signal received by the receiver be x(t). Within the receiver, an additional signal generator creates a signal which is similar to that created in the transmitter (1). ^x_ ðtÞ ¼ ^dðtÞxðt sÞ ^eðtÞ½xðt sÞ3 k 0 ½^xðtÞ xðtÞ;
ð3Þ
where ^xðtÞ; ^dðtÞ and ^eðtÞ are estimates of x(t), d and e, respectively and k0 > 0. The hope is that by a properly designed demodulator, the errors j ^xðtÞ xðtÞ j! 0; j ^dðtÞ d j! 0; j ^eðtÞ e j! 0. Therefore, the unknown information signals
chaotic signal generator
modulator
demodulator
detected information signal
chaotic signal generator
information signal
Fig. 1. Block diagram of a chaotic communications system. The region between the transmitter and the receiver is a hostile environment.
E.-W. Bai et al. / Chaos, Solitons and Fractals 23 (2005) 1071–1076
1073
d and e are recovered by their estimates ^dðtÞ and ^eðtÞ, respectively. The following results provide such a demodulator for the general system shown in (1). Consider the system (1) and the receiver described by the following equation: ^x_ ðtÞ ¼
n X
^hj ðtÞfj ðxðt sj ÞÞ k 0 ð^xðtÞ xðtÞÞ;
ð4Þ
k 0 P 0;
j¼1
where ^xðtÞ and ^hj ðtÞ are estimates of x(t) and hj, respectively. Define the following variables: 0 1 0 0 1 1 ^ f1 ðxðt s1 ÞÞ h1 h1 ðtÞ B C B B C C B . C B B C C .. eðtÞ ¼ ^xðtÞ xðtÞ; wðtÞ ¼ B hðtÞ ¼ B ... C: C; h ¼ B .. C; ^ . @ A @ @ A A hn
fn ðxðt sn ÞÞ
ð5Þ
^ hn ðtÞ
It follows that x_ ðtÞ ¼ hT wðtÞ
ð6Þ
e_ ðtÞ ¼ k 0 eðtÞ þ ð^hðtÞ hÞT wðtÞ:
ð7Þ
and
Theorem 1. Consider Eqs. (1), (4)–(7) and assume that x(t) is bounded and chaotic so that w(t) is persistently exciting. Let the demodulator be _ ^hðtÞ ¼ CeðtÞwðtÞ
ð8Þ
for any n-dimensional positive definite matrix C = C > 0. Then, ^xðtÞ; eðtÞ and ^ hðtÞ 2 L1 . In addition, as t ! 1, we have T
j eðtÞ j¼j ^xðtÞ xðtÞ j! 0 and j ^hðtÞ h j! 0:
ð9Þ
Proof. Let vðtÞ ¼ ½e2 ðtÞ þ ð^hðtÞ hÞT C1 ð^hðtÞ hÞ=2
ð10Þ
v_ ðtÞ ¼ k 0 e2 ðtÞ þ eðtÞð^hðtÞ hÞT wðtÞ eðtÞð^hðtÞ hÞT wðtÞ ¼ k 0 e2 ðtÞ:
ð11Þ
and
Therefore, e 2 L2 \ L1. This implies that e_ ðtÞ 2 L1 ) eðtÞ ! 0 as t ! 1. The parameter convergence arises from the fact that w(t) is persistently exciting. This completes the proof. h The theorem provides a very general result. For the example (2) given above, both unknown d and e, which are the unknown information signals, can be recovered simultaneously in the receiver from ^ dðtÞ and ^eðtÞ. The same idea applies for n > 2 or when d and e are not constant but represent time varying information signals as long as the convergence rate of the adaptive demodulator is faster than the rate changes in d and e. At this point, it is important to demonstrate that the information signal can be ‘‘hidden’’ within the chaotic signal as it propagates in the hostile environment. Consider the example (2) again. With the initial function W(t) = 0.1 for t 2 [s, 0] = [1, 0], the temporal solution x(t) of (1) with e = 1 and d = 1.65 is shown in Fig. 2a. A similar temporal solution x(t) modulated by a different set of values e = 1 and 8 1:65 0 6 t < 150 > > > > > < 1:55 150 6 t < 200 ð12Þ d¼ > 1:72 200 6 t < 250 > > > > : 1:6 250 6 t < 300 is shown in Fig. 2b. By comparing these two figures, we find that they are almost indistinguishable. In addition, the information cannot be recovered unless the interloper has the proper demodulating scheme.
1074
E.-W. Bai et al. / Chaos, Solitons and Fractals 23 (2005) 1071–1076
Fig. 2. (a) The time history of the output x(t) modulated by two constant information signals and (b) the time history of the output xm(t) modulated by non-constant information signals.
3. Simulation results In all simulations, the example (2) was considered and the delay was arbitrarily chosen to be s = 1. The coefficient k0 was chosen to have the value k0 = 5. The initial conditions for both the transmitter and receiver systems are chosen to be W(t) = 0.1 on the interval [s, 0] for all simulations. We also assign the positive definite matrix C to be
10 0 k1 0 ¼ : C¼ 0 k2 0 5
Fig. 3. The transmitted information and the demodulated signals are displayed.
E.-W. Bai et al. / Chaos, Solitons and Fractals 23 (2005) 1071–1076
1075
Fig. 4. (a) The information signal d(t) and (b) the demodulated signal ^dðtÞ from the receiver signal.
For the first simulation, let the information signals d(t) and e(t) be described by 8 8 1:65 100 6 t < 150 0:9 100 6 t < 150 > > > > > > < <1 1:55 150 6 t < 200 150 6 t < 200 dðtÞ ¼ and eðtÞ ¼ > > 1:72 200 6 t < 250 0:85 200 6 t < 250 > > > > : : 1:6 250 6 t < 300 0:9 250 6 t < 300
ð13Þ
as shown in Fig. 3a and b, respectively. The corresponding demodulated information signals are shown in Fig. 3c and d, respectively. The demodulator recovers the unknown information signal well. It is important to note that the information signals do not have to be in the pulse shape and can assume other forms. Fig. 4 shows a non-pulse shaped modulating signal d(t) and also the demodulated signal ^ dðtÞ for e = 1. The demodulator recovers the unknown non-pulse shaped information signal well.
4. Conclusion In this paper, we have introduced a technique of creating a secure communication system that is based on the modulation of various parameters of an equation that admits chaotic solutions. The propagating chaotic signal that contains the information cannot be easily distinguished from a chaotic signal that contains no information. In addition, the information signal can be recovered from this propagating chaotic signal using techniques from modern control theory. We believe that the technique can be generalized to include other equations or sets of equations that are known to admit chaotic solutions.
References [1] Carroll TL, Pecora LM. Synchronizing chaotic circuits. IEEE Trans Circuits Syst I 1991;38:453–6. [2] Kocarev L, Halle KS, Eckert K, Chua LO, Parlitz U. Experimental demonstration of secure communications via chaotic synchronization. Int J Bifurc Chaos 1992;2:709–13. [3] Cuomo KM, Oppenheim AV, Strogatz SH. Synchronization of Lorenz-based chaotic circuits with applications to communications. IEEE Trans Circuits Syst II 1993;40:626–33. [4] Ogorzalek MJ. Taming chaos––part I: synchronization. IEEE Trans Circuits Syst I 1993;40:693–9. [5] Wu CW, Chua LO. A simple way to synchronize chaotic systems with applications to secure communication systems. Int J Bifurc Chaos 1993;3:1619–27. [6] Yang T, Chua LO. Secure communication via chaotic parameter modulation. IEEE Trans Circuits Syst I 1996;43:817–9. [7] Carroll TL. Communicating with use of filtered, synchronized, chaotic signal. IEEE Trans Circuits Syst I 1997;42:105–10.
1076
E.-W. Bai et al. / Chaos, Solitons and Fractals 23 (2005) 1071–1076
[8] Hasler M, Maistrenko Yu. An introduction to the synchronization of chaotic systems: coupled skew tent maps. IEEE Trans Circuits Syst I 1997;44:856–66. [9] Kolumban G, Kennedy TMP, Chua LO. The role of synchronization in digital communications using chaos––part II: chaotic modulation and chaotic synchronization. IEEE Trans Circuits Syst I 1998;45:1129–39. [10] Dachselt F, Schwarz W. Chaos and cryptography. IEEE Trans Circuits Syst I 2001;48:1498–508. [11] Lonngen KE, Bai EW. On the Uc¸ar prototype model. Int J Eng Sci 2002;40:1855–7. [12] Tang S, Chen HF, Hwang SK, Liu JM. Message encoding and decoding through chaos modulation in chaotic optical communications. IEEE Trans Circuits Syst I 2002;49:646–50. [13] Uc¸ar A. A prototype model for chaos studies. Int J Eng Sci 2002;40:251–8. [14] Uc¸ar A. On the chaotic behaviour of a prototype delayed dynamical system. Chaos, Solitions & Fractals 2003;16:187–94. [15] Li C, Liao X, Yu J. Hopf bifurcation in a prototype delay system. Chaos, Solitions & Fractals 2004;19:779–87. [16] Peng M. Bifurcation and chaotic behaviour in the Euler method for a Uc¸ar prototype delay model. Chaos, Solitions & Fractals 2004;22:483–93.
Secure communication via multiple parameter modulation in a delayed chaotic system Er-Wei Bai a
a,*
, Karl E. Lonngren a, Ahmet Uc¸ar
b
Department of Electrical and Computer Engineering, University of Iowa, Iowa City, IA 52242, USA b Department of Electrical and Electronics Engineering, Firat University, 23119 Elazig, Turkey Accepted 8 June 2004
Abstract The problem of secure communication via parameter modulation in a chaotic system is considered. Information signals are used to modulate various parameters of a chaotic system. The resulting chaotic signal is later demodulated and the information signals are recovered using an adaptive demodulator. The convergence of the demodulator is established and simulation results validate the theoretical predictions. Ó 2004 Elsevier Ltd. All rights reserved.
1. Introduction Synchronization plays a significant role in modern communication system that has received significant attention recently [1–5]. It has been suggested that a chaotic signal could be used to carry information from one location to another in a secure manner. Several techniques have been proposed and analyzed recently including additive masking, chaotic switching, chaos shift keying, and chaotic frequency modulation [6–12]. In particular, parameters [6,7] can be modulated with the information. In such cases, either all of the parameters of the system are known with the additional restriction that they be identical or one of the parameters is modulated. This may impose some limitations to the applicability of the technique. The modulation of multiple parameters of a chaotic system is likely to increase the flexibility and security of the system. In the present paper, we show that it is possible to design a secure communication system using synchronization techniques of two chaotic systems based on modern non-linear control theory. We take an information signal and use it to modulate various parameters of a chaotic system. The resulting transmitted signal consists of the information hidden in the signal from the chaotic system. The recovery of the information at the receiver is achieved with an appropriate demodulator which is described in this paper. In Section 2, we develop the technique of creating a secure communication system along with the necessary mathematical foundation that is required in order to create the system. This system is developed with reference to a particular model that is known to admit chaotic solutions [13]. The model is based on a non-linear delay-differential equation
*
Corresponding author. Tel.: +1 319 335 5949; fax: +1 319 335 6028. E-mail addresses: [email protected] (E.-W. Bai), [email protected] (K.E. Lonngren), aucar1@firat.edu.tr (A. Uc¸ar). 0960-0779/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2004.06.072
1072
E.-W. Bai et al. / Chaos, Solitons and Fractals 23 (2005) 1071–1076
that has been analyzed elsewhere [14–16]. Our choice of this particular model does not impose any limitations on the technique. Using the chosen model, we can easily prove the feasibility mathematically and then present numerical solutions that demonstrate the veracity of the technique in this section. We generalize the results in Section 3. Section 4 contains the concluding comments concerning this method.
2. Secure communications system The secure communication system involves the development of a signal that contains the information that is to remain undetectable by others within a carrier signal. We can ensure the security of this information by inserting it into a chaotic signal that is transmitted to a prescribed receiver who would be able to detect and recover the information from the chaotic signal. In the present application, we propose a secure communication system as shown in Fig. 1. The technique takes the information and modulates certain parameters of a non-linear signal that is generated by a chaotic signal generator. The resulting signal is transmitted through the hostile environment to a receiver. The receiver consists of a chaotic signal generator of the type that was used in the chaotic transmitter and a non-linear controller. This will permit the demodulation of the received signal and the recovery of the information. As the signal is transmitted through the hostile environment, it is secure since it requires that an interloper possesses an identical chaotic signal generator and a non-linear controller in order to intercept the information. Consider a chaotic system that is described by the following first-order delay equation: n X x_ ðtÞ ¼ hj fj ðxðt sj ÞÞ; sj P 0 ð1Þ j¼1
with the initial function x(t) = W(t) for t 2 [maxjsj, 0]. In (1), x(t) is the transmitted signal and the hjÕs are unknown parameters. This equation has been studied and shown to admit chaotic solutions for certain values of the parameters [13]. To illustrate the idea, we consider a simple case of n = 2 and f1 ðxðt sÞÞ ¼ xðt sÞ; f2 ðxðt sÞÞ ¼ xðt sÞ3 ;
ð2Þ 3
x_ ðtÞ ¼ h1 xðt sÞ h2 ½xðt sÞ : Suppose that information signals are two constant signals taking values of d and e, respectively. We set h1 = d, and h2 = e. In other words, the information signals are modulated into the coefficients of the transmitter which produces a chaotic output x(t). In order to recover these two constant information signals based only on the received chaotic signal x(t), we have to develop an adaptive demodulator at the receiver that will actually detect these two constant information signals. This can be accomplished by developing a suitable non-linear demodulator in the receiver, similar to the one that we have previously developed for the synchronization purpose in [11] for the system described by (1). Let the signal received by the receiver be x(t). Within the receiver, an additional signal generator creates a signal which is similar to that created in the transmitter (1). ^x_ ðtÞ ¼ ^dðtÞxðt sÞ ^eðtÞ½xðt sÞ3 k 0 ½^xðtÞ xðtÞ;
ð3Þ
where ^xðtÞ; ^dðtÞ and ^eðtÞ are estimates of x(t), d and e, respectively and k0 > 0. The hope is that by a properly designed demodulator, the errors j ^xðtÞ xðtÞ j! 0; j ^dðtÞ d j! 0; j ^eðtÞ e j! 0. Therefore, the unknown information signals
chaotic signal generator
modulator
demodulator
detected information signal
chaotic signal generator
information signal
Fig. 1. Block diagram of a chaotic communications system. The region between the transmitter and the receiver is a hostile environment.
E.-W. Bai et al. / Chaos, Solitons and Fractals 23 (2005) 1071–1076
1073
d and e are recovered by their estimates ^dðtÞ and ^eðtÞ, respectively. The following results provide such a demodulator for the general system shown in (1). Consider the system (1) and the receiver described by the following equation: ^x_ ðtÞ ¼
n X
^hj ðtÞfj ðxðt sj ÞÞ k 0 ð^xðtÞ xðtÞÞ;
ð4Þ
k 0 P 0;
j¼1
where ^xðtÞ and ^hj ðtÞ are estimates of x(t) and hj, respectively. Define the following variables: 0 1 0 0 1 1 ^ f1 ðxðt s1 ÞÞ h1 h1 ðtÞ B C B B C C B . C B B C C .. eðtÞ ¼ ^xðtÞ xðtÞ; wðtÞ ¼ B hðtÞ ¼ B ... C: C; h ¼ B .. C; ^ . @ A @ @ A A hn
fn ðxðt sn ÞÞ
ð5Þ
^ hn ðtÞ
It follows that x_ ðtÞ ¼ hT wðtÞ
ð6Þ
e_ ðtÞ ¼ k 0 eðtÞ þ ð^hðtÞ hÞT wðtÞ:
ð7Þ
and
Theorem 1. Consider Eqs. (1), (4)–(7) and assume that x(t) is bounded and chaotic so that w(t) is persistently exciting. Let the demodulator be _ ^hðtÞ ¼ CeðtÞwðtÞ
ð8Þ
for any n-dimensional positive definite matrix C = C > 0. Then, ^xðtÞ; eðtÞ and ^ hðtÞ 2 L1 . In addition, as t ! 1, we have T
j eðtÞ j¼j ^xðtÞ xðtÞ j! 0 and j ^hðtÞ h j! 0:
ð9Þ
Proof. Let vðtÞ ¼ ½e2 ðtÞ þ ð^hðtÞ hÞT C1 ð^hðtÞ hÞ=2
ð10Þ
v_ ðtÞ ¼ k 0 e2 ðtÞ þ eðtÞð^hðtÞ hÞT wðtÞ eðtÞð^hðtÞ hÞT wðtÞ ¼ k 0 e2 ðtÞ:
ð11Þ
and
Therefore, e 2 L2 \ L1. This implies that e_ ðtÞ 2 L1 ) eðtÞ ! 0 as t ! 1. The parameter convergence arises from the fact that w(t) is persistently exciting. This completes the proof. h The theorem provides a very general result. For the example (2) given above, both unknown d and e, which are the unknown information signals, can be recovered simultaneously in the receiver from ^ dðtÞ and ^eðtÞ. The same idea applies for n > 2 or when d and e are not constant but represent time varying information signals as long as the convergence rate of the adaptive demodulator is faster than the rate changes in d and e. At this point, it is important to demonstrate that the information signal can be ‘‘hidden’’ within the chaotic signal as it propagates in the hostile environment. Consider the example (2) again. With the initial function W(t) = 0.1 for t 2 [s, 0] = [1, 0], the temporal solution x(t) of (1) with e = 1 and d = 1.65 is shown in Fig. 2a. A similar temporal solution x(t) modulated by a different set of values e = 1 and 8 1:65 0 6 t < 150 > > > > > < 1:55 150 6 t < 200 ð12Þ d¼ > 1:72 200 6 t < 250 > > > > : 1:6 250 6 t < 300 is shown in Fig. 2b. By comparing these two figures, we find that they are almost indistinguishable. In addition, the information cannot be recovered unless the interloper has the proper demodulating scheme.
1074
E.-W. Bai et al. / Chaos, Solitons and Fractals 23 (2005) 1071–1076
Fig. 2. (a) The time history of the output x(t) modulated by two constant information signals and (b) the time history of the output xm(t) modulated by non-constant information signals.
3. Simulation results In all simulations, the example (2) was considered and the delay was arbitrarily chosen to be s = 1. The coefficient k0 was chosen to have the value k0 = 5. The initial conditions for both the transmitter and receiver systems are chosen to be W(t) = 0.1 on the interval [s, 0] for all simulations. We also assign the positive definite matrix C to be
10 0 k1 0 ¼ : C¼ 0 k2 0 5
Fig. 3. The transmitted information and the demodulated signals are displayed.
E.-W. Bai et al. / Chaos, Solitons and Fractals 23 (2005) 1071–1076
1075
Fig. 4. (a) The information signal d(t) and (b) the demodulated signal ^dðtÞ from the receiver signal.
For the first simulation, let the information signals d(t) and e(t) be described by 8 8 1:65 100 6 t < 150 0:9 100 6 t < 150 > > > > > > < <1 1:55 150 6 t < 200 150 6 t < 200 dðtÞ ¼ and eðtÞ ¼ > > 1:72 200 6 t < 250 0:85 200 6 t < 250 > > > > : : 1:6 250 6 t < 300 0:9 250 6 t < 300
ð13Þ
as shown in Fig. 3a and b, respectively. The corresponding demodulated information signals are shown in Fig. 3c and d, respectively. The demodulator recovers the unknown information signal well. It is important to note that the information signals do not have to be in the pulse shape and can assume other forms. Fig. 4 shows a non-pulse shaped modulating signal d(t) and also the demodulated signal ^ dðtÞ for e = 1. The demodulator recovers the unknown non-pulse shaped information signal well.
4. Conclusion In this paper, we have introduced a technique of creating a secure communication system that is based on the modulation of various parameters of an equation that admits chaotic solutions. The propagating chaotic signal that contains the information cannot be easily distinguished from a chaotic signal that contains no information. In addition, the information signal can be recovered from this propagating chaotic signal using techniques from modern control theory. We believe that the technique can be generalized to include other equations or sets of equations that are known to admit chaotic solutions.
References [1] Carroll TL, Pecora LM. Synchronizing chaotic circuits. IEEE Trans Circuits Syst I 1991;38:453–6. [2] Kocarev L, Halle KS, Eckert K, Chua LO, Parlitz U. Experimental demonstration of secure communications via chaotic synchronization. Int J Bifurc Chaos 1992;2:709–13. [3] Cuomo KM, Oppenheim AV, Strogatz SH. Synchronization of Lorenz-based chaotic circuits with applications to communications. IEEE Trans Circuits Syst II 1993;40:626–33. [4] Ogorzalek MJ. Taming chaos––part I: synchronization. IEEE Trans Circuits Syst I 1993;40:693–9. [5] Wu CW, Chua LO. A simple way to synchronize chaotic systems with applications to secure communication systems. Int J Bifurc Chaos 1993;3:1619–27. [6] Yang T, Chua LO. Secure communication via chaotic parameter modulation. IEEE Trans Circuits Syst I 1996;43:817–9. [7] Carroll TL. Communicating with use of filtered, synchronized, chaotic signal. IEEE Trans Circuits Syst I 1997;42:105–10.
1076
E.-W. Bai et al. / Chaos, Solitons and Fractals 23 (2005) 1071–1076
[8] Hasler M, Maistrenko Yu. An introduction to the synchronization of chaotic systems: coupled skew tent maps. IEEE Trans Circuits Syst I 1997;44:856–66. [9] Kolumban G, Kennedy TMP, Chua LO. The role of synchronization in digital communications using chaos––part II: chaotic modulation and chaotic synchronization. IEEE Trans Circuits Syst I 1998;45:1129–39. [10] Dachselt F, Schwarz W. Chaos and cryptography. IEEE Trans Circuits Syst I 2001;48:1498–508. [11] Lonngen KE, Bai EW. On the Uc¸ar prototype model. Int J Eng Sci 2002;40:1855–7. [12] Tang S, Chen HF, Hwang SK, Liu JM. Message encoding and decoding through chaos modulation in chaotic optical communications. IEEE Trans Circuits Syst I 2002;49:646–50. [13] Uc¸ar A. A prototype model for chaos studies. Int J Eng Sci 2002;40:251–8. [14] Uc¸ar A. On the chaotic behaviour of a prototype delayed dynamical system. Chaos, Solitions & Fractals 2003;16:187–94. [15] Li C, Liao X, Yu J. Hopf bifurcation in a prototype delay system. Chaos, Solitions & Fractals 2004;19:779–87. [16] Peng M. Bifurcation and chaotic behaviour in the Euler method for a Uc¸ar prototype delay model. Chaos, Solitions & Fractals 2004;22:483–93.