Synchronization and chaotic masking scheme based on occasional coupling

Synchronization and chaotic masking scheme based on occasional coupling

O¨ mer Morgu¨l*

Department of Electrical and Electronics Engineering, Bilkent University, 06533 Bilkent, Ankara, Turkey 共Received 18 April 2000兲

We present a synchronization and a related chaotic masking scheme for discrete-time systems. This method is based on occasional coupling of transmitter and receiver systems. We show that the synchronization may be achieved and the message can be recovered with acceptable error under certain conditions. Then we show that the proposed schemes are robust with respect to noise and parameter mismatch. We also present some simu-lation results.

PACS number共s兲: 05.45.⫺a, 43.72.⫹q

I. INTRODUCTION

In the last decade the synchronization of chaotic systems has received a great deal of attention, see 关1–18兴. One pos-sible application of synchronization is the possibility of us-ing chaotic signals for secure communication 共see 关4,5,7兴兲. There are various synchronization schemes 关9–11,15–18兴, and in most of these the synchronized system consists of two parts: a generator of chaotic signals, which is called the mas-ter共or drive兲 system, and a receiver, which is called the slave

共or response兲 system. A chaotic signal generated by the

mas-ter system may be used as an input to the slave system to synchronize the common signals of both systems. After syn-chronization, one may add the message to the chaotic signal used for synchronization and send this signal as an input to the slave system. This is called chaotic masking, and under certain conditions, one may recover the original message

关2,3兴. An extensive list of references for various aspects of

chaotic systems may be found in Ref. 关1兴.

In this paper, we will consider the discrete-time chaotic systems. Synchronization of such systems, particularly coupled maps, has been investigated by many researchers. In Refs. 关6,8兴, synchronization properties of coupled maps, in-cluding the coupled tent maps, were investigated. In Ref.

关12兴, coupled logistic maps were considered. An

observer-based synchronization scheme for discrete-time systems was given in关23兴 共see 关9,10兴 for continuous-time case兲. In Refs.

关6,21兴 various synchronization schemes and their robustness

properties were given. In Ref.关22兴 some secure communica-tion schemes based on synchronizacommunica-tion were proposed.

Recently, a new synchronization scheme based on occa-sional coupling and its application to communication for continuous-time systems has been given in Refs.关13,14兴. In this paper, we will apply the same idea to discrete-time sys-tems and show that similar results hold under certain condi-tions. We will assume that a synchronization scheme for which the synchronization is achieved exponentially fast is available. The occasional synchronization scheme proposed in this paper consists of the application of synchronization and autonomous phases periodically. In the the synchroniza-tion phases, the exponential synchronizasynchroniza-tion scheme men-tioned above is used and in the autonomous phases, the

re-sponse system is switched to a replica of the drive system. In the case of message transmission, the message is masked by the drive signal and sent to the receiver only in the autono-mous phases. We will show that under certain conditions, it is possible to achieve synchronization, and in the case of message transmission, it is possible to recover the message with acceptable error. In particular, we will show that with this technique, any message of any length can be transmitted in the ideal case. Moreover, we will show that this technique is robust with respect to noise and parameter mismatch. When such nonidealities are present, we will show that there is a maximum allowable message length for successful mes-sage recovery, and if the length of the mesmes-sage exceeds this length, we can divide the message into submessages—each of which having a length smaller than the maximum allow-able length—and send each submessage in one message transmission interval.

This paper is organized as follows. In the next section, we will introduce our synchronization scheme and the related message transmission scheme, and prove their basic proper-ties in the ideal case. In Sec. III, we will give some robust-ness results with respect to noise and parameter mismatch. In Sec. IV, we will present some simulation results, and finally we will give some concluding remarks.

II. OCCASIONAL COUPLING

We consider discrete-time systems in this work. Let the chaotic master system be given as follows:

u共k⫹1兲⫽ f„u共k兲,␮…, y共k兲⫽h„u共k兲…, k⫽0,1,2, . . . ,

共2.1兲

where u苸Rn is the state,␮苸Rp is parameter vector, f :Rn

⫻Rp_→Rn _{and h:R}n_→Rd _{are functions, and y苸R}d _{is the} measurable output of this system that will be used for syn-chronization. Let the slave system used for synchronization be given as

w共k⫹1兲⫽g„w共k兲,y共k兲,␮…, k⫽0,1,2, . . . , 共2.2兲 where g:Rn⫻Rd⫻Rp→Rn is an appropriate function. Let e⫽u⫺w denote the synchronization error. We assume that the error decays exponentially to zero, that is for some M

⬎0, 0⬍␳⬍1, the following holds for any k⬎k0and e(k0): *Email address: morgul@ee.bilkent.edu.tr

PRE 62

储e共k兲储⭐M␳(k⫺k₀)_储e共k

0兲储, 共2.3兲

where 储•储 is any norm in Rn. In case Eq. 共2.1兲 is valid for k0⭐k⬍K⫺1, then we require that Eq. 共2.3兲 be valid for k0⬍k⭐K. In this case we say that the synchronization is exponential. Note that we may take M⭓1 in Eq. 共2.3兲, with-out loss of generality. In some cases Eq.共2.3兲 might hold for sufficiently small e(k0), i.e., for储e(k0)储⬍r for some r⬎0, in which case we say that the synchronization is locally ex-ponential.

We note that some synchronization schemes proposed in the literature are exponential. For example, in Ref. 关9兴, an exponential synchronization scheme for logistic maps is pro-posed, and it was shown that this scheme is robust with respect to noise and parameter mismatch. In 关5兴, a different synchronization scheme 共type 2 in the notation of Ref. 关5兴兲 was applied to a skew-tent map, and similar results were obtained. In Refs. 关10,11兴, an observer-based synchroniza-tion scheme for continuous-time systems was proposed and it was shown that the proposed scheme yields exponential synchronization that is robust with respect to noise and pa-rameter mismatch. The same methodology could be ex-tended to discrete-time systems, see e.g.,关23兴. However, we will not pursue this direction.

In Refs.关12,13兴, a communication scheme for continuous-time systems based on occasional coupling of synchronized systems was proposed. We will apply this methodology to discrete-time systems. Let us rewrite Eq.共2.2兲 in the follow-ing form:

w共k⫹1兲⫽ f„w共k兲,␮…⫹s共k兲G„w共k兲,y共k兲,␮…, 共2.4兲

where G(w,y ,␮)⫽g(w,y,␮)⫺ f (w,␮), and s(k)⫽0,1 is the switching signal. When s(k)⫽1, Eq. 共2.4兲 reduces to Eq.

共2.2兲, and when s(k)⫽0, Eq. 共2.4兲 becomes a copy of Eq. 共2.1兲. As in 关12,13兴, our chaotic masking scheme is based on

changing the switching signal s between 0 and 1, periodi-cally. The periods in which s⫽1 and s⫽0 are used for syn-chronization and message transmission, respectively. More precisely, let Ts and Tmbe the integers that denote the syn-chronization and message transmission intervals, respec-tively. Then, for j⫽1,2, . . . our synchronization scheme is as follows:

i. ( jth synchronization phase兲 For ( j⫺1)(Ts⫹Tm)⭐k

⬍ jTs⫹( j⫺1)Tm, 关i.e., when k „mod(Ts⫹Tm)…苸关0,Ts)兴, use the master system given by Eq. 共2.1兲 and the slave sys-tem given by Eq.共2.4兲 with s(k)⫽1. The signal trasmitted to the slave system is y in this period.

ii. ( jth autonomous phase兲 For jT_s⫹( j⫺1)T_m⭐k⬍ j(T_s

⫹Tm),关i.e., when k„mod(Ts⫹Tm)…苸关Ts,Ts⫹Tm)兴, use the master system given by Eq.共2.1兲 and the slave system given by Eq.共2.4兲 with s(k)⫽0. 共Note that in this phase Eq. 共2.4兲 becomes a replica of Eq.共2.1兲, which is an autonomous sys-tem兲.

Note that in the synchronization phase, the error decays exponentially to zero, as given by Eq. 共2.3兲, and in the au-tonomous phase it may increase, also exponentially fast. However, by arranging Tsand Tm, it may still be possible to obtain an error that decreases exponentially fast at the

begin-ning of synchronization periods, and the error in the message recovery will be small 关see Eq. 共2.19兲兴. This is the basic rationale in our scheme.

Theorem 1: Consider the system given by Eqs.共2.1兲 and

共2.4兲, and the synchronization scheme given above. Assume

that the function f (•,␮) is Lipschitz, i.e., the following holds:

储 f 共u,␮兲⫺ f 共w,␮兲储⭐k1储u⫺w储, 共2.5兲 for some k₁⬎0. Assume that Eq. 共2.3兲 holds in the synchro-nization phases. If Ts⬎0 and Tm⬎0 are chosen as

Ts⬎⫺ ln M ln␳ , Tm⬍⫺ Tsln␳⫹ln M ln k1 , 共2.6兲

then the error储e(k)储 decays to zero. Moreover, this decay is exponential, i.e., the following holds for some Mˆ⬎0, 0⬍␥

⬍1:

储e共k兲储⭐Mˆ␥k_{储e共0兲储. 共2.7兲}

Proof: Let us define the following:

T_js⫽共 j⫺1兲共T_s⫹T_m兲, Tm_j⫽Ts_j⫹T_s, j⫽1,2, . . . ,

共2.8兲

i.e., T_js and Tm_j denote the beginning of the jth synchroniza-tion and autonomous phases, respectively. Since Eq. 共2.3兲 holds in the j th synchronization phase, we have the following: 储e共k兲储⭐M␳(k⫺T_js)_储e共T j s_{兲储, T} j s_⬍k⭐T j m . 共2.9兲

At the jth autonomous phase we have

储e共k⫹1兲储⫽储 f共u共k兲,␮兲⫺ f 共w共k兲,␮兲储⭐k1储e共k兲储,

Tm_j⭐k⬍Ts_j⫹1, 共2.10兲 hence we have 储e共k兲储⭐k₁(k⫺Tm_j) 储e共Tj m_{兲储, T} j m_⬍k⭐T j⫹1 s _{. 共2.11兲} Note that if k1⬍1, then the exponential decay is obvious from Eqs. 共2.9兲 and 共2.11兲. Hence we assume k1⬎1 in the sequel. From Eq.共2.9兲 we obtain

储e共Tj

m_{兲储⭐M␳}T_s储e共T j

s_{兲储. 共2.12兲}

Hence we can rewrite Eq. 共2.11兲 as

储e共k兲储⭐M␳T_sk 1 (k⫺T_jm) 储e共Tj s_{兲储, T} j m_⬍k⭐T j⫹1 s . 共2.13兲 Note that T1

s_{⫽0, hence from Eq. 共2.13兲 we obtain}

储e共Tj_⫹1 s _{兲储⭐␣储e共T} j s_兲储⭐␣j_{储e共0兲储, 共2.14兲} where␣⬎0 is defined as ␣⫽M␳T_sk 1 T_m . 共2.15兲

By using Eq. 共2.6兲 we obtain

ln␣⫽ln M⫹Tsln␳⫹Tmln k1⬍0, 共2.16兲 hence ␣⬍1. Note that since M⭓1 and ␳⬍1, we have Ts

⬎0 in Eq. 共2.6兲, and the first inequality in Eq. 共2.6兲 is

re-quired to guarantee Tm⬎0. By using Eq. 共2.14兲 in Eqs. 共2.9兲 and共2.13兲 we obtain 储e共k兲储⭐M_{␣ ␣}j_{储e共0兲储, T} j s_⬍k⭐T j m_{, 共2.17兲} 储e共k兲储⭐␣j_{储e共0兲储, T} j m_⬍k⭐T j⫹1 s _{. 共2.18兲}

By using Eq. 共2.8兲 we obtain j⭓(k⫹T_m)/(T_s⫹T_m) in Eq.

共2.17兲 and j⭓k/(Ts⫹Tm) in Eq.共2.18兲. By using these in-equalities, respectively, in Eqs. 共2.17兲 and 共2.18兲, and by using the fact that ␣⬍1, we obtain Eq. 共2.7兲 with ␥

⫽␣1/(T_s⫹Tm)_{⬍1, and Mˆ⫽max兵M␥}T_m/␣,1其.

Based on the synchronization scheme given above, we propose the following message transmission scheme. Again, let j⫽1,2, . . . , and let m be the message to be transmitted. Then, our message transmission scheme is as follows:

共i兲 ( jth synchronization phase兲 same as the jth

synchro-nization phase in the synchrosynchro-nization scheme.

共ii兲 ( jth message transmission phase兲 same as the jth

au-tonomous phase in the synchronization scheme. The signal sent to the receiver is the masked message y⫹m in this phase.

共iii兲 共message recovery兲 In jth message transmission

phase, the recovered message mˆ is given as

mˆ共k兲⫽y共k兲⫹m共k兲⫺h„w共k兲…. 共2.19兲

We have the following result for our scheme.

Theorem 2: Consider the systems given by Eqs.共2.1兲 and

共2.4兲, and the message transmission scheme given above.

Assume that Eq. 共2.3兲 holds in the synchronization phases and Eq. 共2.5兲 holds. Moreover, let h be Lipschitz, i.e., the following holds for some k₂⬎0:

储h共u兲⫺h共w兲储⭐k2储u⫺w储. 共2.20兲

Let储e(0)储⭐r for some r⬎0 and let⑀⬎0 be given. Then for any message m of length Tm, there exists a synchronization interval Ts such that the following holds in the message transmission period:

储mˆ共k兲⫺m共k兲储⭐⑀. 共2.21兲

Proof. It can easily be shown that the estimates 共2.9兲–

共2.23兲 are valid. By using Eqs. 共2.19兲, 共2.20兲, and 共2.18兲 we

obtain the following in the j th message transmission phase:

储mˆ共k兲⫺m共k兲储⭐k2␣j储e共0兲储, 共2.22兲 where␣ is given by Eq.共2.15兲. From Eq. 共2.22兲 we see that Eq. 共2.21兲 holds if the following is satisfied:

ln M⫹T_sln␳⫹T_mln k₁⭐1 j ln ⑀ rk2 , j⫽1,2, . . . . 共2.23兲

If ln⑀/rk2⬍0, then Eq. 共2.23兲 is satisfied provided that the following holds:

ln M⫹Tsln␳⫹Tmln k1⭐ln ⑀ rk2

. 共2.24兲

If ln⑀/rk2⭓0, then Eq. 共2.23兲 is satisfied provided that the following holds:

ln M⫹Tsln␳⫹Tmln k1⭐0. 共2.25兲 Note that 0⬍␳⬍1, hence ln␳⬍0. Therefore Tsln␳→⫺⬁ as Ts→⬁. Hence, for any Tm⬎0, one can find a Ts⬎0 such that Eq.共2.24兲 or 共2.25兲 holds.

III. ROBUSTNESS RESULTS

In the previous section, we considered the ideal case. In this section, we will show that the proposed scheme is robust with respect to noise and parameter mismatch. Note that the development of synchronization and message transmission schemes are similar, hence we will consider only the robust-ness of the message transmission scheme in this section. Ro-bustness of the synchronization scheme can easily be shown by performing similar calculations. We note that similar re-sults were given in Refs.关10–13兴 for the observer-based syn-chronization schemes for continuous-time systems, and it was noted that these robustness results are consequences of exponential synchronization. We expect that similar results should hold for the discrete-time systems, and in this section we will prove such a robustness result by using exponential synchronization.

We will assume that the slave system 共2.2兲 has the fol-lowing form:

w共k⫹1兲⫽g„y共k兲⫹n共k兲,w共k兲,␮⬘…, 共3.1兲 where n is a 共random兲 noise term added to the observation y and␮

⬘

共2.3兲, hence robustness is a consequence of exponential

syn-chronization. This result can be extended to the M⬎1 case, but the proof involves some advanced Lyapunov stability results and will not be pursued here.

Theorem 2: Let the system given by Eqs.共2.1兲 and 共2.2兲 satisfy Eq. 共2.3兲 with M⭐1. Assume that g(y,w,␮) is Lip-schitz in y and ␮, i.e., the following hold for some k₃⬎0, k4⬎0:

储g共y1,w,␮兲⫺g共y2,w,␮兲储⭐k3储y1⫺y2储, 共3.2兲

储g共y,w,␮1兲⫺g共y,w,␮2兲储⭐k4储␮1⫺␮2储. 共3.3兲 Now consider the system given by Eqs. 共2.1兲 and 共3.1兲 and assume that 储n储⭐nm for some nm⬎0 and define ⌬␮⫽␮⬘

⫺␮. Assume that the solutions of Eqs.共2.1兲 and 共3.1兲 remain bounded. Then there exist constants c1⬎0, c2⬎0, and c3 such that the following estimate holds:

储e共k兲储⭐c1nm⫹c2储⌬␮储⫹c3␳k. 共3.4兲 Proof: Note that Eq. 共3.1兲 can be rewritten as

w共k⫹1兲⫽g„y共k兲,w共k兲,␮…⫹关g„y共k兲⫹n共k兲,w共k兲,␮⬘…

⫺g„y共k兲,w共k兲,␮⬘…兴⫹关g„y共k兲,w共k兲,␮⬘…

⫺g„y共k兲,w共k兲,␮…兴. 共3.5兲

Hence the error e now satisfies

e共k⫹1兲⫽ f„u共k兲,␮…⫺g„y共k兲,w共k兲,␮…

⫺关g„y共k兲⫹n共k兲,w共k兲,␮⬘…⫺g„y共k兲,w共k兲,␮⬘…兴 ⫺关g„y共k兲,w共k兲,␮

⬘

By using Eqs. 共2.1兲–共2.3兲 and Eqs. 共3.2兲 and 共3.3兲 in Eq.

共3.6兲, we obtain

储e共k⫹1兲储⭐␳储e共k兲储⫹k1nm⫹k2储⌬␮储. 共3.7兲 By using Eq. 共3.7兲 repeatedly, we obtain

储e共k兲储⭐ k3 1⫺␳nm⫹ k4 1⫺␳储⌬␮储 ⫹

冉

冊

which has the same form as Eq. 共3.4兲.

Note that when s(k)⫽1, i.e., in the synchronization inter-val, our scheme uses Eqs.共2.1兲 and 共3.1兲. Hence, Theorem 2 proves that in the synchronization interval, the synchroniza-tion error e remains bounded, hence our scheme is robust with respect to noise and parameter mismatch in this period. For simplicity, let us define e_⬁ as follows:

e_⬁⫽ k3 1⫺␳nm⫹

k4

1⫺␳储⌬␮储. 共3.9兲

Note that e(k)→e_⬁as k→⬁, hence e_⬁gives an asymptotic bound on the error. Moreover, this bound depends linearly on noise level and parameter mismatch, hence it decreases,

共increases兲 linearly as the noise level and/or parameter

mis-match decreases共increases兲. From a practical point of view, if T_s is sufficiently large, we may expect that the error reaches this bound at the end of each synchronization period. Next, we will consider the robustness in the message transmission interval. Note that, in this case Eq. 共2.4兲 takes the following form (s⫽0):

w共k⫹1兲⫽ f„w共k兲,␮⬘…, 共3.10兲

Theorem 3: Consider the systems given by Eqs.共2.1兲 and

共3.10兲. Assume that f (u,•) is Lipschitz, i.e., the following

holds for some k5⬎0:

储 f 共u,␮1兲⫺ f 共u,␮2兲储⭐k5储␮1⫺␮2储. 共3.11兲 Let e_⬁be the error bound given by Eq.共3.9兲 and assume that at the end of each synchronization interval we have 储e储

⭐e⬁⫹⑀1 for some sufficiently small ⑀1 共Note that, by Eq.

共3.8兲, this is the case if Ts is sufficiently large兲. Let ⑀⬎0 satisfy the following:

k₂„k₁共e_⬁⫹⑀₁兲⫹k₄储⌬␮储…⬍⑀. 共3.12兲

Then there exists a maximum allowable message transmis-sion interval T⭓1 such that Eq. 共2.21兲 is satisfied for any Tm⭐T.

Proof: By using Eqs. 共2.1兲 and 共3.10兲 we obtain e共k⫹1兲⫽关 f„u共k兲,␮…⫺ f „w共k兲,␮…兴

⫹关 f„w共k兲,␮…⫺ f „w共k兲,␮⬘…兴. 共3.13兲

By using Eqs. 共2.5兲 and 共3.11兲 in Eq. 共3.13兲 we obtain the following in each message transmission period:

储e共k⫹1兲储⭐k1储e共k兲储⫹k4储⌬␮储. 共3.14兲 Since 储mˆ(k)⫺m(k)储⭐k2储e(k)储, and since 储e储⭐e_⬁⫹⑀1 at the beginning of each message transmission period, by using Eq. 共3.14兲 we obtain the desired result.

Note that in the ideal case, for any length Tm⬎0, we can use our scheme provided that Ts is sufficiently big, and the main reason for this is that we can reduce the error to any level. But in the nonideal case, we cannot guarantee to re-duce the error below a certain level (e_⬁) that depends on the noise level and parameter mismatch, hence as a result of that we have an upper bound for Tm. If the message length is bigger than this bound, we may divide the message in parts and send each part in one message transmission period.

IV. SIMULATION RESULTS

First we choose the logistic equation for the master sys-tem and use the synchronization scheme proposed in 关9兴. Hence, the master and slave systems in our synchronization scheme are given as follows:

u共k⫹1兲⫽␮u共k兲„1⫺u共k兲…, y共k兲⫽h„u共k兲…⫽u共k兲,

共4.1兲

w共k⫹1兲⫽␮⬘w共k兲„1⫺w共k兲…⫹s共k兲关␮⬘„1⫺y共k兲

⫺n共k兲⫺w共k兲…⫺␳兴„y共k兲⫹n共k兲⫺w共k兲…, 共4.2兲

where n(k) is the共random兲 noise, and s(k) is the switching signal such that s(k)⫽1 in the synchronization period and s(k)⫽0 in the message transmission period. Note that h(x)

⫽x in this case. Due to the noise term, we may have w(k ⫹1)⬎1 (⬍0), in which case we set w(k⫹1)⫽1 „w(k ⫹1)⫽0… to guarantee boundedness of w. Let Ts and Tm denote the synchronization and message transmission period lengths. Let m(k) be the message to be transmitted. For sim-plicity we will assume that 0⭐m⭐1. Switching signal s(k) can be given as

s共k兲⫽

再

1 when k „mod共T_s⫹T_m兲…苸关0,T_s兲 0 when k „mod共Ts⫹Tm兲…苸关Ts,Ts⫹Tm兲.

共4.3兲

The signal transmitted to the slave system mscan be given as ms(k)⫽y(k) when s(k)⫽1 and ms(k)⫽0.5„y(k)⫹m(k)… when s(k)⫽0, or in short:

Note that we have 0⭐m_s⭐1, which may result in better masking. In the message transmission period, we have

mˆ共k兲⫽2ms共k兲⫺h„w共k兲…,

k „mod共Ts⫹Tm兲…苸关Ts,Ts⫹Tm兲. 共4.5兲 Simple calculation shows that

m_s共k兲⫺mˆ共k兲⫽ce共k兲,

k „mod共Ts⫹Tm兲…苸关Ts,Ts⫹Tm兲, 共4.6兲

where c⫽1. We note that Eq. 共4.5兲 redefines the recovered message mˆ , which is first introduced in Eq. 共2.19兲. The ap-parent difference between Eqs.共4.5兲 and 共2.19兲 is due to the form of the transmitted signal ms given by Eq. 共4.4兲. Note that in the message transmission phase we have m_s(k)

⫽0.5„y(k)⫹m(k)… as explained above, and hence Eqs. 共4.5兲

and共2.19兲 will have the same form in these phases. We also emphasize that mˆ represents the recovered message. We also note that in the Figs. 1–3, we used the symbol mr for the recovered message instead of mˆ for some technical reasons. We simulated this system for two cases. In the first

simu-FIG. 1. Simulation results for the logistic map: ideal case. 共a兲 Transmitted signal ms, 共b兲

Trans-mitted message m,共c兲 Recovered message mˆ ,共d兲 ln兩e(k)兩.

lation we choose the ideal case with ␮⫽␮⬘⫽4, ␳⫽0.1, n

⫽0, Ts⫽5, Tm⫽13. The message m is chosen as m(k)

⫽兩sin(k)兩 for k „mod(Ts⫹Tm)…苸关Ts,Ts⫹Tm). The Lips-chitz constants that appear in Eqs.共2.5兲, 共2.20兲, 共3.2兲, 共3.3兲, and共3.11兲 can easily be found as k1⫽k3⫽4, k2⫽1, k4⫽k5

⫽0.25. The results of this simulation are shown in Fig. 1.

Here, Fig. 1共a兲 and Fig. 1共b兲 show ms and m, and apparently the message m is well-masked in m_s. Figure 1共c兲 shows the recovered message and Fig. 1共d兲 shows the synchronization error in logarithmic scale共i.e., ln 兩e(k)兩 versus k). 共Note that in this case the error becomes extremely small, which neces-sitates the use of logarithmic scale to show meaningful re-sults.兲 As can be seen, although the error increases in the message transmission periods, overall it decreases to zero exponentially.

In the second simulation we choose the nonideal case with ␮⫽4, ␮

⬘

关0,0.001兴(nm⫽0.001). As for the message we use the word ‘‘EARTHQUAKE,’’ coded by using Baudot code共see 关19兴兲. Here, each letter is represented by a five-digit code. Since m(k)苸兵0,1其, the recovered message mˆ can be corrected by using simple comparison as follows:

mc共k兲⫽

再

1 if mˆ共k兲⭓0.55

0 if mˆ共k兲⭐0.45. 共4.7兲

This also increases the tolerable error level 关i.e., ⑀ in Eqs.

共2.24兲 and 共3.12兲兴. The results of this simulation are shown

in Fig. 2. Figures 2共a兲 and 2共b兲 show msand m, and as can be seen the message is well-masked. The synchronization error e is shown in Fig. 2共c兲 with normal scale and in 2共e兲 with logarithmic scale (ln兩e(k)兩 vs k). The received and corrected messages are shown in Figs. 2共d兲 and 2共f兲, respectively. As can be seen, after correction, the message is reconstructed without error.

The main reason for relatively small T_m 共or small ratio Tm/Ts) in the above simulations is the large Lipschitz con-stant k1. To increase this ratio, we need chaotic systems with smaller k1. An example of such a system may be given by a tent map as follows:

f共u,␮兲⫽

再

u when 0⭐u⭐0.5

␮⫺␮u when 0.5⭐u⭐1, 共4.8兲

FIG. 2. Simulation results for the logistic map: Nonideal case.共a兲 Transmitted signal ms,共b兲 Transmitted message m, 共c兲 Error e(k), 共d兲

and it can be shown that this system is chaotic for␮⬎1 共see

关20兴兲. We also have k1⫽␮ for this example. The master system is given as

u共k⫹1兲⫽ f„u共k兲,␮…. 共4.9兲

Note that in this case we have 0⭐m2⭐u⭐m1⭐1. Hence, we scale u to obtain the measured signal y as follows:

y共k兲⫽h„u共k兲…⫽u共k兲⫺m2 m1⫺m2

, 共4.10兲

hence we have 0⭐y⭐1. For the synchronization, we use the scheme proposed in关6,8兴, hence the slave system is given by

w共k⫹1兲⫽ f 关w共k兲⫹s共k兲␦„z共k兲⫹n共k兲…,␮⬘兴,

z共k兲⫽u共k兲⫺w共k兲, 共4.11兲

where s(k) is given by Eq.共4.3兲 and n(k) is a random noise. It can easily be shown that Eq. 共2.3兲 is satisfied with ␳

⫽␮(1⫺␦) 共see 关5兴 for a similar computation for a skew-tent map兲. The transmitted message ms(k) and the recovered

message mˆ (k) are given by Eqs.共4.4兲 and 共4.5兲, respectively. Note that Eq. 共4.6兲 is satisfied in this case for c⫽1/(m1

⫺m2).

We simulated this system with ␮⫽1.4, ␮⬘⫽1.39 (⌬␮

⫽0.01), ␳⫽0.1 (␦⫽0.93), Ts⫽10, Tm⫽10, and n is a ran-dom noise, uniformly distributed in 关0,0.01兴 (nm⫽0.01). The message m is chosen as the sentence ‘‘CHAOS IS BEAUTIFUL,’’ again coded by using Baudot code. Since m(k)苸兵0,1其, after message recovery, we can use the mes-sage correction as given by Eq. 共4.7兲. The results of this simulation are shown in Fig. 3. Figures 3共a兲 and 3共b兲 show ms and m, and as can be seen the message is well-masked. The synchronization error e is shown in Fig. 2共c兲 with nor-mal scale and in 2共e兲 with logarithmic scale (ln兩e(k)兩 vs. k). The received and corrected messages are shown in Figs. 2共d兲 and 2共f兲, respectively. As can be seen, after correction, the message is reconstructed without error.

Comment 1: The usage of alternating synchronization and message transmission phases and the fact that the synchroni-zation signal is only sent in the former phases while the message is only sent in the latter may be useful in certain applications. If only a synchronization scheme is used, there will not be any message to send, hence the message

trans-FIG. 3. Simulation results for the tent map: Nonideal case.共a兲 Transmitted signal ms, 共b兲 Transmitted message m, 共c兲 Error e(k), 共d兲

mission phases may be used for some other purposes, e.g., time multiplexing may be possible. For example, by care-fully selecting the lengths of these intervals, it may be pos-sible to synchronize e.g., two chaotic drive systems with their corresponding response parts by using a single commu-nication channel. In such a case, the synchronization signals of the first and second chaotic drive systems will be sent to the corresponding response systems through the channel in the synchronization and message transmission phases, re-spectively. This approach may even be extended to synchro-nize more than two chaotic drive systems by using a single channel. However, this point requires careful investigation. As for the chaotic masking scheme, both analog共e.g., non-quantized兲 and digital messages may be used for message transmission; see the first and second simulations. However, the main application might be the transmission of digital messages through analog communication channels, since such signals are more error tolerant, see the second and third simulations. Also by using the idea of time multiplexing pre-sented above it may be possible to send different messages to different response systems by using a single channel. How-ever, this point also needs careful investigation.

Comment 2: One disadvantage of the proposed scheme is the fact that the message is only sent in the message trans-mission phases, which reduces the efficiency in using the channel. The quantity␩⫽Tm/(Ts⫹Tm) may be used to de-termine the efficiency. Since the useful information 共mes-sage兲 is only sent in the message transmission phases, ␩ could also be used as an indicator of the carrying capacity of the proposed scheme共i.e., the rate of transmission of useful information versus the total information兲. Obviously ␩⬍1, and as ␩ increases, so does the carrying capacity and the efficiency in using the channel. Note that ␩ is larger in the ideal case, and it depends on some factors including the tol-erable error level, the noise level, and the parameter mis-match in the nonideal case. In the first simulation presented above, we have␩⫽0.72 共ideal case兲, whereas in the second simulation we obtained ␩⫽0.33 共nonideal case兲, which shows a sharp decrease in efficiency. In the third simulation we obtained ␩⫽0.5. By using the tent map given by Eq.

共4.8兲 and the parameters given in the third simulation, except

for Ts and Tm, we obtained efficiencies as large as ␩

⫽0.75 in certain simulations. We note that ␩ may be im-proved by using different chaotic systems, however this point also requires further investigation.

V. CONCLUSION

In this paper, we consider a synchronization and a related message transmission scheme by using synchronized chaotic systems. As in most synchronization schemes, we assume that a master system generates a chaotic signal that is used as an input in the slave system for synchronization. We as-sumed that a synchronization scheme for which the synchro-nization is achieved exponentially fast is available. The oc-casional synchronization scheme proposed in this paper consists of the application of synchronization and autono-mous phases periodically. In the synchronization phases, the exponential synchronization scheme mentioned above is used and in the autonomous phases, the response system is switched to an autonomous system that is a replica of the drive system. In the case of message transmission, the mes-sage is masked by the drive signal and sent to the receiver only in the autonomous phases. We showed that under cer-tain conditions, it is possible to achieve synchronization, and in the case of message transmission, it is possible to recover the message with acceptable error. We also proved that the proposed scheme is robust with respect to noise and param-eter mismatch under certain conditions. Note that this robust-ness result is quite general and is due to the exponential synchronization. Hence our results also imply that any scheme that yields exponential synchronization is also robust with respect to noise and parameter mismatch under some conditions. We also presented some simulation results, which indicate that the proposed scheme could be used in some applications. These simulations suggest that the tech-nique is particularly suitable for transmission of digital sig-nals. In this case, the tolerable error level is quite large共e.g., half the message magnitude兲 and this increases the maximum allowable message length in the nonideal case. Moreover, by using a simple comparison, the message can be recovered exactly.

We do not investigate the security of our scheme, and do not claim any level of security. But we note that our results are independent of message level, whereas in most of the chaotic masking schemes, the message level is required to be sufficiently lower than that of the chaotic carrier. This point may be considered as an advantage of our scheme.

We also did not consider the synchronization of switching signal s(k) between the master and slave systems. Since this signal is periodic, it can easily be generated in master and slave systems separately. Other schemes may be possible, but since researching such schemes is not our main aim, we did not investigate this problem in detail.

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