- Email: [email protected]
Information transfer via implicit encoding with delay time modulation in a time-delay system
Physics Letters A 376 (2012) 2663–2667
Contents lists available at SciVerse ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Information transfer via implicit encoding with delay time modulation in a time-delay system Won-Ho Kye Korean Intellectual Property Office, Government Complex Daejeon Building 4, 189, Cheongsa-ro, Seo-gu, Daejeon 302-701, Republic of Korea
a r t i c l e
i n f o
Article history: Received 19 April 2012 Received in revised form 29 June 2012 Accepted 17 July 2012 Available online 20 July 2012 Communicated by A.R. Bishop
a b s t r a c t A new encoding scheme for information transfer with modulated delay time in a time-delay system is proposed. In the scheme, the message is implicitly encoded into the modulated delay time. The information transfer rate as a function of encoding redundancy in various noise scales is presented and it is analyzed that the implicit encoding scheme (IES) has stronger resistance against channel noise than the explicit encoding scheme (EES). In addition, its advantages in terms of secure communication and feasible applications are discussed. © 2012 Elsevier B.V. All rights reserved.
Synchronization is characterized by a loss of exponential instability or neutrality in the transverse direction x1 − x2 due to a weak interaction in coupled chaotic systems [1,2]. It is considered to be one of the most fundamental phenomena in nonlinear dynamics [1–4]. As one of the challenging applications of the phenomenon, chaos communication realized by loading the message in a chaotic signal and extracting the message on the synchronization manifold has been extensively investigated [5,6]. However, the report that the message hidden in a chaotic signal and even in a hyperchaotic signal can be extracted from nonlinear dynamic forecasting [7] led researchers to consider higher-dimensional chaotic systems such as time-delay systems. Meanwhile, it was shown that time-delay systems [8] possess characteristics of high-dimensional hyperchaos, despite a small number of degrees of freedom depending on the delay time. Furthermore, time delay systems can be easily implemented in an electronic circuit. Owing to these intriguing characteristics, timedelay systems are considered plausible candidates for communication systems [8–12]. However, it was reported that delay time can also be detected from a chaotic signal with appropriate measures such as one step prediction error, autocorrelation, and filling factor, and that the reconstructed phase space of the time-delay system collapses into a low-dimensional manifold [13,14]. On the basis of phase space reconstruction, it was demonstrated that the message masked by the signal of the time-delay system can be extracted even in the presence of a message signal of small amplitude [14,15]. In these situations, a time-delay system with delay time modulation (DTM) [16–19], in which the delay time is modulated by the state variable or time, has been proposed. The time-delay sys-
E-mail address: [email protected]. 0375-9601/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physleta.2012.07.015
tem with DTM has been used as a model for describing interacting systems whose network topology or boundary is changing in time such as neural networks [20], social networks [21], optical systems with a moving boundary [19], and electromagnetic two-body systems [22]. It was originally introduced for the purpose of whipping out an imprint of the static delay time encoded in a time series [16]. Since the delay time varies with time in DTM, the particular delay time cannot be determined and the complexity of the attractor is significantly increased depending on the modulation amplitude [17]. In addition, the report that robust synchronization can be established between two coupled chaotic systems with DTM [16] has attracted extensive attention to time-delay systems as an ideal candidate for secure communication. It was also recently demonstrated that a message can be transferred by loading it on the modulated delay time [18]. The time-delayed system with delay time modulation is described by
x˙1 = f x1 (t ), x1 (t − τ ) ,
(1)
τ = G (x1 , t ),
(2)
where G (x1 , t ) is a modulation function. For communication application, the receiver is introduced as follows:
x˙2 = f x2 (t ), x1 (t − τ ) ,
(3)
where in the receiver the modulation function is not necessary, because the delayed signal is generated in the transmitter only and it is possible to establish a synchronized state by feeding the delayed signal into the system. In a previous investigation [18] an encoding method in which the message m is loaded on the delay time was developed as follows:
τ = G (x1 ) = γ g (x1 ) + γ m(t ),
(4)
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Fig. 1. Bifurcation diagram of logistic map with DTM for several values of the modulation amplitude at
where γ is a scaling parameter and m is a message. We refer to the scheme defined by Eq. (4) as the explicit encoding scheme (EES). To decode the message, on the synchronization manifold, the receiver should preserve the previous images of its own state variable x2 for the time interval [t − τmax , t ] in the delay buffer denoted by {x2 [i ]} N , where τmax is the maximum value of the delay time τ , N = [τmax /δt ], i ∈ {0, 1, . . . , N − 1} and δt is a sampling step. Here, [x] denotes the largest integer less than x, which is updated at every step of time evolution by replacing the state variable of the current time slice with the oldest one in the delay buffer [18]. Even though the receiver has built the delay buffer {x2 [i ]} N based on the state variable x2 , after a short transient time, the receiver obtains the same copy of the delay buffer {x1 [i ]} N , since the coupled systems are on the synchronization manifold. For this reason, the receiver can identify the delay time index i ∗ by comparing the transmitted signal from the transmitter x1 (t − τ ) with the previous images preserved in the delay buffer {x2 [i ]} N such that x2 [i ∗ ] = x1 (t − τ ), where we take τ = i ∗ δt and the superscript ∗ denotes the value determined by the receiver. If the channel noise ξ is considered, the equality in the above equation should be interpreted as the best matching solution within the noise amplitude. Here, the message is decoded by m∗ = i ∗ − g (x2 [0]), where the scaling parameter is taken as γ = δt for simplicity and the state variable x1 has been replaced by x2 , since the systems are assumed on the synchronization manifold. In this Letter we study a new encoding scheme and analyze its characteristics focusing on the transmission efficiency in the presence of channel noise. In the new encoding scheme, delay time τ is described by the modulation function and the message m as follows:
τ = G (x1 ) = γ g x1 (t − γ m) ,
(5)
where as one see the message is encoded in the delay time implicitly. We refer to the scheme defined by Eq. (5) as the implicit encoding scheme (IES). The receiver needs to have a delicate decoding mechanism that enables high communication efficiency even in the presence of a high level of channel noise. To decode the message m, the receiver needs to determine the index of the delay time i ∗ in the same manner as described above.
α = 0.4: (a) Λ = 1, (b) Λ = 10, (c) Λ = 100, and (d) Λ = 250.
The receiver should then solve the equation with the determined value τ ∗ and the delay buffer
τ ∗ = γ g x2 (t − γ m) ,
(6)
where x1 is replaced with x2 because the system is assumed on the synchronization manifold. The above equation should be understood as the discredited form: k∗ = g (x2 [m]) in a real implementation, where k∗ = τ ∗ /δt, γ = δt and x2 [m] is the m-th image preserved in the delay buffer. The above equation is not solvable because the explicit dependence between the message and state variable is not known. However the possible number state of message m represented by m ∈ {0, 1, . . . , n − 1} is finite, and hence it is possible to find the matching solution by scanning preserved images of the state variables x2 [m] in Eq. (6) even though the explicit dependence between m and τ is not known, in contrast with the case of EES (see Eq. (4)). As mentioned above, in the case of channel noise being considered, the receiver should find the m that minimizes the following value with the delay buffer:
∗ k − g x2 [m] ,
(7)
where x denotes the absolute value of x. To demonstrate the proposed encoding scheme based on DTM, we consider two coupled logistic maps. The transmitter and its modulation function are introduced by
x1 (n + 1) = λ¯x1 (n) 1 − x¯ 1 (n) ,
τ = Λx1 (n − m) ,
(8) (9)
where x¯ 1 = (1 − α )x1 (n) + α x1 (n − τ ) and we take the binary representation for the message such that m ∈ {0, 1}. Fig. 1 shows a bifurcation diagram with various values of modulation amplitude. If the trajectory is in the fixed point x1 (n) → x∗ the delayed signal x1 (n − τ ) is also converse to the fixed point. For this reason, the transmitter has the same fixed point with that of an ordinary logistic map independent of the modulation amplitude Λ in 0 < λ < 3 (this region is not shown in Fig. 1). Outside of the fixed point, the delayed signal plays a role of perturbation of the order of α x1 (n − τ ), and as the modulation amplitude Λ increases, the periodic orbit starts to lose stability and periodic windows are destroyed and the attractor grows (compare (c) with (d) in Fig. 1).
W.-H. Kye / Physics Letters A 376 (2012) 2663–2667
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Fig. 2. (a) shows the contour plot of conditional Lyapunov exponent as a function of coupling strength and modulation amplitude (α , Λ) at λ = 3.9, where the filled circle shows the positive region of conditional Lyapunov exponent and the open square shows the negative region; (b) and (c) show the time series of x1 (n) − x2 (n) at α = 0.28 and α = 0.32 with Λ = 100, respectively.
Fig. 3. Time series of x1 (n), delayed signal x1 (n − τ ), delay time
τ , encoded random message m, decoded message m∗ at λ = 3.9, Λ = 250, and α = 0.4.
The receiver is introduced as follows:
x2 (n + 1) = λ¯x2 (n) 1 − x¯ 2 (n) ,
(10)
where x¯ 2 = (1 − α )x2 (n) + α x1 (n − τ ). We emphasize that the transmitted signal from the transmitter is just x1 (n − τ ), which is not explicitly dependent on the message m and the signal x1 (n − τ ) is temporally shuffled shots of the state variable x1 . Accordingly, the eavesdropper could not rearrange the shots, because the value of the delay time τ is completely unknown. In order to analyze the synchronization threshold, we consider the difference dynamics such that X (n + 1) = λ(1 − α )(1 − (¯x1 (n) + x¯ 2 (n)) X (n), where X (n) = x1 (n) − x2 (n). The conditional Lyapunov exponent λc which determines the synchronization threshold, can be obtained by following the standard procedure: λc = limn→∞ (1/n) ln | X (n)/ X (0)| [3]. Fig. 2 shows the contour plot of the conditional Lyapunov exponent on (α , Λ) plane (see (a)) and the time series of x1 (n) − x2 (n) near the synchronization threshold (see (b) and (c)). One see that the coupled maps with DTM shows relative larger synchronization regime, which is
desirable characteristic to implement communication systems, for 0.3 < α regardless of modulation amplitude (Λ). In a real environment, noise is unavoidable and thus it is important to analyze the effect of noise on the proposed scheme for practical applications. The channel noise can be considered by introducing uniform noise ξ(n) into the receiver such that x¯ 2 = (1 − α )x2 (n) + α (x1 (n − τ ) + ξ(n)), where ξ(n) detunes the synchronization manifold. To quantify the effect of the noise, we introduce two measures, the encoding redundancy and information transfer rate. The encoding redundancy ρ is defined by the number of iteration steps to encode 1 bit and the information transfer rate η is defined by the number of successfully transmitted bits divided by the total number of transmitted bits [23]. Fig. 3 shows the temporal behaviors of several state variables of coupled logistic maps in which m is a random transmitted message and m∗ is a decoded message. As one can see here, the time series presented in Fig. 3 corresponds to the ρ = 1 and η = 1 case since the random message m is encoded in every iteration step n and there is no error bit up to 500 steps.
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Fig. 4. Information transfer rate η as a function encoding redundancy at Λ = 100 (left column (a), (b), (c), and (d)) and Λ = 250 (right column (e), (f), (g), and (h)) in different channel noises: diamond is for ξ 10−2 , square ξ 10−3 , circle ξ 10−4 , and triangle ξ = 0.0. Each datum point is the result of 106 × ρ bits transfer. The filled symbols show the data in the case of IES defined in Eq. (5) and the opened symbols show those in EES defined in Eq. (4).
We define the critical value of the encoding redundancy ρc at which the information transfer rate η becomes unity such that η(ρc ) = 1. Fig. 4 shows the information transfer rate η as a function of the encoding redundancy ρ . In the absence of noise, EES shows better performance in the sense that ρc for EES is smaller than that of IES (see (a) and (e) in Fig. 4). However, the trends begins to be reversed if the noise is turned on. In the presence of a relatively small noise amplitude ξ = 10−4 ((b) and (f) in Fig. 3), the two schemes show comparable behavior for the information transfer rate with a critical value of encoding redundancy ρc = 7.0. With higher channel noise of ξ = 10−3 , it is interesting that IES begins to show a better information transfer rate than that of EES ((c) and (g) in Fig. 3). In severe channel noise of the order of ξ = 10−2 ((d) and (h) in Fig. 4), the information transfer rate of IES increases linearly and eventually it has a value of η = 0.9 and η = 0.7 in Λ = 100 and Λ = 250, respectively, while in EES the value of the transfer rate is close to 0.5, at which there is no actual information transfer, like random walk or a coin toss [23]. The complexity of the attractor depending on the modulation amplitude [17] likely induces the error rate in determining the minimized solution of the measure defined in Eq. (7), which accounts for the information transfer rate being smaller in Λ = 250 than that of Λ = 100. It is worth discussing why IES acquires better resistance against strong channel noise. In EES, if the synchronized manifold is detuned by x1 = x2 + ξ , the delay time and the message are connected by τ = g (x2 ) + g (x2 )ξ + m(t ) + O (ξ 2 ), where Eq. (4) is expanded for the small noise amplitude ξ . In IES, Eq. (5) is expanded for small channel noise ξ such that τ = g (x2 (t − m)) + g (x2 )ξ + O (ξ 2 ). In both schemes the additive term g (x2 )ξ appears. However, it is important that the additive term originating from noise ξ does not directly contribute to changing the value of the message m in IES, while in EES the additive term directly does. After performing total differentiation in the above equations, we obtain dm = −[ ∂∂xm2 ]−1 for IES and dm = g (x2 ) for EES, where dξ dξ we have used the chain rule along the dτ = 0 surface. By applying the above equation to coupled maps in Eqs. (8)–(10), we can obtain a rough estimation for EES with regard to the sensitivity of a decoded message with respect to the noise such m that ξ ∼ Λ, which explains why the higher modulation amplitude degrades the information transformation rate in EES (compare open symbols of left columns with those of right columns in Fig. 3). We emphasize that the DTM based encoding scheme has the following characteristics: (i) the eavesdropper cannot build a correct delay buffer {x2 [i ]} N because the transmitted signal x1 (t − τ )
via a public channel is temporally shuffled; (ii) the eavesdropper cannot determine the static delay time, which is necessary to reconstruct the phase space [14], because the delay time is varying with the time; and (iii) IES (Eq. (5)) provides not only higher resistance against channel noise but also security against eavesdropping, since the message is completely mixed with the state variable (Eq. (5)), while the message only shifts the delay time in EES (Eq. (4)). In conclusion, we have proposed a new encoding scheme based on DTM in a time delay system, in which the message is encoded on the delay time implicitly. By demonstrating the scheme in coupled maps, it is shown that the proposed scheme has strong resistance against channel noise compared to the previously proposed EES [18] and even in a severely noisy environment, the information can be transmitted. By analyzing the sensitivity of the message with respect to various channel noises, we have explained why IES acquires stronger resistance against channel noise than EES. We anticipate that the proposed DTM based encoding scheme can be used in real communication systems. Acknowledgements The author thanks Heesoo son and H.S. Kim for reading this manuscript and engaging in valuable discussions. References [1] H. Fujisaka, T. Yamada, Prog. Theor. Phys. 69 (1983) 32; V.S. Afraimovich, N.N. Verichev, M.I. Rabinovich, Radiophys. Quantum Electron. 29 (1986) 747. [2] L.M. Pecora, T.L. Carroll, Phys. Rev. Lett. 64 (1990) 821. [3] A. Pikovsky, M. Rosenblum, J. Kurths, Synchronization: A Universal Concept in Nonlinear Science, Cambridge University Press, Cambridge, UK, 2001. [4] S. Boccaletti, J. Kurth, G. Osipov, D.L. Valladares, C. Zhou, Phys. Rep. 336 (2002) 1, and references therein. [5] K.M. Cuomo, A.V. Oppenheim, Phys. Rev. Lett. 71 (1993) 65; P. Colet, R. Roy, Opt. Lett. 19 (1994) 2056. [6] Shyam Sundar, Ali A. Minai, Phys. Rev. Lett. 85 (2000) 5456; Shihong Wang, Jinyu Kuang, Jinghua Li, Yunlun Luo, Huaping Lu, Gang Hu, Phys. Rev. E 66 (2002) 065202(R). [7] K.M. Short, A.T. Parker, Phys. Rev. E 58 (1998) 1159, and references therein. [8] K. Pyragas, Phys. Rev. E 58 (1998) 3067; R. He, P.G. Vaidya, Phys. Rev. E 59 (1999) 4048; L. Yaowen, G. Gguangming, Z. Hong, W. Yinghai, Phys. Rev. E 62 (2000) 7898. [9] J. Fort, V. Méndez, Phys. Rev. Lett. 89 (2002) 178101; D. Wu, S. Zhu, Phys. Rev. E 73 (2006) 051107. [10] T. Heil, I. Fischer, W. Elsässer, J. Mulet, C.R. Mirasso, Phys. Rev. Lett. 86 (2001) 795; R.M. Nguimdo, et al., Phys. Rev. Lett. 107 (2011) 034103. [11] D.V. Ramana Reddy, A. Sen, G.L. Johnston, Phys. Rev. Lett. 85 (2000) 3381.
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[12] V.S. Udaltsov, J.-P. Goedgebuer, L. Larger, W.T. Rhodes, Phys. Rev. Lett. 86 (2001) 1892; C.-M. Kim, et al., Phys. Rev. Lett. 88 (2001) 014103; D. Rontani, M. Sciamanna, A. Locquet, D.S. Citrin, Phys. Rev. E 80 (2009) 066209. [13] M.J. Bünner, Th. Meyer, A. Kittel, J. Parisi, Phys. Rev. E 56 (1997) 5083; R. Hegger, M.J. Bünner, H. Kantz, A. Giaquinta, Phys. Rev. Lett. 81 (1998) 558. [14] B.P. Bezruchko, A.S. Karavaev, V.I. Ponomarenko, M.D. Prokhorov, Phys. Rev. E 64 (2001) 056216; V.I. Ponomarenko, M.D. Prokhorov, Phys. Rev. E 66 (2002) 026215; C. Zhou, C.-H. Lai, Phys. Rev. E 60 (1999) 320. [15] J.D. Farmer, Physica D 4 (1982) 366; K.M. Short, A.T. Parker, Phys. Rev. E 58 (1998) 1159. [16] W.-H. Kye, M. Choi, M.-W. Kim, S.-Y. Lee, S. Rim, C.-M. Kim, Y.-J. Park, Phys. Lett. A 322 (2004) 338;
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Contents lists available at SciVerse ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Information transfer via implicit encoding with delay time modulation in a time-delay system Won-Ho Kye Korean Intellectual Property Office, Government Complex Daejeon Building 4, 189, Cheongsa-ro, Seo-gu, Daejeon 302-701, Republic of Korea
a r t i c l e
i n f o
Article history: Received 19 April 2012 Received in revised form 29 June 2012 Accepted 17 July 2012 Available online 20 July 2012 Communicated by A.R. Bishop
a b s t r a c t A new encoding scheme for information transfer with modulated delay time in a time-delay system is proposed. In the scheme, the message is implicitly encoded into the modulated delay time. The information transfer rate as a function of encoding redundancy in various noise scales is presented and it is analyzed that the implicit encoding scheme (IES) has stronger resistance against channel noise than the explicit encoding scheme (EES). In addition, its advantages in terms of secure communication and feasible applications are discussed. © 2012 Elsevier B.V. All rights reserved.
Synchronization is characterized by a loss of exponential instability or neutrality in the transverse direction x1 − x2 due to a weak interaction in coupled chaotic systems [1,2]. It is considered to be one of the most fundamental phenomena in nonlinear dynamics [1–4]. As one of the challenging applications of the phenomenon, chaos communication realized by loading the message in a chaotic signal and extracting the message on the synchronization manifold has been extensively investigated [5,6]. However, the report that the message hidden in a chaotic signal and even in a hyperchaotic signal can be extracted from nonlinear dynamic forecasting [7] led researchers to consider higher-dimensional chaotic systems such as time-delay systems. Meanwhile, it was shown that time-delay systems [8] possess characteristics of high-dimensional hyperchaos, despite a small number of degrees of freedom depending on the delay time. Furthermore, time delay systems can be easily implemented in an electronic circuit. Owing to these intriguing characteristics, timedelay systems are considered plausible candidates for communication systems [8–12]. However, it was reported that delay time can also be detected from a chaotic signal with appropriate measures such as one step prediction error, autocorrelation, and filling factor, and that the reconstructed phase space of the time-delay system collapses into a low-dimensional manifold [13,14]. On the basis of phase space reconstruction, it was demonstrated that the message masked by the signal of the time-delay system can be extracted even in the presence of a message signal of small amplitude [14,15]. In these situations, a time-delay system with delay time modulation (DTM) [16–19], in which the delay time is modulated by the state variable or time, has been proposed. The time-delay sys-
E-mail address: [email protected]. 0375-9601/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physleta.2012.07.015
tem with DTM has been used as a model for describing interacting systems whose network topology or boundary is changing in time such as neural networks [20], social networks [21], optical systems with a moving boundary [19], and electromagnetic two-body systems [22]. It was originally introduced for the purpose of whipping out an imprint of the static delay time encoded in a time series [16]. Since the delay time varies with time in DTM, the particular delay time cannot be determined and the complexity of the attractor is significantly increased depending on the modulation amplitude [17]. In addition, the report that robust synchronization can be established between two coupled chaotic systems with DTM [16] has attracted extensive attention to time-delay systems as an ideal candidate for secure communication. It was also recently demonstrated that a message can be transferred by loading it on the modulated delay time [18]. The time-delayed system with delay time modulation is described by
x˙1 = f x1 (t ), x1 (t − τ ) ,
(1)
τ = G (x1 , t ),
(2)
where G (x1 , t ) is a modulation function. For communication application, the receiver is introduced as follows:
x˙2 = f x2 (t ), x1 (t − τ ) ,
(3)
where in the receiver the modulation function is not necessary, because the delayed signal is generated in the transmitter only and it is possible to establish a synchronized state by feeding the delayed signal into the system. In a previous investigation [18] an encoding method in which the message m is loaded on the delay time was developed as follows:
τ = G (x1 ) = γ g (x1 ) + γ m(t ),
(4)
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Fig. 1. Bifurcation diagram of logistic map with DTM for several values of the modulation amplitude at
where γ is a scaling parameter and m is a message. We refer to the scheme defined by Eq. (4) as the explicit encoding scheme (EES). To decode the message, on the synchronization manifold, the receiver should preserve the previous images of its own state variable x2 for the time interval [t − τmax , t ] in the delay buffer denoted by {x2 [i ]} N , where τmax is the maximum value of the delay time τ , N = [τmax /δt ], i ∈ {0, 1, . . . , N − 1} and δt is a sampling step. Here, [x] denotes the largest integer less than x, which is updated at every step of time evolution by replacing the state variable of the current time slice with the oldest one in the delay buffer [18]. Even though the receiver has built the delay buffer {x2 [i ]} N based on the state variable x2 , after a short transient time, the receiver obtains the same copy of the delay buffer {x1 [i ]} N , since the coupled systems are on the synchronization manifold. For this reason, the receiver can identify the delay time index i ∗ by comparing the transmitted signal from the transmitter x1 (t − τ ) with the previous images preserved in the delay buffer {x2 [i ]} N such that x2 [i ∗ ] = x1 (t − τ ), where we take τ = i ∗ δt and the superscript ∗ denotes the value determined by the receiver. If the channel noise ξ is considered, the equality in the above equation should be interpreted as the best matching solution within the noise amplitude. Here, the message is decoded by m∗ = i ∗ − g (x2 [0]), where the scaling parameter is taken as γ = δt for simplicity and the state variable x1 has been replaced by x2 , since the systems are assumed on the synchronization manifold. In this Letter we study a new encoding scheme and analyze its characteristics focusing on the transmission efficiency in the presence of channel noise. In the new encoding scheme, delay time τ is described by the modulation function and the message m as follows:
τ = G (x1 ) = γ g x1 (t − γ m) ,
(5)
where as one see the message is encoded in the delay time implicitly. We refer to the scheme defined by Eq. (5) as the implicit encoding scheme (IES). The receiver needs to have a delicate decoding mechanism that enables high communication efficiency even in the presence of a high level of channel noise. To decode the message m, the receiver needs to determine the index of the delay time i ∗ in the same manner as described above.
α = 0.4: (a) Λ = 1, (b) Λ = 10, (c) Λ = 100, and (d) Λ = 250.
The receiver should then solve the equation with the determined value τ ∗ and the delay buffer
τ ∗ = γ g x2 (t − γ m) ,
(6)
where x1 is replaced with x2 because the system is assumed on the synchronization manifold. The above equation should be understood as the discredited form: k∗ = g (x2 [m]) in a real implementation, where k∗ = τ ∗ /δt, γ = δt and x2 [m] is the m-th image preserved in the delay buffer. The above equation is not solvable because the explicit dependence between the message and state variable is not known. However the possible number state of message m represented by m ∈ {0, 1, . . . , n − 1} is finite, and hence it is possible to find the matching solution by scanning preserved images of the state variables x2 [m] in Eq. (6) even though the explicit dependence between m and τ is not known, in contrast with the case of EES (see Eq. (4)). As mentioned above, in the case of channel noise being considered, the receiver should find the m that minimizes the following value with the delay buffer:
∗ k − g x2 [m] ,
(7)
where x denotes the absolute value of x. To demonstrate the proposed encoding scheme based on DTM, we consider two coupled logistic maps. The transmitter and its modulation function are introduced by
x1 (n + 1) = λ¯x1 (n) 1 − x¯ 1 (n) ,
τ = Λx1 (n − m) ,
(8) (9)
where x¯ 1 = (1 − α )x1 (n) + α x1 (n − τ ) and we take the binary representation for the message such that m ∈ {0, 1}. Fig. 1 shows a bifurcation diagram with various values of modulation amplitude. If the trajectory is in the fixed point x1 (n) → x∗ the delayed signal x1 (n − τ ) is also converse to the fixed point. For this reason, the transmitter has the same fixed point with that of an ordinary logistic map independent of the modulation amplitude Λ in 0 < λ < 3 (this region is not shown in Fig. 1). Outside of the fixed point, the delayed signal plays a role of perturbation of the order of α x1 (n − τ ), and as the modulation amplitude Λ increases, the periodic orbit starts to lose stability and periodic windows are destroyed and the attractor grows (compare (c) with (d) in Fig. 1).
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Fig. 2. (a) shows the contour plot of conditional Lyapunov exponent as a function of coupling strength and modulation amplitude (α , Λ) at λ = 3.9, where the filled circle shows the positive region of conditional Lyapunov exponent and the open square shows the negative region; (b) and (c) show the time series of x1 (n) − x2 (n) at α = 0.28 and α = 0.32 with Λ = 100, respectively.
Fig. 3. Time series of x1 (n), delayed signal x1 (n − τ ), delay time
τ , encoded random message m, decoded message m∗ at λ = 3.9, Λ = 250, and α = 0.4.
The receiver is introduced as follows:
x2 (n + 1) = λ¯x2 (n) 1 − x¯ 2 (n) ,
(10)
where x¯ 2 = (1 − α )x2 (n) + α x1 (n − τ ). We emphasize that the transmitted signal from the transmitter is just x1 (n − τ ), which is not explicitly dependent on the message m and the signal x1 (n − τ ) is temporally shuffled shots of the state variable x1 . Accordingly, the eavesdropper could not rearrange the shots, because the value of the delay time τ is completely unknown. In order to analyze the synchronization threshold, we consider the difference dynamics such that X (n + 1) = λ(1 − α )(1 − (¯x1 (n) + x¯ 2 (n)) X (n), where X (n) = x1 (n) − x2 (n). The conditional Lyapunov exponent λc which determines the synchronization threshold, can be obtained by following the standard procedure: λc = limn→∞ (1/n) ln | X (n)/ X (0)| [3]. Fig. 2 shows the contour plot of the conditional Lyapunov exponent on (α , Λ) plane (see (a)) and the time series of x1 (n) − x2 (n) near the synchronization threshold (see (b) and (c)). One see that the coupled maps with DTM shows relative larger synchronization regime, which is
desirable characteristic to implement communication systems, for 0.3 < α regardless of modulation amplitude (Λ). In a real environment, noise is unavoidable and thus it is important to analyze the effect of noise on the proposed scheme for practical applications. The channel noise can be considered by introducing uniform noise ξ(n) into the receiver such that x¯ 2 = (1 − α )x2 (n) + α (x1 (n − τ ) + ξ(n)), where ξ(n) detunes the synchronization manifold. To quantify the effect of the noise, we introduce two measures, the encoding redundancy and information transfer rate. The encoding redundancy ρ is defined by the number of iteration steps to encode 1 bit and the information transfer rate η is defined by the number of successfully transmitted bits divided by the total number of transmitted bits [23]. Fig. 3 shows the temporal behaviors of several state variables of coupled logistic maps in which m is a random transmitted message and m∗ is a decoded message. As one can see here, the time series presented in Fig. 3 corresponds to the ρ = 1 and η = 1 case since the random message m is encoded in every iteration step n and there is no error bit up to 500 steps.
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Fig. 4. Information transfer rate η as a function encoding redundancy at Λ = 100 (left column (a), (b), (c), and (d)) and Λ = 250 (right column (e), (f), (g), and (h)) in different channel noises: diamond is for ξ 10−2 , square ξ 10−3 , circle ξ 10−4 , and triangle ξ = 0.0. Each datum point is the result of 106 × ρ bits transfer. The filled symbols show the data in the case of IES defined in Eq. (5) and the opened symbols show those in EES defined in Eq. (4).
We define the critical value of the encoding redundancy ρc at which the information transfer rate η becomes unity such that η(ρc ) = 1. Fig. 4 shows the information transfer rate η as a function of the encoding redundancy ρ . In the absence of noise, EES shows better performance in the sense that ρc for EES is smaller than that of IES (see (a) and (e) in Fig. 4). However, the trends begins to be reversed if the noise is turned on. In the presence of a relatively small noise amplitude ξ = 10−4 ((b) and (f) in Fig. 3), the two schemes show comparable behavior for the information transfer rate with a critical value of encoding redundancy ρc = 7.0. With higher channel noise of ξ = 10−3 , it is interesting that IES begins to show a better information transfer rate than that of EES ((c) and (g) in Fig. 3). In severe channel noise of the order of ξ = 10−2 ((d) and (h) in Fig. 4), the information transfer rate of IES increases linearly and eventually it has a value of η = 0.9 and η = 0.7 in Λ = 100 and Λ = 250, respectively, while in EES the value of the transfer rate is close to 0.5, at which there is no actual information transfer, like random walk or a coin toss [23]. The complexity of the attractor depending on the modulation amplitude [17] likely induces the error rate in determining the minimized solution of the measure defined in Eq. (7), which accounts for the information transfer rate being smaller in Λ = 250 than that of Λ = 100. It is worth discussing why IES acquires better resistance against strong channel noise. In EES, if the synchronized manifold is detuned by x1 = x2 + ξ , the delay time and the message are connected by τ = g (x2 ) + g (x2 )ξ + m(t ) + O (ξ 2 ), where Eq. (4) is expanded for the small noise amplitude ξ . In IES, Eq. (5) is expanded for small channel noise ξ such that τ = g (x2 (t − m)) + g (x2 )ξ + O (ξ 2 ). In both schemes the additive term g (x2 )ξ appears. However, it is important that the additive term originating from noise ξ does not directly contribute to changing the value of the message m in IES, while in EES the additive term directly does. After performing total differentiation in the above equations, we obtain dm = −[ ∂∂xm2 ]−1 for IES and dm = g (x2 ) for EES, where dξ dξ we have used the chain rule along the dτ = 0 surface. By applying the above equation to coupled maps in Eqs. (8)–(10), we can obtain a rough estimation for EES with regard to the sensitivity of a decoded message with respect to the noise such m that ξ ∼ Λ, which explains why the higher modulation amplitude degrades the information transformation rate in EES (compare open symbols of left columns with those of right columns in Fig. 3). We emphasize that the DTM based encoding scheme has the following characteristics: (i) the eavesdropper cannot build a correct delay buffer {x2 [i ]} N because the transmitted signal x1 (t − τ )
via a public channel is temporally shuffled; (ii) the eavesdropper cannot determine the static delay time, which is necessary to reconstruct the phase space [14], because the delay time is varying with the time; and (iii) IES (Eq. (5)) provides not only higher resistance against channel noise but also security against eavesdropping, since the message is completely mixed with the state variable (Eq. (5)), while the message only shifts the delay time in EES (Eq. (4)). In conclusion, we have proposed a new encoding scheme based on DTM in a time delay system, in which the message is encoded on the delay time implicitly. By demonstrating the scheme in coupled maps, it is shown that the proposed scheme has strong resistance against channel noise compared to the previously proposed EES [18] and even in a severely noisy environment, the information can be transmitted. By analyzing the sensitivity of the message with respect to various channel noises, we have explained why IES acquires stronger resistance against channel noise than EES. We anticipate that the proposed DTM based encoding scheme can be used in real communication systems. Acknowledgements The author thanks Heesoo son and H.S. Kim for reading this manuscript and engaging in valuable discussions. References [1] H. Fujisaka, T. Yamada, Prog. Theor. Phys. 69 (1983) 32; V.S. Afraimovich, N.N. Verichev, M.I. Rabinovich, Radiophys. Quantum Electron. 29 (1986) 747. [2] L.M. Pecora, T.L. Carroll, Phys. Rev. Lett. 64 (1990) 821. [3] A. Pikovsky, M. Rosenblum, J. Kurths, Synchronization: A Universal Concept in Nonlinear Science, Cambridge University Press, Cambridge, UK, 2001. [4] S. Boccaletti, J. Kurth, G. Osipov, D.L. Valladares, C. Zhou, Phys. Rep. 336 (2002) 1, and references therein. [5] K.M. Cuomo, A.V. Oppenheim, Phys. Rev. Lett. 71 (1993) 65; P. Colet, R. Roy, Opt. Lett. 19 (1994) 2056. [6] Shyam Sundar, Ali A. Minai, Phys. Rev. Lett. 85 (2000) 5456; Shihong Wang, Jinyu Kuang, Jinghua Li, Yunlun Luo, Huaping Lu, Gang Hu, Phys. Rev. E 66 (2002) 065202(R). [7] K.M. Short, A.T. Parker, Phys. Rev. E 58 (1998) 1159, and references therein. [8] K. Pyragas, Phys. Rev. E 58 (1998) 3067; R. He, P.G. Vaidya, Phys. Rev. E 59 (1999) 4048; L. Yaowen, G. Gguangming, Z. Hong, W. Yinghai, Phys. Rev. E 62 (2000) 7898. [9] J. Fort, V. Méndez, Phys. Rev. Lett. 89 (2002) 178101; D. Wu, S. Zhu, Phys. Rev. E 73 (2006) 051107. [10] T. Heil, I. Fischer, W. Elsässer, J. Mulet, C.R. Mirasso, Phys. Rev. Lett. 86 (2001) 795; R.M. Nguimdo, et al., Phys. Rev. Lett. 107 (2011) 034103. [11] D.V. Ramana Reddy, A. Sen, G.L. Johnston, Phys. Rev. Lett. 85 (2000) 3381.
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