Applications of Symbolic Dynamics in Chaos Synchronization

Transcription

1 1014 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 44, NO. 10, OCTOBER 1997 [5] T. Kapitaniak, M. Sekieta, and M. Ogorzalek, Monotone synchronization of chaos, Int. J. Bifurc. Chaos, vol. 6, no. 1, pp , [6] L. Kocarev, A. Shang, and L. O. Chua, Transitions in dynamical regimes by driving: a unified method of control and synchronization of chaos, Int. J. Bifurc. Chaos, vol. 3, no. 2, pp , [7] T. Kapitaniak, Controlling chaos. San Diego, CA: Academic, [8] K. M. Cuomo, A. V. Oppenheim, and S. H. Strogatz, Synchronization of Lorenz-based chaotic circuits with applications to communications, IEEE Trans. Circuits Syst. II, vol. 40, pp , Oct [9] L. Kocarev, K. S. Halle, K. Eckert, L. O. Chua, and U. Parlitz, Experimental demonstration of secure communications via chaotic synchronization, Int. J. Bifurc. Chaos, vol. 2, no. 3, pp , [10] U. Feldmann, M. Hasler, and W. Schwarz, Communication by chaotic signals: The inverse system approach, Int. J. Circuit Theory Applicat., vol. 24, no. 4, pp , [11] C. W. Wu and L. O. Chua, A simple way to synchronize chaotic systems with applications to secure communication systems, Int. J. Bifurc. Chaos, vol. 3, no. 6, pp , [12] C. W. Wu and L. O. Chua, A unified framework for synchronization and control of dynamical systems, Int. J. Bifurc. Chaos, vol. 4, no. 4, pp , [13] Ö. Morgül and E. Solak, Observed based synchronization of chaotic systems, Phys. Rev. E, vol. 54, no. 5, pp , [14] J. H. Peng, E. J. Ding, M. Ding, and W. Yang, Synchronizing hyperchaos with a scalar transmitted signal, Phys. Rev. Lett., vol. 76. no. 6, pp , [15] L. Kocarev and U. Parlitz, General approach for chaotic synchronization with applications to communication, Phys. Rev. Lett., vol. 74. no. 25, pp , [16] U. Parlitz, L. Kocarev, T. Stojanovski, and H. Preckel, Encoding messages using chaotic synchronization, Phys. Rev. E, vol. 53, no. 5, pp , [17] K. Pyragas, Predictable chaos in slightly perturbed unpredictable chaotic systems, Phys. Lett. A, vol. 181, no. 3, pp , [18] B. Friedland, Advanced Control System Design. Englewood Cliffs, NJ: Prentice Hall, [19] A. Isidori, Nonlinear Control Systems. New York: Springer, [20] R. Marino and P. Tomei, Nonlinear Control Design. Englewood Cliffs, NJ: Prentice-Hall, [21] J. J. Slotine and W. Lie, Applied Nonlinear Control. Englewood Cliffs, NJ: Prentice-Hall, [22] C. T. Chen, Linear System Theory and Design. Holt, Rinehart and Winston, [23] O. E. Rössler, An equation for hyperchaos, Phys. Lett., vol. 71A, no. 2 3, pp , [24] T. Matsumoto, L. O. Chua, and K.Kobayashi, Hyperchaos: laboratory experiment and numerical confirmation, IEEE Trans. Circuits Syst., vol. 33, pp , Nov [25] A. Tamasevicius, A. Namajunas, and A. Cenys, Simple 4D chaotic oscillator, Inst. Elect. Eng. Electron. Lett., vol. 32, no. 11, pp , [26] M. Gotz, U. Feldmann, and W. Schwarz, Synthesis of higher dimensional Chua circuits, IEEE Trans. Circuits Syst. I, vol. 40, pp , Nov [27] A. Rogers, P. Sheridan, and O. Feely, Synthesis of higher order Chua circuits, in Proc. Fourth Int. Workshop on Nonlinear Dynamics of Electron. Syst. (NDES 96), Seville, Spain, 1996, pp [28] W. L. Brogan, Modern Control Theory. Englewood Cliffs, NJ: Prentice-Hall, Applications of Symbolic Dynamics in Chaos Synchronization Toni Stojanovski, Ljupčo Kocarev, and Richard Harris Abstract We give the relationship between symbolic dynamics and chaos synchronization. If the capacity of the channel which onedirectionally connects two chaotic systems with the aim of synchronizing them is larger than Kolmogorov Sinai entropy of the driving system, then the synchronization error can be made arbitrarily small. Index Terms Chaos, channel capacity, symbolic dynamics, synchronization. I. INTRODUCTION In this brief, we analyze nonlinear discrete-time dynamical systems whose chaotic evolution is governed by deterministic equations [1]. Despite of the absence of stochastic terms in the governing equations it is usually said that the long-term behavior of chaotic systems is unpredictable. Such an unpredictability is due to a unique property of chaotic systems, namely exponential sensitivity to changes in initial states. Any uncertainty in the knowledge of the initial state gets amplified by the chaotic nature of the dynamical system, and eventually reaches the chaotic attractor s size thus preventing the long-term prediction. Still, two or more chaotic systems when suitably coupled can successfully synchronize [2] [4], that is, their trajectories tend to each other. As early as in one of the pioneering works on chaos synchronization [4] it was pointed out that the reproducibility of chaotic trajectories through synchronized chaotic motion in addition to the unpredictability and random-like appearance of chaotic trajectories might be interesting for secure communications applications. Indeed, numerous papers have been published on the subject since then. Making a complete reference is almost impossible and we only point to several papers with extensive references [5] [9]. The issue of influence of the capacity of a communication channel on the synchronization between two chaotic systems connected by the channel has not been addressed yet. Channel capacity is equal to the maximal amount of information that can be conveyed through a channel per unit time where the maximization is done over all possible channel input signals. In simple terms, both analog and digital communication channels have finite capacity, while chaos synchronization methods require that the driving signal is transmitted to the response circuit without any distortions including noise addition or amplitude quantization. Having in mind that chaotic signals take values from a continuous set, this is virtually a requirement for a channel with infinite capacity which is impossible to be satisfied. When a coarse-graining of the state space is introduced, for example, by a measurement process, then the deterministic behavior of a chaotic system on a microscopic (continuous) scale is turned into a stochastic behavior on a macroscopic (coarse-grained) scale [1], Manuscript received January 16, 1997; revised June 16, This paper was recommended by Guest Editor M. P. Kennedy. T. Stojanovski and R. Harris are with the CATT Centre, RMIT University, Carlton Victoria 3053, Australia ( tonci@catt.rmit.edu.au; richard@catt.rmit.edu.au). L. Kocarev is with the Department of Electrical Engineering, Sv. Kiril i Metodij University, Skopje, Macedonia ( lkocarev@cerera.etf.ukim.edu.mk). Publisher Item Identifier S (97) /97$ IEEE

2 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 44, NO. 10, OCTOBER [10], [11]. The macroscopic states are usually denoted with letters, and the coarse-grained trajectory is a sequence of letters. The resulting macroscopic dynamics is called symbolic dynamics. This way the chaotic system is turned into an information source [12] and can be analyzed by means of information theory. Viewing chaotic systems as information sources, and applying results from symbolic dynamics and information theory, we will be able to prove that chaos synchronization is possible via channels with finite capacity. II. CHAOS SYNCHRONIZATION In this brief, we are interested in discrete-time dynamical systems xn+1 = f(xn); xn 2SR N : (1) Unless otherwise stated we shall assume that the dynamical system (1) exhibits chaotic motion, the transient behavior has finished and the motion has settled on the chaotic attractor. Our discussion is valid for invertible as well as noninvertible maps f. Assume that another dynamical system is one-directionally driven by (1) y n+1 = g(y n ;d n ); y n 2SR N d n =q(x n ) (2) where d n is the driving signal, and g(y n ;d n )describes the driving connection and satisfies g[y n ; q(y n )] = f(y n ) [7]. Throughout this brief we assume that g and q are properly chosen so that the driven system (2) synchronizes with (1), that is, lim n!1 kx n 0 y n k =0for all (x 1;y 1 )2S2S. For the sake of simplicity we assume that the basin of attraction of the synchronized motion is S2S, and this does not reduce the generality of the forthcoming discussion. All existing synchronization methods have been proposed and analyzed assuming that the driving signal can be transmitted to the driven system with no distortion of any kind (noisy disturbance, quantization, attenuation, band limitation, or nonlinear distortions). However, in real channels the noise is always added to the transmitted signal. Therefore, we are interested only in a synchronized motion which is resistant to the negative influence of noise. As shown in [13] and [14] the synchronized motion in the real chaotic systems may be interrupted with outbreaks of desynchronized bursty behavior provoked by the noise. A necessary and sufficient condition for the occurrence of the synchronization is the negativity of the transverse Lyapunov exponents (TLE) [4], that is, asymptotic stability of the coupled systems to perturbations which are transverse to the synchronization manifold x = y. The TLE s are averaged over the whole attractor and their negativity does not exclude existence of phase points lying on the attractor whose TLE s computed over a finite number of iterations may take one or more positive values. The noise permanently sets the motion of the coupled systems off the synchronization manifold. Whenever the coupled systems state gets close to such a locally unstable phase point, the noise disturbance gets amplified by the instability of the phase point thus provoking a brief high-magnitude (comparable to the size of the attractor) departure from the synchronization manifold. Synchronized motion which is free of brief intermittent peaks in presence of noise is named a high-quality synchronized motion (HQSM) [13], [14]. A HQSM is attainable in many cases of coupled chaotic systems [13] [15]. In the same references the reader can also find examples of coupled chaotic systems which synchronize when there is no noise, but do not achieve a HQSM. III. SYMBOLIC DYNAMICS Any set of m disjoint regions = fc i g i=m i=1 which covers the state space S of (1) is called a partition, that is i=m = fc i g i=m i=1 ; C i \C j =;for i 6= j; i=1 C i = S: Then we assign a unique symbol i = (C i );i2m={1, 2, 111;m} to every region C i of the partition. The process of partitioning the state space and assigning symbols to every region from the partition is called symbolic dynamics. Denote with 9= 1 M the space of j=1 all infinite length sequences X 1 1 = X 1X 2 111X j 111 with X j 2 M. This way we obtain a map : S! 9defined as (x 1)=X 1 1,f j01 (x 1)2C X ; for j 1 which assigns a sequence X to every point x 1 2S, where X j is the symbol generated at time j. The motion of the dynamic system (1) in the continuous (microscopic) state space is deterministic and is governed by (1). Having known the initial state x 1 and the map f one can compute the future evolution of (1). On the contrary, its motion in the partitioned (macroscopic) space is stochastic and the trajectories are sequences of symbols. On the basis of the knowledge of the past coarse-grained trajectory of (1) we can predict its future macroscopic states only in probabilistic terms. Different states which belong to a same region C X at time j, due to the chaotic nature of the dynamical system, may go to different regions at time j +1. Partitioning of the state space turns the deterministic chaotic system into an information source which can be analyzed in terms of information theory. For the newly obtained information source one can compute the entropies H n = 0 X P (X n 1 )logp(x n 1 ) with P (X n 1 ) being the probability of occurrence of the trajectory subsequence (word) X n 1 = X 1 111X n. H n quantifies the average uncertainty when predicting words of length n. Throughout the paper we use logarithms with base 2 and the amount of information will be expressed in bits. The conditional entropy of the (n +1)th symbol in the macroscopic trajectory when the previous n symbols are known is equal to h n = H n+1 0 H n for n =1;2;111 and h 0 = H 1. The source entropy of (1) for a given partition is defined by h = lim n!1 h n = lim n!1 1 n H n : The Kolmogorov Sinai (KS) entropy of (1) is the supremum of the source entropy over all possible partitions h KS =suph : If h = h KS, then is a generating partition. An interesting property of a generating partition is that the corresponding map is injective, that is, x 0 6= x 00 ) (x 0 ) 6= (x 00 ). From the viewpoint of the h KS value, chaotic systems are similar to stochastic systems since in both cases h KS > 0. This means that the predictability of chaotic as well as stochastic systems is bounded by an always present uncertainty h KS. For a given partition we can define its refinement n at stage n as consisting of the following disjoint regions C X = x 2Sjx2 n j=1 f 0j+1 C X where f 0j+1 C X = fx 2Sjf j01 (x)2c X gfor j =1;111;n. If is a generating partition, then lim n!1 diam ( n )=0, with

3 1016 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 44, NO. 10, OCTOBER 1997 diam( n ) being the maximum diameter of all the regions that n consists of. For a particular initial state x 1 the first n symbols X n 1 specify one region of n to which x 1 belongs. The next symbol Xn+1 brings the positive amount of information hn about x 1, and points to the region of n+1 which contains x 1. This additional information is expressed through the fact that diam ( n+1 ) < diam ( n ) for n 1 which means that every new symbol Xn+1 from the sequence (x 1) specifies x 1 with higher and higher precision. Probably the most trivial and illustrative example is Bernoulli shift x n+1 = B(x n)=2x n mod 1 with the generating partition [0, 1 2 )! 0, [ 1,1)! 1. It can be easily shown that: 1) the physical ergodic 2 probability measure [10] is constant in the range [0, 1); 2) the coarsegrained Bernoulli shift is an information source without memory, that is, h n = h 0 for n 1; and 3) h n =1[bit=symbol] for n 0. If the initial state x 1 is written in a binary notation x 1 =0:a 1 a 2 a where a 1;a 2;a 3;111 2 {0, 1}, then (x 1) = a 1a 2a The nth refinement n consists of 2 n regions [i=2 n ; (i +1)=2 n )i = 0;1;111; 2 n 0 1. The first n symbols from the sequence (x 1 ) specify one region of n to whom x 1 belongs, say [l=2 n ; (l+1)=2 n ). Then a n+1 determines whether x 1 belongs to either the left or the right half of [l=2 n ; (l +1)=2 n ). Both halves have probability 1=2, and a n+1 brings 1 [bit] information about x 1. IV. RELATIONSHIP BETWEEN SYMBOLIC DYNAMICS AND CHAOS SYNCHRONIZATION Analog communication channels permit their input and output signals to take values from infinite alphabets and therefore are suitable for transmission of chaotic signals. Since a noise free transmission is not possible, it seems that many coupled chaotic systems can not achieve a HQSM when locally unstable phase points exist. Today communication systems are dominantly digital and one naturally wonders whether a HQSM is possible between chaotic systems which are connected via a digital communication channel. Digital communication channels allow their input signals to take values from a finite alphabet. If one wants to connect two chaotic systems over a digital communication channel, a quantization of the driving signal is necessary. The process of quantization can be regarded to as an addition of quantization noise Q(d n ) 0 d n to the signal d n thus producing a quantized signal Q(d n). The quantization noise can produce brief and high-magnitude departures from the synchronization manifold similarly to the additive noise in analog communication channels. It seems that digital communications channels are also useless if one wants to achieve a HQSM in cases where locally unstable phase points exist. One may hope that the quality of the synchronized motion can be increased by means of lowering the noise level for analog channels, or increasing the number of quantization levels and thus decreasing the quantization noise for digital channels. Unfortunately, both cases require increased channel capacity. However, channel capacity is determined by some inflexible parameters, such as channel bandwidth and noise power, and its increase is difficult. Here we show that a synchronized motion with an arbitrarily small synchronization error, that is, a HQSM is achievable when the channel capacity is larger than a finite value, namely, h KS of (1). When (2) tends to synchronize with (1) it actually adjusts its trajectory so that it appears as if their initial states y 1 and x 1 were identical. So, the process of synchronization may be viewed as a generation (at the driving side) of the driving signal d n = q(x n ) which contains information about the current x n and the initial state x 1 of the driving system, and a computation (at the driven side) of x n and x 1 on basis of d n. As previously said, when a generating partition is used every new symbol refines the partition, and lim n!1 diam ( n ) = 0. On the basis of the knowledge of the sequence X 1 1 one can determine the initial state x 1 of the system which produces the sequence. The more symbols one receives, the more information one gets about x 1. Then the precision of determination of x 1 is higher. Since the determination of x 1 is also a goal of the synchronization methods, the sequence X 1 1 can be used to drive and synchronize another identical dynamical system. For a generating partition, a chaotic system transforms into an information source that generates information at rate h KS. According to the noisy channel theorem [12], if the source entropy h KS is smaller than the channel capacity C, then the data generated by the source can be transmitted over the channel with negligible probability of error. If h KS <C, then the sequence (x 1) can be transmitted over the channel with arbitrarily small probability of error, and x 1 can be determined with arbitrarily high precision at the driven (receiving) side. The way the sequence X 1 1 can be used for synchronization purposes is almost obvious. If the symbol X n is used to evaluate x n, then the precision of evaluation is diam () at worst, that is, after receiving X n one can only determine the region of the partition to whom x n belongs. Clearly, better evaluation of x n can be done if not only X n is known but also several other symbols following X n. The more one waits, the more symbols one receives and the better the evaluation of x n is. If x n is evaluated after X n+k01 n are received, then the precision of evaluation of x n is diam ( k ) at worst. The driven system (2) can be synchronized with (1) in the following way. On the basis of X n+k01 n one can compute an evaluation ^x n of x n, and then an estimation of the driving signal ^d n = q(^x n ). Then ^d n can be used to drive (2). The driven system (2) will not perfectly synchronize with (1) but the difference between their states can be controlled. If the driven system is patient enough to wait for arbitrarily many iterations k, then ^d n can be arbitrarily close to d n, the synchronization error kx n 0 y n k can be made arbitrarily small and a HQSM is possible. This synchronization procedure has certain drawbacks: the computation of ^x n from X n+k01 n might be a demanding task; the synchronization error does not tend to zero; and there is a synchronization (decoding) delay. Still its good properties are worthwhile: the synchronization error can be kept as small as possible by using longer decoding delays k and thus higher refinements k ; the driving signal can be transmitted with negligible probability of error and a HQSM can be achieved over a channel with finite capacity C > h KS. Obviously, only generating partitions are useful for synchronization purposes. A nongenerating partition produces sequences with entropies h <h KS, which can be transmitted over channels with capacity C<h KS, but such sequences do not map to unique initial states. The channel capacity has been defined for channels with continuous-valued inputs and outputs as well. When the driving signal for a driven chaotic system is contaminated by a noise, then the noise presence can be interpreted as a transmission of the driving signal through a channel with a limited capacity that depends on the noise level. For example, the capacity of a band-limited channel with average-power constraint and signal-independent additive white Gaussian noise is C = W log(1 + P s=p n) [12], where W is the channel bandwidth, P s is the allowed average power of the input signal, and P n is the average power of the noise. The nonquantized driving signal can be interpreted as coming from a partition with diam ()! 0. Thus is a generating partition and every new state x n brings h = h KS amount of information about the initial state x 1. If C>h KS, then the transmitted sequence x 1 1 can be determined with arbitrarily high precision at the channel output which actually means an ability to produce a HQSM. We can summarize as follows: Using a generating partition, a HQSM is possible over channels

4 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 44, NO. 10, OCTOBER Fig. 1. Dependence of erms on c for (3) and (4). The curve s parameter is r. erms is averaged over iterations. whose capacity C is bigger than the KS entropy h KS of the driving chaotic system. Once we know under what conditions a HQSM can be accomplished, the next question is: How can we achieve a HQSM and reduce the influence of the noise on the synchronized motion? In terms of information theory the how question can be rephrased as: How can one achieve error-free transmission at information rates arbitrarily close to the channel capacity? In his masterpiece [12] Shannon showed that there must exist at least one encoding of the sequence generated by an information source which allows error-free transmission of the sequence when the channel capacity is larger than the source entropy. Since then intensive research has been undertaken to achieve the bounds set by Shannon, the most significant result probably being the trellis-coded modulation (TCM) [16]. Information theory suggests that the transmission is most resistant to the noise-provoked detection errors when the sequences of transmitted (channel) symbols are very different from each other. Motivated by this suggestion, TCM introduces redundant bits in the sequence of information bits, and codes the newly obtained sequence of bits with symbols from the channel alphabet while trying to maximize the Euclidean distance between the sequences of channel symbols. TCM offers significant improvements of the robustness of digital transmission against additive noise, compared to uncoded modulation. More on this subject can be found in [16]. In short, the how question has not been answered entirely yet and therefore the universal path to a HQSM is not known at present. Still, the welldeveloped results from information theory could be applied when one wants to improve the quality of synchronization. We believe that the application of the knowledge accumulated in the information theory is the best if not the only way to a HQSM. Example: In this example, we successfully reduce the noise influence and achieve a HQSM between two coupled systems which otherwise are very susceptible to the noise influence. Consider two Bernoulli shifts xn+1 = B(xn) =2xnmod 1 (3) yn+1 = B 1 (yn; ^xn) =2yn +c sin 4 ^x n 0 sin 4 y n mod 1 (4) where the driving signal ^xn is computed from the received signal zn = xn + wn, c is the coupling strength, and wn is a sequence of independent noise samples uniformly distributed in the range Fig. 2. Dependence of erms and emax on k for different r. The curve s parameter is r = 0.1, 0.01, 0.001, ; Higher curves correspond to larger r. c = 3.0. The minimization in (5) is done over points from [0, 1). erms and emax are computed for iterations of (3) and (4). (0r; +r). If there is no noise (r = 0), then the ptle of (3) pand (4) is ln j2 0 c=p 2j, and they synchronize for 2 <c<3 2. In order to determine the quality of the synchronized motion we measured e rms and e max as a function of c 2 [0, 6] and for different r, where e rms is the average value of the synchronization error en = jxn 0 ynj averaged over a certain number of iterations, and e max is the maximal value of en observed during the same number of iterations. The maximal error e max reveals the magnitudes of the possible intermittent bursts, and values of e max small compared to the attractor size are an indication of a HQSM. At first, we examine the quality of the synchronized motion when ^xn = zn, that is, there is no attempt to reduce the noise influence (see Fig. 1). As expected [17], the small values for e rms coincide with the region of c where <0. Gauthier and Bienfang [14] have suggested a criterion for occurrence of a HQSM: the largest TLE for every invariant set embedded within the synchronization p manifold is negative. It can be easily shown that for 4= 2 <c<12= there are no transversely unstable periodic orbits of (3) and (4) that lie on the manifold x = y. However, our measurements indicated that e max can be arbitrarily close to 1 for all values of c and r>0. The criterion from [14] is not applicable to (3) and (4) because of the discontinuity of B(x) at x = 0.5. If xn =0:50where 0 <1and yn 0.5, then xn+1 =102 while the noise wn can cause yn+1 to be very small ( 0) and thus en+1 1. That the probability of this event is nonzero follows from the ergodicity of (3) and the randomness of the noise wn. So, a HQSM does not occur when ^xn is computed only on basis of the knowledge of zn through ^xn = zn. We now improve the quality of the synchronized motion through enlargement of the difference between the possible transmitted sequences, which is along the main idea of TCM. Assume the state xnis known to be either x 0 n or x 00 n, where x 0 n and x 00 n are very close to each other. If ^xn is to be computed, that is, the choice between x 0 n and x 00 n is to be made only on basis of zn, then the probability of error detection is high. However, due to the chaotic nature of (3) the distance between the next states B i (xn) 0 and B i (xn) 00 increases with i for small i>0. So, the choice between B i (xn) 0 and B i (xn), 00 and consequently between x 0 n and x 00 n can be made with smaller probability of error if zn+i = xn+i + wn+i is also known. The noise-disturbed trajectory zn n+k01 becomes more similar to one of the possible choices x 0 n;b(xn);111;b 0 k01 (xn) 0 and x 00 n;b(xn); ;B k01 (xn) 00 as k increases, and one can distinguish with higher confidence between them. So, if at the driven side one waits longer and collects more noisy symbols zn n+k01, then

5 1018 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 44, NO. 10, OCTOBER 1997 one gathers more information about xn and can produce a better estimation ^xn. As previously said, finding the optimal computation of ^xn for given k is not an easy task. Therefore, we choose a very simple computation which emerges straightforwardly from our goal to reduce e max where ^xn = B i 2(y n ;u)= min u2[0; 1) max i=0; 1; 111;k01 jbi 2(y n;u)0z n+ij (5) u; for i =0 B i01 [B 1 (y n ;u)]; for i 1 that is, for ^x n we choose the value u 2 [0, 1) that produces the most similar trajectory to the noisy received symbols, where the criterion of similarity is the maximal distance between the trajectories. Fig. 2 shows the dependence of e rms and e max on k for different r. As shown in Fig. 2, increasing the number of symbols k which take part in the computation of ^x n decreases e max as well as e rms and improves the quality of the synchronized motion. According to our simulations e max did not depend on r for the examined range of r values and therefore in Fig. 2 there is only one curve representing e max. [8] U. Parlitz, L. Kocarev, T. Stojanovski, and H. Preckel, Encoding messages using chaotic synchronization, Phys. Rev. E, vol. 53, pp , [9] U. Feldmann, M. Hasler, and W. Schwarz, Communication by chaotic signals: The inverse system approach, Int. J. Circuit Theory Applicat., vol. 24, pp , [10] J. P. Eckmann and D. Ruelle, Ergodic theory of chaos and strange attractors, Rev. Modern Phys., vol. 57, pp , [11] W. Ebeling, J. Freund, and K. Rateitschak, Entropy and extended memory in discrete chaotic dynamics, Int. J. Bifurc. Chaos, vol. 6, pp , [12] C. E. Shannon, A mathematical theory of communication, Bell Syst. Tech. J., vol. 27, pp , [13] J. F. Heagy, T. L. Carroll, and L. M. Pecora, Desynchronization by periodic orbits, Phys. Rev. E, vol. 52, pp. R1253 R1256, [14] D. J. Gauthier and J. C. Bienfang, Intermittent loss of synchronization in coupled chaotic oscillators: Toward a new criterion for high-quality synchronization, Phys. Rev. Lett., vol. 77, pp , [15] T. Stojanovski, L. Kocarev, U. Parlitz, and R. Harris, Sporadic driving of dynamical systems, Phys. Rev. E, vol. 55, pp , [16] G. Ungerboeck, Trellis-coded modulation with redundant signal sets, Part I: Introduction, IEEE Commun. Mag., vol. 25, pp. 5 11, [17] H. G. Schuster, S. Martin, and W. Martienesen, New method for determining the largest Liapunov exponent of simple nonlinear systems, Phys. Rev. A, vol. 33, pp , V. CONCLUSION For generating partitions: 1) a chaotic system turns into an information source whose source entropy is equal to the KS entropy h KS and 2) there is an injective mapping between initial states of the chaotic system and coarse-grained trajectories. According to the noisy channel theorem [12], the coarse-grained trajectory can be transmitted with negligible probability of error over a channel with capacity C > h KS. In other words, if C > h KS, then using generating partitions one can produce arbitrarily small synchronization error, i.e., a HQSM. Achieving error-free transmission at information rates arbitrarily close to the channel capacity C has been a hot area of research in the last almost 50 years and significant results have been produced amongst which probably the most significant is the TCM. We believe that the problem of improving the synchronized motion which is very sensitive to the noise influence, and establishing a HQSM can be solved with application of the knowledge from information theory. The example from Section IV contributes to this aim, and illustrates how the principles of the TCM can be employed in achieving a HQSM. REFERENCES [1] H. G. Schuster, Deterministic Chaos: An Introduction. Weinheim: VCH, [2] H. 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Rulkov Abstract Using a low frequency nonlinear electrical circuit, we experimentally demonstrate an efficient nonlinear control method based on our theoretical developments. The method works in a state space for the circuit which is reconstructed from observations of a single voltage. Assuming small control variations from the uncontrolled state, the method is fully nonlinear and one step optimal. It requires no knowledge of local state space linearizations of the dynamics near the target state. Starting from various initial states within the basin of attraction of the circuit attractor, we control to a period one and to a period two target orbit. Each target orbit is an unstable periodic orbit of the uncontrolled system. Index Terms Chaos, control, nonlinear, optimal. I. INTRODUCTION Nonlinear dynamical systems in a regime where their uncontrolled motion can be chaotic are unstable everywhere on their attractor Manuscript received January 10, 1997; revised October 6, This work was supported in part by the U.S. Department of Energy by Grant DE-FG03-90ER14138 and Grant DE-FG03-95ER14516 and by the Australian Research Council and the National Science Foundation under Grant NCR The work of A. I. Mees was supported by The University of Western Australia. This paper was recommended by Guest Editor M. P. Kennedy. H. D. I. Abarbanel is with the Department of Physics and the Marine Physical Laboratory, Scripps Institution of Oceanography, University of California at San Diego, La Jolla, CA USA ( hdia@hamilton.ucsd.edu). L. Korzinov and N. F. Rulkov are with the Institute for Nonlinear Science, University of California at San Diego, La Jolla, CA USA ( korz@hamilton.ucsd.edu; rulkov@hamilton.ucsd.edu). A. I. Mees is with the Department of Mathematics, The University of Western Australia, Nedlands, Perth 6907, Western Australia, on leave at the Institute for Nonlinear Science, University of California at San Diego, La Jolla, CA USA ( alistair@maths.uwa.edu.au). Publisher Item Identifier S (97) /97$ IEEE