562 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 44, NO. 6, JUNE 1997 Express Letters Digital Coding via Chaotic Systems Toni Stojanovski, Ljupco Kocarev, and Ulrich Parlitz Abstract—The concept of time-discontinuous coupling of two identical chaotic systems called sporadic coupling is addressed. The conditions for the occurrence of synchronization between two sporadically coupled chaotic systems are given. Amplitude quantization of the driving signal in addition to the sporadic coupling allows digital exchange of digital information signals between the sporadically coupled chaotic systems. Index Terms—Chaos, chaos synchronization, digital coding. I. INTRODUCTION Even though first papers analyzing synchronized motion of chaotic systems date back to 1983 [1] and 1986 [2], it is Pecora and Carroll’s work [3], [4] that has provoked intensive research on synchronized chaos. The unique properties of chaotic systems: randomness, aperi- odicity, and broad-band spectrum of their solutions in alliance with the possibility for their synchronization, that is, the predictability of the chaotic trajectory and its reproducibility in other chaotic systems, have motivated scientists from many scientific disciplines to analyze possible applications of synchronized chaos. Numerous published papers have addressed possible use of synchronized chaos in the area of analog communications claiming security of the proposed communication systems. However, today communications are almost exclusively digital. Motivated by this challenge—digital communications—we discuss in this letter the question of exchange of digital information signals between two synchronized continuous chaotic systems connected via a digital communication channel. Such a communication can be accomplished only if the driving signal, which also contains the encoded information, is discretized in both the time and the amplitude domain. The idea of time-discontinuous coupling of chaotic systems is already known [5]–[7]. In [5], it was reported that dynamical systems driven by sequences of random forces may exhibit asymptotically stable motion, i.e., independence of the initial conditions in terms of [5], provided that the time interval between the driving forces is smaller than a certain threshold. As announced in [6] and [7], in order to synchronize two chaotic systems it is sufficient to influence the state of the driven system only at discrete times. Here is the layout of our presentation. In Section II, we describe the concept of sporadic coupling and show that the synchronization of time-continuously coupled chaotic systems implies their synchroniza- tion also when they are time-discontinuously coupled for sufficiently small coupling periods. Section III examines the effect of amplitude Manuscript received March 26, 1996; revised Nobember 4, 1996. This paper was recommended by Associate Editor T. Endo T. Stojanovski was with the Department of Electrical Engineering, “Sv. Kiril i Metodij” University, Skopje, Republic of Macedonia. He is now with the CATT Centre, RMIT University, Melbourne, Australia (e-mail: tonci@cerera.etf.ukim.edu.mk). L. Kocarev is with the Department of Electrical Engineering, “Sv. Kiril i Metodij” University, Skopje, Republic of Macedonia (e-mail: lkocarev@cerera.etf.ukim.edu.mk). U. Parlitz is with the Drittes Physikalisches Institut, Universit¨ at G¨ ottingen, D-37073 G¨ ottingen, Germany (e-mail: ulli@physik3.gwdg.de). Publisher Item Identifier S 1057-7122(97)03480-6. quantization of the driving signal, and describes our digital coding based on chaotic systems. We close the paper with the conclusion and a brief summary of open problems. II. SPORADIC COUPLING OF CHAOTIC SYSTEMS At first we describe the method for synchronization of chaotic systems through driving of an asymptotically stable subsystem [3]. Consider two identical -dimensional chaotic systems (1) and (2) The basin of attraction of the considered chaotic attractor of (1) is a set . Decompose the state vector of (2) into two parts and , and the vector field into and . Then we can rewrite (2) as (3) (4) Consider (4) as a nonautonomous system whose driving term is . Let us assume that the driven system (4) is asymptotically stable with respect to the driving signal . It means that there exists a set , called a region of asymptotic stability of (4), such that for all initial conditions , that is, the solutions of (4) whose initial conditions belong to tend to each other. Then, a simple substitution of for in (3) leads to a driven version of (2) and (5) that synchronizes with (1), i.e., when . The difference exponentially vanishes with the time and the speed of convergence to 0 is determined by the conditional Lyapunov exponents (CLE) [4] of (5). Driving of (2) with the signal as described by (5) is applied continuously as time goes on. As already shown in [5]–[7], the con- tinuous coupling is not a necessity, that is, the synchronized motion between (1) and (2) can be achieved through time-discontinuous cou- pling. In other words, under certain conditions in order to synchronize two chaotic systems it is sufficient to couple them only at certain times and not continuously as time goes on. The concept of time-continuous coupling applied to (2) can be concisely described as follows: (6) where denotes a periodic se- quence of Dirac pulses with period ; is a sequence of -dimensional time-equidistant samples of the signal , i.e., ; and is a sequence of the samples of the signal immediately prior the times , i.e., 1057–7122/97$10.00  1997 IEEE