Generating Chaos in Chua s Circuit via Time-Delay Feedback

IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 48, NO. 9, SEPTEMBER 1 1151 model for out-of-sample drive signals and express this error in terms of signal-to-noise in column five. We calculate the signal-to-noise ratio using SNR = log 1 std(actual) std(errors) db: (13) We show the analogous results obtained by reconstructing and testing static models on the same data in the remaining columns. Typical results are presented in Fig. (a) (d) where we show sections of the time series produced by simulating the models compared to the actual measured values of the device. We show the results obtained by simulating the models reconstructed using the C 3, C 6, C 9 and C 1 data sets. Good agreement is seen in the figures as expected from the numbers given in Table I. These simulations are also superior to the results we obtained using the best static models we could reconstruct. In most of the cases examined, the long-term solutions are not sensitive to the initial seed value and, in the case of periodic drives, they appear to converge to a unique solution. [11] I. J. Leontaritis and S. A. Billings, Input-Output parametric models for nonlinear systems part I: Deterministic nonlinear systems, Int. J. Control, vol. 41, no., pp. 33 38, 1985. [1] S. Haykin and J. Principe, Making sense of a complex world, IEEE Signal Processing Mag., pp. 66 81, May 1998. [13] J. Stark, Delay Embeddings of Forced Systems: I deterministic forcing, J. Nonlinear Science, vol. 9, pp. 55 33, 1999. [14] G. F. Franklin, Feedback Control of Dynamic Systems. Reading, MA: Addison-Wesley, 1994. [15] D. M. Walker and N. B. Tufillaro, Phase space reconstruction using input-output time-series data, Phys. Rev. E, vol. 6, no. 4, pp. 48 413, 1999. [16] D. M. Walker, R. Brown, and N. B. Tufillaro, Constructing transportable behavioral models for nonlinear electronic devices, Phys. Letts A, vol. 55, no. 4/6, pp. 36 4, 1999. [17] K. Judd and A. I. Mees, On selecting models for nonlinear time series, Physica D, vol. 8, no. 4, pp. 46 444, 1995. [18] J. S. Lee and L. E. Miller, CDMA Systems Engineering Handbook. Boston, MA: Artech, 1998. [19] B. T. Murphy, Diode and transistor self-analogues for circuit analysis, Bell Syst. Techn. J., no. 4, pp. 487 5, 1968. [] J. King, Matlab for Engineers. San Francisco, CA: Addison-Wesley, 1998. V. CONCLUSION We have shown how to construct stable, free-running, input-output models for a class of electronic devices and circuits having fading memory and (in the absence of a drive signal) converge to a constant solution. The models are built from band-limited, spread-spectrum excitations and such excitations provide a sufficiently rich training set to make accurate predictions of periodic or similar spread spectrum drive signals. Adding additional local and global properties appears to be a promising avenue for research in building stable and accurate black-box models. ACKNOWLEDGMENT The authors wish to thank R. Brown for some early discussions about this investigation. Generating Chaos in Chua s Circuit via Time-Delay Feedback Xiao Fan Wang, Guo-Qun Zhong, Kit-Sang Tang, Kim F. Man, and Zhi-Feng Liu Abstract A time-delay chaotification approach can be applied to the Chua s circuit by adding a small-amplitude time-delay feedback voltage to the circuit. The chaotic dynamics of this newly derived time-delay Chua s circuit is studied by theoretical analysis, verified by computer simulations as well as by circuit experiments. Index Terms Chaos, stability, time delay. REFERENCES [1] H. Kantz and T. Schrieber, Nonlinear Time Series Analysis. Cambridge, U.K: Cambridge Univ. Press, 1997. [] N. Tufillaro, D. Usikov, L. Barford, D. Walker, and D. Schreurs, Measurement driven models of nonlinear electronic components, in Dig. 54th ARFTG Conf., Boston, MA, June. [3] K. Judd and M. Small, Towards long-term prediction, Physica D, vol. 136, no. 1/, pp. 31 44,. [4] L. A. Aguirre and S. A. Billings, Dyanamical effects of overparametrization in nonlinear models, Physica D, vol. 8, no. 1,, pp. 6 4, 1995. [5] L. Chua, C. Desoer, and E. Kuh, Linear and Nonlinear Circuits. San Francisco, CA: McGraw-Hill, 1987. [6] S. Boyd and L. Chua, Fading memory and the problem of approximating nonlinear operators with Volterra series, IEEE Trans. Circuits Syst., vol. CAS-3, pp. 115 1161, Nov. 1985. [7] R. Harber and H. Unbchauen, Structure identification of nonlinear dyanamic systems, A survey on input/output approaches, Automatica, vol. 6, no. 4, pp. 651 677, 199. [8] L. A. Aguirre, P. F. Donoso-Garcia, and R. Santos-Filho, Use of a priori Information in the Identification of Global Nonlinear Models A Case Study Using a Buck Converter, IEEE Trans. Circuits Syst. I, vol. 47, pp. 181 185, July. [9] N. A. Gershenfeld, The Nature of Mathematical Modeling. Cambridge, U.K: Cambridge Univ. Press, 1999. [1] M. Casdagli, A dynamical systems approach to modeling input-output systems, in Nonlinear Modeling and Forecasting, SFI Studies in the Sciences of Complexity, Proc, M. Casdagli and S. Eubank, Eds. Reading, MA: Addison-Wesley, 199, vol. XII. I. INTRODUCTION Chua s circuit is one of the physical systems for which the presence of chaos (in the sense of Shil nikov) has been established experimentally, confirmed numerically, and proven mathematically. In recent years, Chua s circuit has become a standard model for studying chaos in systems described by finite-dimensional ordinary differential equations [1]. Synchronization of chaotic Chua s circuit with application to secure communication has also been investigated. However, a classic Chua s circuit is a third-order continuous-time autonomous system which can only produce low-dimensional chaos with one positive Lyapunov exponent. Manuscript received January 4, ; revised August 1, and December 1,. This work was supported in part by the Research Grants Council, Hong Kong, under Competitive Earmarked Research Grant 94565, and in part by the City University of Hong Kong under Strategic Grant 7956. This paper was recommended by Associate Editor M. Di Bernardo. X. F. Wang was with the Department of Automatic Control, Nanjing University of Science & Technology, Nanjing, 194, China. He is now with the Department of Mechanical Engineering, University of Bristol, Bristol, U.K. (e-mail: X.Wang@bristol.ac.uk). G.-Q. Zhong, K.-S. Tang, K. F. Man and Z.-F. Liu are with the Department of Electronic Engineering, City University of Hong Kong, Kowloon, Hong Kong. Publisher Item Identifier S 157-71(1)7711-X. 157 71/1$1. 1 IEEE

115 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 48, NO. 9, SEPTEMBER 1 Fig. 1. The time-delay-feedback Chua s circuit. On the other hand, it has been known that even a very simple first-order system with a time-delay feedback can produce very complex chaotic behaviors [] [9]. Mathematically, continuous-time systems with time-delay feedback can be described by delay differential equations that possess of infinite-dimensional state spaces and have the possibilities of producing high-dimensional hyperchaotic attractors with a large number of positive Lyapunov exponents. This property has already stimulated the work on both analysis and design, and we have also witnessed a number of physically implemented chaotic time-delay systems for secure communication which claimed to have low detectability [1] [13]. Recently, in the pursuit of anticontrol of chaos (also called chaotification) [14] [18], we have seen a general approach to making a continuous-time autonomous system chaotic via time-delay-feedback perturbation on a system parameter, or an exogenous time-delay-feedback input [18]. This work aims at building a practical chaotic time-delay circuit based on a standard Chua s circuit. This can be achieved by adding a small-amplitude time-delay voltage feedback in a nominal Chua s circuit. Asymptotic analysis shows that, if the nominal Chua s circuit has an exponentially stable point, then the corresponding time-delay-feedback Chua s circuit can be chaotic within a neighborhood of the equilibrium point for arbitrarily given small feedback amplitude. Both computer simulations and circuit experiments show that the time-delay-feedback Chua s circuit can create different type of chaotic attractors, even if the nominal Chua s circuit has stable multiperiod orbits. Therefore, this newly derived time-delay-feedback Chua s circuit can be confirmed and used as an robust chaos generator. II. TIME-DELAY FEEDBACK CHUA S CIRCUIT The nominal Chua s circuit consists of a linear inductor L, two linear resistors R and R, two linear capacitors C 1 and C and a nonlinear resistor-the Chua s diode N R (see Fig. 1). The state equations of Chua s circuit are given by dt = 1 (G(v C 1 v 1 ) f (v 1 )) dv dt = 1 (i C 3 G(v v 1)) di 3 dt = 1 L (v + R i 3 ) where v 1, v and i 3 are the voltage across C 1, the voltage across C and the current through L, respectively. G =1=R and f (v 1)=G b v 1 + 1 (Ga G b)fjv 1 + Ej jv 1 Ejg (1) is the v-i characteristic of the nonlinear resistor with a slope equal to G a in the inner region and G b in the outer region and E is the breakpoint voltage of the Chua s diode. Because of the piecewise-linear nature of N R, the vector field of Chua s circuit may be decomposed into three distinct affine regions: v 1 < E, jv 1 j E and v 1 > E. We call these the D1, D, and D 1 regions, respectively. For a fixed set of parameters and initial conditions, the trajectories of the Chua s circuit could converge to a stable equilibrium point, a limit cycle, or a lowdimensional chaotic attractor with one positive Lyapunov exponent. To make a nonchaotic Chua s circuit chaotic via perturbation with arbitrarily given amplitude, we add a time-delay voltage feedback in the nominal Chua s circuit, as shown in Fig. 1. The new time-delayfeedback Chua s circuit can be described by the following equations: dt = 1 (G(v v 1) f (v 1)) C 1 dv dt = 1 (i C 3 G(v v 1 )) di 3 dt = 1 (v + Ri3 + w(v1(t ))) L where the time delay term is taken as () w(v 1 (t )) = " sin(v 1 (t )) (3) where " and are two positive constants and represents the delay time. Clearly, the maximum amplitude of the time-delay term is ", i.e. jw(v 1 (t ))j ": (4) We will show that for arbitrarily given ">, the time-delay-feedback Chua s circuit () can be chaotic for sufficiently large and, even if the corresponding nominal Chua s circuit (1) has stable period orbits. III. ANALYSIS OF CHAOS IN TIME-DELAY FEEDBACK CHUA S CIRCUIT In this section, we suppose that for a fixed set of parameters, the nominal Chua s circuit (1) has a symmetric pair of exponentially stable equilibrium points P + = [v1;v 3 ;i 3 3 3] T D 1 and P =[v1; 3 v; 3 i3] 3 T D 1, where v 3 1 = (GR + 1)(G a G b ) G +(GR +1)G b v 3 = GR GR +1 v3 1

IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 48, NO. 9, SEPTEMBER 1 1153 i 3 3 = G GR +1 v3 1: Due to the symmetry of P + and P, we will consider the equilibrium point P + only. We will present an asymptotic analysis to show in such a way that, for the same set of circuit parameters and any given small constant ">, the corresponding time-delay-feedback Chua s circuit is chaotic within a neighborhood of P + for sufficiently large constant and delay time. Denote x(t) = [v 1 (t) v (t)i 3 (t)] T and x(t) = x(t) P +. According to nonlinear system theory [19], since P + is an exponentially stable point of the nominal Chua s circuit (1), there exists a Lyapunov function V (t; x) of (1) that satisfies d 1kxk V (t; x) d kxk _V j (1) d 3 kxk (5) @V d @ x 4kxk for all (t; x) [; 1) D, where D = fx < n jkxk <rg and d 1, d, d 3, d 4 and r are positive constants. Denote f (x) = g(t; x) = Equation () can be rewritten as 1 C 1 (G(v v 1 ) f (v 1 )) 1 C (i 3 G(v v 1 )) 1 L (v + R i 3 ) L 1 w(v 1(t )) _x = f (x) +g(t; x) : By viewing g(t; x) as a perturbation term, we have the following boundedness result which can be derived from Lemma 5. in [19]. Lemma 1: Let P + be an exponentially stable equilibrium point of the nominal Chua s circuit (1) and V (t) be a Lyapunov function that satisfies (5). Suppose that jw(v 1(t ))j "< d 3 d 4 d 1 d Lr for some positive constant <1. Then, for all initial conditions x(t) satisfy kx(t)k < d 1=d r, t, the solution of the time-delayfeedback Chua s circuit () is bounded and there exists a finite time t 1 >, such that where Furthermore, if kx(t)k b" 8t t 1 b = d 4 Ld 3 d d 1 : "< v3 1 E b then x(t) D 1, i.e., v 1(t) >Efor all t t 1. Lemma 1 can be viewed as a robustness property of the nominal Chua s circuit that possesses an exponentially-stable equilibrium point at P +. It shows that small-amplitude perturbation will not result in large steady-state deviations from the equilibrium point. That is, if " is small enough and the initial conditions are near P +, then the trajectories x(t) of the time-delay-feedback Chua s circuit () will lie in a neighborhood of P + which belongs to D 1. Furthermore, we show that the trajectories can be chaotic within the neighborhood, if and are large enough. : In the region D 1,wehave f (v 1)=G b v 1 + G a G b ; df (v 1 ) = G b : From the first and second equation of (1), we get (where we assume E = 1) v =RC 1 dt +(RG b +1)v 1 +(G a G b )R (6) d v 1 i 3 =RC 1 C dt +(RC G b + C 1 + C ) dt + G b v 1 + G a G b : (7) Denote v 1 (t) =v 1 (t) v 3 1 and w(v 1 (t )) = w(v 1 (t )) = w(v 1(t )+v 3 1), we can derived from (6), (7) and the third equation of (1) that d 3 v 1 dt 3 + d v 1 dt + 1 dt + v 1 (t) = w(v 1 (t )) (8) where = G + G b(gr +1) LC 1C 1 = LGG b + C 1 (1 + R G)+(G + G b )R C LC 1 C = GLC1 + LC(G + G b)+r C 1C LC 1C = G : LC 1 C Therefore, in the region D 1, the () is equivalent to the time delay (8). Since P + is an exponentially-stable equilibrium point of (1), s 3 + s + 1 s + is a Hurwitz stable polynomial. Therefore, (8) can be viewed as a third-order stable linear differential equation with a timedelay feedback w(v 1(t )). Recently, an approximate relationship between a nth-order linear differential equation with a time-delay feedback and its corresponding discrete map [18] has been established. This relationship is a generalization of the results in [4], [9]. Lemma : Consider a system described by the time-delay differential equation y (n) (t)+ n1y (n1) (t)+111+ 1 _y(t)+ y(t) = w(y(t )) (9) where f i g n1 i= and are constants with 6= and s n + n1s n1 + 111 + 1s + is a Hurwitz polynomial. Denote y(m; ^t) = y(m + ^t). For a sufficiently large and large ^t <,wehave y(m; ^t) w y(m 1; ^t) (1) with y (k) (m; ^t) for m =; 1 ;...and k =1; ;... ;n 1. Denote v 1 (m; ^t) = v 1 (m + ^t) for t = m + ^t. According to Lemma, for sufficiently large time delay and large ^t <,wehave v 1 (m; ^t) w(^v 1 (m 1; ^t)) which implies that, if (= )w is a chaotic map, then the output v 1(t) of the time-delay (8) would be chaotic. From the choice of w, we get y k+1 = w(y k )=" sin((y k + v 3 1 )) (11) where " =( = )" 6=. It can be proved that the map is chaotic in the sense of Li-Yorke [] for sufficiently large.

1154 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 48, NO. 9, SEPTEMBER 1 Fig.. Several different types of chaotic attractors of the time-delay-feedback Chua s circuit with circuit parameter given in (1) and R = 195. (a) " =:, =:. (b) " =:, =:5. (c) " =:5, =:5. (d) " =:5, =. Moreover, according to [1], the Lyapunov dimension d L of the corresponding chaotic attractor is nearly equal to the delay time divided by the correlation time of the time-delay-feedback driving force w(v 1), i.e., d L = =. Therefore, the time-delay-feedback Chua s circuit can produce high-dimensional chaotic attractors for a large value of delay time. IV. SIMULATION RESULTS In our computer simulations, the following values are fixed. (See equation (1) at the bottom of the page.) In the case R = 195, the nominal Chua s circuit has a pair of stable equilibrium points Fig. 3. A stable-period orbit of the Chua s circuit with circuit parameter given in (1) and R = 191. In summary, if the nominal Chua s circuit has an exponentially-stable equilibrium point, then for an arbitrarily given small constant " 6=, the corresponding time-delay-feedback Chua s circuit can be chaotic within a neighborhood of the equilibrium point for sufficiently large and. To further reinforce the finding, both computer simulations and circuit experiments can be shown that the result obtained can be generalized to the case the nominal Chua s circuit has stable multiperiod orbits. P + =[3:4478:96 :18] T P =[3:4478 :96:18] T : Fig. (a) (d) shows several different types of chaotic attractors of the time-delay circuit for different values of " and. The maximum Lyapunov exponents of the four attractors are approximately equal to.11,.16,.19, and.1, respectively. Now, we take R = 191. The nominal Chua s circuit has two stable period orbits, as shown in Fig. 3. Fig. 4(a) (d) shows several different types of chaotic attractors of the time-delay-feedback Chua s circuit with R = 191 and different values of " and. The corresponding maximum Lyapunov exponents are approximately equal to.8,.15,.18, and.3, respectively. C 1 =1nF C = 1 nf L =18:68 mh R =16 G a = :75 ms G b = :41 ms E =1V =:1 (1)

IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 48, NO. 9, SEPTEMBER 1 1155 Fig. 4. Several different types of chaotic attractors of the time-delay-feedback Chua s circuit with circuit parameter given in (1) and R = 191. (a) " = :4, =:5. (b) " =:, =:5. (c) " =:5, =3. (d) " =1, =1. Fig. 5. Circuit implementation of time delay. V. EXPERIMENTAL RESULTS To supplement the computer simulation results, a physical circuit implementation was developed. In our experimental setup, the following circuit parameters are fixed C 1 = 11: F C =18:99 nf L =18:68 mh R =14:78 Ga = :87 ms G b = :44 ms E = 1 V = :4 96: (13) The sine function is implemented by using the universal trigonometric function converter IC, AD639. Implementation of time delay is shown in Fig. 5. For R = 18, the nominal Chua s circuit has a pair of stable equilibrium points [Fig. 6(a)]. Fig. 6(b) (d) shows a spiral chaotic attractor, a double scroll chaotic attractor and a full-developed chaotic attractor of the time-delay-feedback Chua s circuit with (b) " =:45, =:855, (c) " =:65, =:855, and (d) " =:199, =:91, respectively. In the case of R = 17, the Chua s circuit has a pair of stablelimit circles [Fig. 7(a)]. Fig. 7(b) (d) shows a spiral chaotic attractor, a double-scroll chaotic attractor, and a fully-developed chaotic attractor of the time-delay-feedback Chua s circuit with (b) " = :6, = :855, (c) " = :58, = :855 and (d) " = :185, = 1:74, respectively.

1156 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 48, NO. 9, SEPTEMBER 1 ACKNOWLEDGMENT The authors wish to thank the anonymous reviewers for their valuable comments and suggestions. Fig. 6. (a) A stable-equilibrium point of the Chua s circuit. (b) (d) Several different types of chaotic attractors of the time-delay-feedback Chua s circuit with circuit parameter given in (13) and R = 18. (b) " = :45, = :855. (c) " =:65, =:855. (d) " =:199, =:91. (a) (c) (d) Fig. 7. (a) A limit cycle of the Chua s circuit. (b) (d) Several different types of chaotic attractors of the time-delay-feedback Chua s circuit with circuit parameters as given in (13) and R = 17. (b) " = :6, = :855. (c) " =:58, =:855. (d) " =:185, =1:74. VI. CONCLUSION In this paper, a new time-delay-feedback Chua s circuit is established by adding a small-amplitude voltage time-delay-feedback to an ordinary Chua s circuit. This time-delay-feedback Chua s circuit is different from the time-delayed Chua s circuit proposed by Sharkowsky et al. [] which is derived from the classical Chua s circuit by substitution of the lumped LC resonator with an ideal transmission line. Chaotic dynamics in the time-delay-feedback Chua s circuit were investigated via asymptotic analysis, computer simulations, and circuit experiments. We will further investigate the complex behavior of this new circuit and it s application to chaos-based secure communication. (b) REFERENCES [1] L. O. Chua, C. W. Wu, A. Huang, and G.-Q. Zhong, A universal circuit for studying and generating chaos-part I: Routes to chaos, IEEE Trans. Circuits Syst. I, vol. 4, pp. 73 74, Oct. 1993. [] M. C. Mackey and L. Glass, Oscillation and chaos in physiological control systems, Science, vol. 197, no. 3, pp. 87 89, 1977. [3] J. D. Farmer, Chaotic attractors of an infinite-dimensional dynamical systems, Physica D, vol. 4, no., pp. 366 393, 198. [4] K. Ikeda and K. Matsumoto, High-dimensional chaotic behavior in systems with time-delay feedback, Physica D, vol. 9, pp. 3 35, 1987. [5] T. Chevalier, A. Freund, and J. Ross, The effects of a nonlinear delayed feedback on a chemical reaction, J. Chem. Phys., vol. 95, no., pp. 38 316, 1991. [6] O. Diekmann, S. A. van Gils, S. M. verduyn Lunel, and H.-O Walther, Delay Equations: Functional-, Complex-, and Nonlinear Analysis. Berlin, Germany: Springer-Verlag, 1995. [7] V. S. Anischenko, Dynamical Chaos Models and Experiments, Singapore: World Scientific, 1995. [8] H. Lu and Z. He, Chaotic behaviors in first-order autonomous continuous-time systems with delay, IEEE Trans. Circuits Syst. I, vol. 43, pp. 7 7, 1 1996. [9] P. Celka, Delay-differential equations versus 1D-map: Application to chaos control, Physica D, vol. 14, no. 1, pp. 17 147, 1997. [1] B. Mensour and A. Longtin, Synchronization of delay-differential equations with application to private communication, Phys. Lett. A, vol. 44, no. 1, pp. 59 7, 1998. [11] K. Paragas, Transmission of signals via synchronization of chaotic time-delay systems, Int. J. Bifurcation Chaos, vol. 8, no. 5, pp. 1839 184, 1998. [1] A. Tamasevicius, A. Cenys, A. Namajunas, and G. Mykolaitis, Synchronizing hyperchaos in infinite-dimensional systems, Chaos Solitons Fractals, vol. 9, no. 8, pp. 143 148, 1998. [13] R. He and P. G. Vaidya, Time delayed chaotic systems and their synchronization, Phys. Rev E., vol. 59, no. 4, pp. 448 451, 1999. [14] X. F. Wang and G. Chen, On feedback anticontrol of discrete chaos, Int. J. Bifurcation Chaos, vol. 9, no. 7, pp. 1435 144, 1999. [15], Chaotification via arbitrarily small feedback controls: Theory, method and applications, Int. J. Bifurcation Chaos, vol. 1, no. 3, pp. 549 57,. [16], Chaotifying a stable LTI system by tiny feedback control, IEEE Trans. Circuits Syst. I, vol. 47, pp. 41 415, Mar.. [17] X. F. Wang and Chen, Chaotifying a stable map via smooth small-amplitude high-frequency feedback control, Int. J. Circuit Theory Appl., vol. 8, no. 3, pp. 35 31,. [18] X. F. Wang, G. Chen, and Yu, Chaotification of continuous-time systems via time-delay feedback, Chaos, vol. 1, no. 4, pp. 771 779,. [19] H. Khalil, Nonlinear Systems, ed. New York: Macmillan, 1996. [] T. Y. Li and J. A. Yorke, Period three implies chaos, Amer. Math. Monthly, vol. 8, no. 4, pp. 481 485, 1975. [1] M. L. Berre, E. Ressayre, A. Tallet, H. M. Gibbs, D. L. Kaplan, and M. H. Rose, Conjecture on the dimension of chaotic attractors of delayed-feedback dynamical systems, Phys. Rev. A, vol. 35, no. 9, pp. 4 43, 1987. [] A. N. Sharkowsky, Y. Maistrenko, P. Deregel, and L. O. Chua, Dry turbulence from a time-delayed chua s circuit, J. Circuits Syst. Comput., vol. 3, no. 7, pp. 645 668, 1993.