Synchronization and control in small networks of chaotic electronic circuits A. Iglesias Dept. of Applied Mathematics and Computational Sciences, Universi~ of Cantabria, Spain Abstract In this paper, a very recent chaos control method that stabilizes chaotic systems by introducing small perturbations in the system variables [l] and a new synchronization method that reproduces the driving signal with a single connection [2] are simultaneously applied to small networks of chaotic electronic circuits (the Chua circuits [3]).\ire stabilize a Chua circuit (which acts as drive of the cascade) by applying the chaos control method in [l]. and its regular behavior induces, by synchronizing with other circuits of the cascade. a regular behavior in the network. The paper explores the interaction between, on one hand: the analytical results and the numerical sirnulations and, on the other hand, the behavior of the real network of circuits. In particular, we show that all the results obtained from both sources agree. Finally, some interesting applications are discussed. 1 Introduction Recently, a new chaos control method that stabilizes chaotic systems by introducing small perturbations in the system variables has been suggested [l] and several features of its application to discrete and continuous dynamical systems have been discussed [4, 5; 6: 7, 81. On the other hand, several recent studies have shown the possibility of synchronizing chaotic systems [g]. An interesting extension of this method allows us to reproduce the driving signal with a single connection [2]: increasing thus the number of potential connections of a given system. In this paper we analyze the int,eresting effects obtained when both methods are simultaneously applied to small networks of chaotic electronic circuits (the Chua circuits [3]). \Ye stabilize a Chua circuit (whicll acts
as drive of the cascade) by applying the chaos control method in [l], and its regular behavior induces, by synchronizing with other circuits of the cascade, a regular behavior in the network. All this work has been carried out through two different (but complementary) approaches: 1. The electronic approach: assembling the real network of circuits and modifying the parameter values of some accesible electronic component. 2. The analytic & numerical approach: deriving the symbolic equations of the circuit and reproducing the different behaviors by computer. The structure of the paper is as follows: firstly, in Section 2, we introduce our model, the Chua circuit. The section opens with the description of the electronic components and their characteristics, as required by the first approach. Then, the Kirchoff laws are applied to derive the dimensionless equations of the circuit, used for the second approach. The analysis of the circuit behavior as a function of an accesible parameter is achieved in Section 3. This analysis allows us to choose parameter values leading to a chaotic regime, used to check the chaos control method applied in Section 4. In Section 5 a generalization of the Pecora & Carroll method [g] for synchronizing chaotic systems is applied to small networks of Chua circuits. Then, the combination of the control and synchronization methods, with very important implications in reusing chaotic circuits, is considered in Section 6. Finally, the paper closes with the main conclusions of this work. 2 The model: The Chua circuit Since its discovery in 1983, the Chua circuit has attracted a big much interest due to its simplicity, robustness and low cost, being extremely useful for the developn~ent and testing of experimental methods for investigating nonlinear systems and chaotic dynamics. In this section we describe the electronic components of the circuit and its equations. 2.1 Electronic components The Chua circuit is a simple electronic circuit exhibiting a wide variety of bifurcation and chaotic phenomena. Such simplicity is given by the fact that it contains 4 linear elelnents (one resistor, one inductor and two capacitors) and only one simple nonlinear element, called the Chua diode (see Figure 1). In addition, Table 1 reports the specification of these components, that is, their operation and tolerance values. However, as we will shown in Section 3, almost every chaotic and bifurcation phenomena that has been reported in the literature, as perioddoubling bifurcation, intermittency route to chaos, Hopf-like bifurcations, crisis route to chaos, metastable chaos, etc. have been observed in the Chua circuit.
Conlprtational Methods and E.vperinw~tul Measwes 80 1 Figure 1: The Chua circuit: it consists of four linear elements (a inductor L, two capacitors Cl and C2 and a resistor R) and one simple nonlinear element NR (the Chua diode). Table 1: Components of the Chua circuit. Components Induct or Capacitor 1 Capacitor 2 Resistor Nonlinear element Name L c1 c2 R NR 2.2 Equations for the Chua circuit In this section the equations for the Chua circuit are derived. From Figure 2(left) it must be noticed that: 1. the inductor L and the capacitor C2 are in parallel, so both support the same voltage 2. in the Chua diode, the intensity is given by: 1 INR = ~ VNR) = GBVN~ + -(GB- 2 GA)(IVN~ where GB and GA are given by Figure 2(right). - 11 + lhr + 11) (I) Applying the Kirchoff laws at the knots 1 and 2 (Figure 2(left)) and 1 1 taking into account the equalities G = - - = g and IN, (t) = VC, (t) g, R' N, we get
802 Cornputatiot~al Metl~ods and E.rper.imenta1 Measwes Figure 2: (left) Scheme to derive the equations of the Chua circuit; (right) Intensity vs. voltage of the nonlinear element of the Chua circuit. C dvc z (t) L = IL(t) - G(V& (t)- Vcl (t)) dt Eqns (1)-(4) describe the Chua circuit. 2.3 Dimensionless equations To perform numerical simulations by computer, dimensionless equations are more adequate. Applying the following changes of variables to the eqns (1)-(4) we obtain the following system of dimensionless differential equations: X = a[y -X - f(x)] y=x-y+z (5) i = -/3y d where '= - and dt is the 3-segment piecewise-linear characteristic of the nonlinear resistor (the Chua diode) and a, P, a and b are the circuit parameters. Note that the system of eqns (5) and (6) is equivalent to eqns (1)-(4).
Comnputatior~al Methods and E.vperinzental Measures 803 Figure 3: Behavior of the Chua circuit when varying the a parameter: (a) a = 6.5: fixed point; (b) cu = 8.0: period-l orbit; (c) cu = 8.2: period-2 orbit; (d) a = 8.42: period-4 orbit; (e) a = 8.5: spiral chaotic attractor; (f) a = 8.9: double-scroll chaotic attractor. 3 Analyzing the Chua circuit behavior In this section, we described the system behavior for the parameter values that will be used in the rest of the. paper.. It can be shown that, for certain values of the circuit parameters, such as,b = 14.28,~ = -1.14 and b = -0.71, the system (5)-(6) has, at least, three equilibrium points. For these values (that will be fixed for the rest of the paper) the Chua circuit exhibits a very rich behaviour as a function of the a parameter. Beginning with a = 6.5 the circuit presents a fixed point regime (Figure 3(a)), while for a = 8.0 the system destabilizes with a Hopf bifurcation into a limit cycle, an stable period-l orbit (see Figure 3(b)). Increasing the value of the parameter a, the system exhibits a period doubling bifurcation cascade, with a period-2 orbit for a = 8.2 (Figure 3(c)), period-4 orbit
for cu = 8.42 (Figure 3(d)), etc... All these periodic orbits can be visually appreciated after a brief transient which increases with the a: parameter, presenting each time a longer lifetime, finally falling into two spiral chaotic Rossler-type attractor for cu = 8.5, such that any trajectory from one side of the spiral attractor can never go to the other side, which implies that every simulation shows, depending on the initial conditions, only one of these spiral attractors, as in Figure 3(e). As usually in the period doubling route to chaos, periodic windows appear coexisting with the chaotic regime as, for example, a periodic-3 orbit for cu = 8.583, a period-6 orbit for cu = 8.595 and so on. Increasing further the value of the parameter cu until 8.9 the twin attractors of the spiral Chua circuit collided with each other, giving rise to the crisis plienomenon, with the birth to the odd-symmetric chaotic attractor, called double-scroll attractor, shown in Figure 3(f), in which the trajectories can go from one side to the other, indicating that a homoclinic bifurcation occurs before the crisis. Of course, many other interesting phenomena, as metastable chaos for cu = 8.986 or semiperiodicity (coexistance of a periodic orbit and the double-scroll attractor) for cu = 8.99, can be found for the Chua circuit. 4 Applying the chaos control method to the circuit In this section, we apply our chaos control method [l] to the Chua circuit following the electronic and the computational approaches. 4.1 Numerical simulations by computer The control method consists in the application every An iterations of a pulse of strength X in the system variables, having the form: with i r 0 (modan). The method can be interpreted by noticing that, depending of the sign of X, some quantity of xi is injected or retired depending of the value of X, at that moment. As a result, an interesting set of phenomena varying the values of the control parameters An and X can be obtained. Thus, varying An from 10 towards 100, the Chua circuit exhibits a sequence of bifurcations from the stable stationary state (An = 10) to a period-doubling cascade (period-:! for An = 50), to a spiral Chua attractor, finally arriving at a double-scroll chaotic attractor (An = 70). At this point, we must remark that to select An or X as the system parameter is not relevant, since there is a relation between both parameters, such that an enlargement of one of these parameters corresponds to an decrease of the another one and viceversa. Since the corresponding pictures are very much alike to those from Figure 3, they are not reproduced here.
4.2 Chaos control in the real circuit Similar results have been obtained in the experimental context by considering the value of the variable resistance R as a control parameter. Indeed, varying R from 2000 Cl continuously towards zero, the Chua circuit exhibits a route, through a Hopf bifurcation, from a stable stationary state (R = 2000 R) to a period-doubling cascade, with period-l (R = 1672 R), period-:! (R = 1664 R), period-4 (R = 1657 R), etc. and finally to a chaotic attractor. For R = 1630 R, we find the spiral attractor and when the parameter R crosses some critical value (in our experiments R FZ 1622 R), a crisis phenomenon gave birth to the double-scroll chaotic attractor. Once again, the pictures from the oscilloscope (with the appropriate scaling) look like the ones from Figure 3, so they are not included here. 5 Applying the synchronization method to a network of circuits The synchronization method applied in this work is a generalization of [g]. In this reference one has a drive-response couple such that a chaotic signal from the drive system is used to force a second response system. In fact, a subsystem of the drive will be used to make the response synchronize with the drive. The basic idea of this construction can be understood by assuming a decomposition of an n-dimensional dynamical system (n = m+p+q) into the following driven and response subsystems y1 = d x, Y) driving subsystem z1 = f (X, Z) response subsystem (9) where X E IRm, y E RP and z E RP. The subsystem (8) is decoupled from subsystem (9) and x(t) is used as the input of (9). Lyapunov exponents of the z-subsystem for a particular x(t) are called condztzonal Lyapunov exponents. In Pecora & Carroll approach [g], stable driven subsystems can be synchronized if their conditional Lyapunov exponents are all negative. When the response system (with respect to a given drive) acts as the drive of a second response system, this connection is being called cascadzng, with very interesting applications in the field of secure communications [10]. However, in [g] the connectivity is limited because not all the possible subsystems are stable from the viewpoint of synchronization. Then, an interesting generalization could be to allow the driving system to enter at one or more terms in the evolution equations of the response. This idea has been considered in [2]. In this modified method, the synchronization has been achieved by introducing the driving signal at a given place of the evolution equations of the response. As a consequence, it is possible to regenerate the input signal within a single connection. It implies the possibility of using
806 Con~putariot~al Methods and E.yer.in~etltal Measures twice the same connection without alternating with another one and, therefore, a number of different networks with different connectivities can be set UP. 5.1 Numerical simulations by computer Three stable connections have been found in the case of the Chua circuits, which are shown in Table 2. Note that the place where the driving signal enters has been underlined for clarity. Table 2: Conditional Lyapunov exponents for various partial connections in a network of Chua circuits. Partial connection Lyapunov exponents In this work, we only consider one of them: taking (5) as the evolution equation for the drive, the response of the second circuit is written as: With this scheme, some different connections can be considered for the case of n oscillators by just replacing eqn (10) by where the index i is used to indicate the ith circuit. A typical configuration is given by the linear geometry with a single connection. We have carried out simulations for this case with n taking values from 2 to 6 and for different values of the parameter a corresponding to regular and chaotic behaviors. In all the situations, synchronization has been obtained for all the circuits of the network. An interesting case is given by a network of chaotic circuits with the same parameter values, for instance cu = 8.9. Deterministic chaos is characterized by the strong dependence of initial conditions. This implies that different initial conditions generate, in a natural way, different behaviors, even for circuits with the same parameter values. Therefore, before applying the synchronization method, their evolution is mutually independent. On the contrary, after applying it the second circuit is almost instantaneously synchronized with the first one, the third circuit is synchronized with the second one a little bit longer and, in general, the ith circuit synchronizes with the previous one after a brief transient that increases with the index of the circuit.
A different configuration is given by the closed loop, in which the last circuit of the network acts as a drive for the first one. The results obtained for the network of chaotic circuits are very similar although the transients are shorter in the sense that the global network takes less time to become totally synchronized. A more interesting scenario consists of a first circuit exhibiting a regular behavior, let us say a = 6.5 and a chaotic regime for the rest of the circuits in a linear cascading. In this case, synchronization is also obtained. This is a relevant result, because it is not expected a regular behavior to dominate the chaotic one and it has many implications that will be discussed in the next section. We finally remark that these results do not apply to the case of closed loops. For this configuration, synchronization is also obtained, but the global network always synchronizes to the chaotic behavior. 5.2 Electronic synchronization All the results of the previous section apply here. It may be surprising because the experimental devices do not exhibit the ideal conditions of the computer simulations (in fact, the last column of Table 1 shows the tolerance errors of the circuit components). It should be noticed, however, that the synchronization method is robust to small perturbations on the parameters of the drive or response systems [2]. 6 Combination of the control and synchronization methods and its application to a network of circuits In practical settings, chaotic behaviors are usually undesirable and one is interested in obtaining a more regular behavior for the system. That is the aim of the chaos control methods, which consist of stabilizing the desired periodic behavior from those that span the chaotic attractor. On the other hand, the synchronization methods allow to extend the individual behavior of a circuit to a network. Furthermore, in the previous section we have established that, under certain conditions (for instance, a linear cascading), the regular behavior of the first circuit extends to a global network of chaotic circuits. And such a regular behavior of the drive can be induced by applying a control method, as shown in Section 4. Therefore, an adequate combination of control and synchronization methods allows us to stabilize a network of chaotic circuits by controlling the first circuit only (provided that they follow the linear cascading or a similar pattern). 7 Conclusions By conclusion, the applications of these ideas include the possibility of reusing sets of chaotic circuits, which exhibited undesirable behaviors before control. This kind of connection can also be applied to the case where
the units represent model neurons. Then, arrays of circuits might be useful as information proccesing systems by synchronizing one to each other, mimicking the behavior observed in physiological studies. Acknowledgements Authors would like to acknowledge the CICYT of the Spanish Ministry of Education (project PB98-0640) and the European Fund FEDER (Contract 1FD97-0309) for partial support of this work. References [l] Matias, h1.a. & Giikmez, J. Stabilization of chaos by proportional pulses in the system variables. Phys. Rev. Lett. 72, pp. 1455-1458, 1994. [2] Gukmez J. & Matias, MA. Modified method for synchronizing and cascading chaotic systems. Phys. Rev. E, 52, pp. 2145-2148, 1995. [3] Chua, L.O., Komuro, M. & Matsumoto, T. The double-scroll family. IEEE Trans. Circuits and Syst., 33, pp. 1073-1118, 1986. [4] Gukmez, J. & Matias, MA. Control of chaos in unidimensional maps. Phys. Lett. A, 181 pp. 29-32, 1993. [5] Gukmez, J., Gutibrrez, J.M., Iglesias, A. 6t hiatias, N.A. Stabilization of periodic and quasiperiodic motion in chaotic systems through changes in the system variables. Phys. Lett. A, 190, pp. 429-433, 1994. [6] Gukmez, J., Gutibrrez, J.M., Iglesias, A. & hlatias, h1.a. Suppression of chaos through changes in the system variables: transient chaos and crises. Physica D, 79, pp. 164-173, 1994. [7] Gutikrrez, J.M., Iglesias, A., GuQmez, J. & hlatias, h1.a. Suppression of chaos through changes in the system variables through Poincark and Lorenz return maps. Int. J. Bzf. Chaos, 6, pp. 1351-1362, 1996. [8] Iglesias, A., Gutikrrez, J.M., Gukmez, J. & hlatias, M.A. Suppression of Chaos through Changes in the System Variables and Numerical Rounding Errors. Chaos, Solitons and Fractals, 7, pp. 1305-1316, 1996. [g] Pecora, L.M. & Carroll, T.L. Synchronization on chaotic systems. Phys. Rev. Lett., 64, 821-824, 1990. [l01 Kocarev, L. J., Halle, K. S., Eckert, K., Chua, L. 0. & Parlitz, U. Experimental demonstration of secure communications via chaotic synchronization. Int. J. of Bifurcatzon and Chaos, 2, pp. 709-713, 1992.