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NATIONAL CONFERENCE ON NONLINEAR SYSTEMS & DYNAMICS 45 Dynamics of two coupled canonical Chua’s circuits K. Thamilmaran and D. V. Senthilkumar Abstract—The chaotic behaviour of the canonical Chua’s circuit is extensively reported in the literature. However, for the first time, two identical canonical Chua’s circuits have been coupled resistively and the resultant synchronization of chaos in them has been studied. The method of linear difference signal has been applied. The corresponding differential equations have been integrated numerically and the synchronization threshold of the coupling parameter has been identified. Prior to synchronization, the coupled oscillator is found to exhibit hyperchaotic behaviour for certain higher strengths of coupling parameter. The Lyapunov exponents of the system and the bifurcation diagrams of the main synchronization regime have also been presented. The results have been confirmed by laboratory experiments. Keywords—Chaos, Hyperchaos, Synchronization and Mutual coupling. II. C IRCUIT REALIZATION OF COUPLED CANONICAL C HUA’ S CIRCUITS AND ITS DYNAMICS : E XPERIMENTAL S TUDY The experimental circuit is shown in Fig. 1. It consists of two canonical Chua’s circuit one being the drive system and the other being the response system. Individually each canonical which has circuit consists of a nonlinear resistive element, the typical characteristic, viz., negative resistance around the origin and positive slopes away from the break point and a , which can be built using off-the linear negative conductor, shelf op-amp. I. I NTRODUCTION N the last decade control and synchronization of chaos in coupled systems have received considerable attention [1]. This is because coupled systems are good models to describe nonlinear phenomena in natural sciences, chemistry, communication engineering, etc [2]. Theoretically, when two chaotic systems which are independently excited, interact mutually through a coupling resistance, the coupled system reaches a state wherein there is a perfect convergence of the state variables of both the systems and mutual sharing of different dynamical features [3]. This state of convergence arising due to adjustment of characteristics of individual chaotic systems is defined as synchronization. Recently, Pecora and Carroll have introduced the concept of synchronized chaos [4] and the method of cascading synchronization [5]. This idea has been in fact successfully tested in a variety of nonlinear dynamical systems, including Lorenz equations, the Rossler system, hysteresis circuits, Chua’s circuit, and so on [6] - [10] . Also, synchronization of chaos has been investigated in two mutually coupled identical nonlinear systems [11], [12]. Very recently, Kyprianidis and Stouboulos have studied coupled four-dimensional autonomous canonical Chua’circuit with an external force [13], by adding one inductor, one linear resistor and one voltage source to the original canonical Chua’s circuit, and so its dynamics is given by a system of ten-coupled first-order differential equations. In this paper we consider the evolution from nonsynchronized self-oscillations to synchronized one with mutual coupling parameter in a system of two coupled canonical Chua’s circuits. The most interesting aspect of this coupled oscillator system is that it is the simplest (six dimension) coupled oscillator reported so far in the canonical Chua’s circuit family that can exhibit hyperchaotic oscillations. K. Thamilmaran and D. V. Senthilkumar are with Centre for Nonlinear Dynamics, Department of Physics, Bharathidasan University, Tiruchirappalli - 620 024, India, e-mail: [email protected],. The work forms part of a Department of Science and Technology, Government of India research project of Prof. M. Lakshmanan. L R - - - L - + v’ 1 C1 C2 - G1 + iL + v 1 v2 + R RC ’ +iL + v’ 2 C1 NR N R - - C2 - - G1 Fig. 1. Circuit realization of the two resistively coupled chaotic circuits ( coupling) . The circuit equations for the subsystems (either drive or response system) when they are uncoupled are easily obtained by applying Kirchoff’s laws to the various branches of the subcircuit of Fig. 1. For example, we get the following set of three coupled first-order differential equations for the left hand subsystem in Fig. 1, where ! ! % & "# ! $ ! '(*)+# (1) -, /.103254 6-78-,9;:=< >.@?BA <CD< B? A < E 2 GFIH3 Similarly, one can write the state equations for the right hand subsystem. In their original work, Chua and Lin [14] [15] have studied the dynamics of the third-order electrical circuit by varyonly and have kept ing the control parameter, capacitor the other parameters fixed. Such a circuit in this case admits perid-doubling bifurcations, intermittency and crisis-induced intermittency [15] behaviour for certain ranges of circuit parameters. They assumed the control parameter to be the capacitor only for both numerical and experimental realizations. In ! ! 46 NATIONAL CONFERENCE ON NONLINEAR SYSTEMS & DYNAMICS ! our present study, we have also verified the same results for inand linear dividual circuits by considering both capacitor resistor as variables. The summarized results such as perioddoubling bifurcations and chaos followed by periodic windows are shown in the computer generated one parameter bifurcation plane as in the Fig. 2, from this we can diagram in the clearly see the route to chaos via period-doubling bifurcations while the numerical and experimental phase portraits of chaotic attractor at =1050 are shown in Figs. 3(a) and 3(b). When the two canonical Chua’s circuits are coupled together the circuit becomes six dimensional and the dynamics of the above circuit now depends on five control parameters, namely the capacitances , the inductances , the linear resistance and the coupling resistance , besides the parameters associated with and nonlinear element . linear negative conductance )- 1 0.6 )D ) ! ) % 0.4 0.2 0 y -0.2 -0.4 ) 6 ; (a) 0.8 -0.6 -0.8 -1 -2 0.5 -1.5 -1 -0.5 0 0.5 1 1.5 2 x 0 -0.5 -1 -1.5 v -2 1 (b) -2.5 -3 -3.5 -4 -4.5 1400 1350 1300 1250 1200 1150 1100 1050 R Fig. 2. Computer simulated one-parmater bifurcation diagram in the plane of the individual third-order canonical Chua’s circuit . v2 To investigate mutual synchronization, we fix the parameters of the canonical Chua’s circuit so that the system exhibits a chaotic attractor, specifically, the so-called double scroll attractor. The following nominal values produce the double scroll: =100 nF, =30.5 nF, =107 mH, =0.68 V, =-0.105 mS, =7.0 mS, =-0.45 mS and =1050 . ! , % ) 7 ? A v1 A. Computer Simulation For computer simulation study, we normalize the state equations of the coupled canonical Chua’s circuit (Fig. 1) given , , in (1) by appropriate rescaling, , , , , , , , , , , and , which give the following set of normalized equations, G ? A& ! H 7 ! ! ? A ! ? A ! ! ! ? A " , F C) 9G " . ! /" ! ! # ! # $ G ! /" ! ! # > ! # $ ! ! ? A ? A& ! ! % where > ; and Fig. 3. Numerical(Fig. (a)) and experimental(Fig. (b)) phase portraits of chaotic attrator of the individual third-order. canonical Chua’s circuit % .103254 H"# 9< . F&< D< F< &CH3 . 02 4 H"# I;9< . F< < F< &I Now the dynamics of (2) depends on the parameters , ! , , H , , ! F ) . The phase portraits of the system composed of identical os (2) cillators are shown in Fig. 4. The fine diagonal line in Fig. 4(c) indicates complete synchronization of the individual oscillators INDIAN INSTITUTE OF TECHNOLOGY, KHARAGPUR 721302, DECEMBER 28-30, 2003 at larger coupling coefficients. Obviously, the system of synchronized chaotic oscillators is not a hyperchaotic one. Visual inspection of the phase portraits at smaller coupling coefficents does not allow one to distinguish between simple chaotic and hyperchaotic states. This can be done by direct calculation of there are two positive the Lyapunov Exponents (LEs). At ! the situation is rather LEs, as expected. Meanwhile at ! & , one obtains hyperchaotic complicated. At a value of ! attractor shown in Fig. 4(a). Phase space trajectories on this attractor are characterised by two positive Lyapunov exponents ( , , , , & , & & & ). At the value of ! & , just before the onset of synchronization, the phase space attractor is shown in Fig. 4(b) whose phase space trajectories char& acterised by two positive Lyapunov exponents ( , , , , & & , ) and further on increasing the the value of ! (at ! & & ), the attractor shown in Fig. 4(c), which represents the onset of synchronization and the corresponging Lya& & , , , punov exponents are ( & , & ). , 0 0 02 F 4 30 2 0 F ! 30 2 0 032 0 0 30 2 0 0 0 0 2 0 4 2 0 2 ! 1 30 2 0 032 0 0 02 0 0 032 0 0& 4 02 0 0 -F 52 4 032 F 02 0 ! 032 0 0 F 0 032 0 0 02 0 0 032 0 4 - F 2 F 0.8 0.6 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.4 0.2 0.2 0.2 x’ x’ 0 -0.2 -0.4 -0.6 -0.8 x’ 0 -0.2 -0.4 -0.4 -0.6 -0.6 -0.8 -0.8 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 -1 1 -0.8 -0.6 -0.4 -0.2 0 0.2 x x x (a) (b) (c) 0.4 0.6 0.8 1 Fig. 4. Typical examples of phase portraits of hyperchaotic attractors and sychronized attractor observed from the coupled differential equations (2) at Vs prodifferent coupling resistances (mutual coupling): shown in jection; (a) , , , , , , , (b) , , , , , and (c) , ( , , , , , . "!$# % !'&(! ) * "+$# "1*&$ "2!(-# ,8. 3 ".+3"! "!$# 0/49 3 . 5": 3,-2 "" 6(# .;" % ""7!!"+ 3! "/0" *) < = ) 73 "!" ,9 * "6 /3A >".6 3 ": :" ?29 " !$B# " %C#'<6 @ "3 ) >> In Fig. 5 the computer generated bifurcation diagrams, is shown for versus the coupling coefficient, ! . F C) 9 F 0&4C0 B. Experimental Synchronization of chaos in the identical oscillators In the present case, the oscillators have been built using the same circuit parameters as given in Fig. 1 and only the coupled linear resistor is varied. When =0, the two canonical circuits are uncoupled and each will follow its own dynamics. Due to sensitive dependence on initial conditions, the trajectories of the two systems evolve independent of each other, i.e. there is no synchronization. When the oscillators are coupled to each ), it has been other and the coupling is strong enough ( shown numerically that the oscillators ”forget” their own initial conditions and after a short transient the two oscillators will synchronize completly, i.e. the trajectories of the two canonical Chua’ circuits follow each other. The experimental illustrations are presented in Fig. 6. In the experiments it is convenient to plane, i.e. the output observe the phase portraits in the signal from one generator against the output signal from another one. In the figure, when two systems synchronize, on the screen of oscilloscope the relation between and is a line through the origin as shown in Fig. 6(c). When the two systems do not synchronize, the relation between and deviates from this line. The greater the coupling resistor whose value is larger than its critical value, the greater is the deviation: that is, the more the systems do not synchronize. When is large enough, the two systems do not synchronize and the relation between and becomes chaotic as shown in Fig. 6(a) and 6(b). In the case of unsynchronized chaotic oscillations the phase portrait (or Lissajous figure) is a very complicated one, indicating ”random” mutual phase and amplitudes of the oscillations (Fig. 6(a)). This chaotic status is called hyperchaotic attractor characterised by more than one postive Lyapunov exponents, which occur in the four or higher dimensional system. Previously, hyperchaos was observed in a network of (both open and closed) five unidirectionally coupled Chua’s circuits [16] [17]. However, in our study we have observed the hyperchaotic behaviour in a coupled third-order canonical Chua’s circuit. Meanwhile the fine diagonal line in Fig. 6(c) proves that the momentary amplitudes and phases of the oscillations from the two oscillators do coincide with each other, i.e. the oscillators are fully synchronized. ) ) ) 0 4 J ) -1 -1 -1 0 -0.2 47 3 ) 2 1 1 1.5 0.8 0.8 0.6 0.6 0.4 1 0.4 0.2 0.2 0.5 0 v’ 1 0 -0.2 0 -0.4 -0.4 ’ x 1− x 1 v’ 1 v’ 1 -0.2 -0.6 -0.6 -0.5 -0.8 -0.8 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -1 v1 -1.5 (a) v1 v1 (c) (b) -2 1 2 3 4 5 6 Rc 7 H<".I 8 9 10 D E Vs GF , in the Fig. 5. Computer simulated bifurcation diagram of case of mutual coupling for Fig. 6. Typical example of experimental phase portraits of hyperchaotic attractors and sychronized attractor observed from the two coupled identical canonical Chua’s oscillators at different coupling resistances: (a) , (b) and (c) Veritcal scale =0.5 V/division; horizontal scale. =0.5 V/division I F KC# 5I F L&' 0I % F M 0I 48 NATIONAL CONFERENCE ON NONLINEAR SYSTEMS & DYNAMICS III. C ONCLUSION In this paper numerical and experimental investigations of a set of coupled canonical Chua’s circuits are carried out. The advantage of our study is that we have observed hyperchaotic oscillations in simplest coupled canonical Chua’s cirucits (maximum of six dimensional phase space), incontrast to which previous studies involving coupled canonical circuits of higher order (more than six) circuits. Yet it exhibits hyperchaos and synchronization of chaotic orbits. This feature it is hoped will be convenient for application in secure communications and spatio temporal chaos. R EFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] L. Kocarev and U. Parlitz, General approach for chaotic synchronization with applications to communication, Phys. Rev. Lett., 74, 5028 (1995). Y. Nishio and A. Ushida, Quasi-synchronization phenomena in chaotic circuits coupled by one resistor, IEEE Trans.Circuits Syst., 43, 491 (1996). V. C. Anishchenko, T. E. Vadivasova, V. V. Astakhov, O .V. Sonsnovtseva, C. W. Wu and L. O. Chua, Dynamics of two coupled Chua’s circuits, Int. J. Bifurcation and chaos, 5, 1677 (1995). L. M. Pecora and T. L. Caroll, Synchronization in chaotic systems, Phys. Rev. Lett., 64, 821 (1991). T. L. Caroll and L. M. Pecora, Synchronizing chaotic circuits, IEEE Trans. Circuits Syst., 38, 453 (1993). S. Anishchenko, T. Kapitaniak, M. A. Safonora and O. V. Sosnovzeva, Birth-double scroll attractor in coupled Chua’s circuit, Phys. Lett. A., 192, 205 (1994). K. Murali and M. Lakshmanan, Synchronizing chaos in driven Chua’s circuit, Int. J. Bifurcation and chaos, 3, 1057 (1993). M. Lakshmanan and K. Murali, Chaos in Nonlinear Oscillators: Controlling Synchronization, World Scientific, Singapore, (1996). L. O. Chua, L. Kocarev, K. Eckert and M. Itoh, Experimental chaos synchronization in Chua’s circuit, Int. J. Bifurcation and chaos, 2, 705 (1992). S. Moro, Y. Nishio and S. Mori, Synchronization phenomena in oscillators coupled by one resistor, IEICE Trans. Fundamentals, E78-A, 244 (1995). K. Murali and M. Lakshmanan, Transmission of signals by synchronization in a van der Pol Duffing oscillaotor, Phys. Rev. E, 48, R1624, (1993). K. Murali and M. Lakshmanan, Effect sinusoidal excitation on the Chua’s circuit, IEEE Trans.Circuits Syst., 39, 264 (1992). I. M. Kyprianidis and I. N. Stouboulos, Synchronization of two resistively coupled nonautonomous and hyperchaotic oscillators, Chaos, Solitons and Fractals, 17, 317 (2003). L. O. Chua and G. N. Lin, Canonical realization of Chua’s circuit family, IEEE Trans. Cir. Syst. 42, 882 (1990). I. M. Kyprianidis, M. L. Petrani, J. A. Kalomiros, A. N. Anagnosto, Crisisinduced intermittency in the third-order electrical circuit, Phys. Rev. E., 52, 2268 (1995). L. Locarev, K. S. Halle, K. Eckert, L. O.Chua and U. Parlitz, Experimental demonstration of secure communications in a chaotic synchronization, Int. J. Bifurcation and chaos, 2, 709 (1992) . T. Kapitaniak and L. O. Chua, Hyperchaotic attractor of unidirectionally coupled Chua’s circuits, Int. J. Bifurcation and chaos, 4, 477 (1994).