Dynamics of Two Resistively Coupled Electric Circuits of 4 th Order

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1 Proceedings of the 0th WSEAS International onference on IRUITS, Vouliagmeni, Athens, Greece, July 0-, 006 (pp9-96) Dynamics of Two Resistively oupled Electric ircuits of 4 th Order M. S. PAPADOPOULOU, I.M. KYPRIANIDIS, and I. N. STOUBOULOS Physics Department Aristotle University of Thessaloniki Thessaloniki, 544 GREEE Abstract: - We have studied the case of chaotic synchronization of two identical nonlinear autonomous electric circuits, which are mutually (bidirectionally) coupled via a linear resistor R. The two circuits can be either υ -coupled or υ -coupled. Both, experimental and simulation results, have shown that chaotic synchronization is possible only in the case of υ -coupling. Key-Words: - Nonlinear circuit, haos, Attractor, Mutual coupling, Reverse period-doubling. Introduction Since the discovery by Pecora and arroll [], that chaotic systems can be synchronized, the topic of synchronization of coupled chaotic circuits and systems has been studied intensely [] and some interesting applications, as in broadband communications systems, have come out of this research [3-5]. Generally, there are two methods of chaos synchronization available in the literature. In the first method, due to Pecora and arroll [], a stable subsystem of a chaotic system could be synchronized with a separate chaotic system, under certain suitable conditions. The second method to achieve chaos synchronization between two identical nonlinear systems is due to the effect of resistive coupling without requiring to construct any stable subsystem [6-8]. According to arroll and Pecora [9], periodically forced synchronized chaotic circuits are much more noise-resistant than autonomous synchronized chaotic circuits. In this paper we have studied the case of two identical fourth-order autonomous nonlinear electric circuits (Fig.) with two active elements, one linear negative conductance and one nonlinear resistor with a symmetrical piecewise linear v - i characteristic (Fig.). Using the capacitances and as the control parameters, we have observed a reverse period-doubling sequence, as well as a crisis phenomenon, when the spiral attractor suddenly widens to a double-scroll attractor [0]. implemented using the circuits shown in Figs.3 and 4 respectively. The real v-i characteristic of the conductance G is shown in Fig.5. Fig.. The fourth-order autonomous nonlinear electric circuit. Fig.. The piecewise-linear and symmetrical v-i characteristic of the nonlinear resistor R N. The Nonlinear ircuit The circuit, we have studied, is shown in Fig. and has two active elements, a nonlinear resistor R N and a linear negative conductance G, which were Fig.3. The nonlinear resistor implementation

2 Proceedings of the 0th WSEAS International onference on IRUITS, Vouliagmeni, Athens, Greece, July 0-, 006 (pp9-96) The Bifurcation diagram of the dynamics of the circuit is shown in Fig.6. We can observe forward and reverse period-doubling cascades, i.e. antimonotonocity [0]. Fig.4. The negative conductance implementation Fig.6. The bifurcation diagram (i L ) p vs. for =5.30nF. Fig.5. The real v-i characteristic of the negative conductance G. The state equations of the circuit are: dυ = ( i L i ) () dt dυ ( G υ +i i ) dt = L + L () dil = ( υ υ Ri L ) (3) dil = ( υ Ri L ) (4) where i=g(υ )=Gυ +0.5(Ga -G b )( υ +B p - υ-b p ) (5) +0.5(G -G )( υ +B - υ -B ) b c p p The values of the circuit parameters are: L =0.mH, L =.5mH, R =.0KΩ, R =08Ω, G n =-0.5mS, G p =-0.5mS, L R =7.5V, G a =-0.833mS, G b =-0.54mS, G c =.9mS, B p =.46V and B p =0.V. 3 Mutual Resistive oupling If two identical circuits are resistively coupled, complex dynamics can be observed, as well as chaotic synchronization, as the coupling parameter R is varied. The two circuits can be either υ - coupled or υ -coupled, as we can see in Figs.7 and 8 respectively. 3. υ -coupling The system of the two identical 4 th order nonlinear circuits υ -coupled via the linear resistor R is shown in Fig.7. The state equations of the system are: dυ υ υ = + il g( υ ) (6) dt R dυ = { G υ il il} dt (7) di = υ +υ Ri { } L L (8) dil = { υ Ri } (9) L dυ υ υ = + il g( υ ) dt R (0) dυ = { G υ il il} dt ()

3 Proceedings of the 0th WSEAS International onference on IRUITS, Vouliagmeni, Athens, Greece, July 0-, 006 (pp9-96) di = { υ +υ Ri } L L () dil = { υ Ri L} (3) dil = { υ Ri } L where g(υ j ) is given by Eq. (4) () where g(υ j)=gcυ j+0.5(ga -G b )( υ j+b p - υj-b p ) (4) +0.5(G -G )( υ +B - υ -B ) and j=,. b c j p j p Fig.8. The υ -coupled system of the two identical 4 th order nonlinear circuits. Fig.7. The υ -coupled system of the two identical 4 th order nonlinear circuits. 3. υ -coupling The system of the two identical 4 th order nonlinear circuits υ -coupled via the linear resistor R is shown in Fig.8. The state equations of the system are: dυ { i L g υ dt ( ) = } (5) dυ υ υ = Gυ il il dt R (6) dil = { υ +υ Ri L} (7) dil = { υ Ri } (8) L dυ = { il g( υ )} (9) dt dυ υ υ = Gυ il il dt R (0) dil = { υ +υ Ri L} () 4 Dynamics of the oupled System We have studied the evolution of the two identical resistively coupled autonomous and nonlinear 4 th order electric circuits from nonsynchronized to synchronized oscillations, when they exhibit chaotic behavior. By synchronization of chaos we mean, that the chaotic oscillations of individual circuits are identical to each other. Using the bifurcation diagram of Fig.6, we have chosen the values of where chaotic behavior is observed and have studied the possibility of chaotic synchronization by plotting the diagrams of υ -υ vs. R / R, the coupling factor. haotic synchronization was never observed in the case of υ -coupling. 4. haotic attractors The chaotic attractors of the two circuits, when they are uncoupled, are shown in the following figures. In Figs. 9 and 0 we can see the theoretical phase portraits for =5.3nF and =8.0nF and =8.4nF respectively, while in Figs. -4 the experimental chaotic attractors are shown, for the two identical circuits.

4 Proceedings of the 0th WSEAS International onference on IRUITS, Vouliagmeni, Athens, Greece, July 0-, 006 (pp9-96) Fig.9. haotic attractor for =5.30nF and =8.00nF. Fig.3. haotic attractor for =5.30nF and =8.00nF for the second circuit (i L vs. υ ). Fig.0. haotic attractor for =5.30nF and =8.40nF. Fig.4. haotic attractor for =5.30nF and =8.40nF for the second circuit (i L vs. υ ). Fig.. haotic attractor for =5.30nF and =8.00nF for the first circuit (i L vs. υ ). 4. The case of υ -coupling We have studied the possibility of chaotic synchronization when the initial conditions of the coupled circuits belong to the same basin of attraction. The diagrams of υ -υ vs. R /R are shown in Figs. 5 and 6 correspond to the case, when the initial conditions of the coupled circuits belong in the same basin of attraction. In Figs. 7-9 we can see the experimental results from the coupled system for =5.30nF, =8.00nF and various values of the coupling resistor R. In Figs. 0- we can see the experimental results for =5.30nF and =8.40nF, as the coupling resistance R is varied. We observe how the system passes from nonsynchronized states to synchronized ones, depending on the coupling resistance R. Fig.. haotic attractor for =5.30nF and =8.40nF for the first circuit (i L vs. υ ).

5 Proceedings of the 0th WSEAS International onference on IRUITS, Vouliagmeni, Athens, Greece, July 0-, 006 (pp9-96) Fig.5. The diagram of υ -υ vs. R /R, for =5.30nF and =8.00nF. Fig.9. υ -coupling for =5.30nF, =8.00nF and R =.3KΩ (υ vs. υ ). haotic synchronization. Fig.6. The diagram of υ -υ vs. R /R, for =5.30nF and =8.40nF. Fig.0. υ -coupling for =5.30nF, =8.40nF and R = KΩ (υ vs. υ ). Fig.7. υ -coupling for =5.30nF, =8.00nF and R = (υ vs. υ ). Fig.. υ -coupling for =5.30nF, =8.40nF and R =40KΩ (υ vs. υ ). Fig.8. υ -coupling for =5.30nF, =8.00nF and R =49KΩ (υ vs. υ ). Fig.. υ -coupling for =5.30nF, =8.40nF and R =.5KΩ (υ vs. υ ). haotic synchronization.

6 Proceedings of the 0th WSEAS International onference on IRUITS, Vouliagmeni, Athens, Greece, July 0-, 006 (pp9-96) 5 onclusion In this paper we have shown the process of chaotic synchronization using computer simulation and experiment. We have studied the dynamics of the individual nonlinear autonomous electric circuit and determined the parameter s region, where the circuit exhibits chaotic behavior. Furthermore, we observed the chaotic synchronization of two identical circuits, which are mutually coupled via a linear resistor R, when the initials conditions of the coupled circuits belong to the same basin of attraction. Both, experimental and simulation results have shown that chaotic synchronization is possible in the case of υ -coupling and not possible in the case of υ - coupling. In the first case, the coupling is established at the node of the linear part of the autonomous circuit, while in the second case the coupling is established at the node of the nonlinear part. The transferred energy from the coupling branch is not enough to synchronize the coupled system in the second case, as it has been verified by simulation and experiment. This is also enhanced in the case when conductance G n becomes nonlinear. The increased complexity of the system does not allow synchronization, even in the case of υ - coupling configuration []. Acknowledgements: This work has been supported by the research program EPEAEK II, PYTHAGORAS II, code number 8083, of the Greek Ministry of Education and E.U. Fundamentals of digital communications, IEEE Transactions ircuits Systems-I, Vol.44, 997, pp [6] Murali, K. and Lakshmanan, M., Drive-response scenario of chaos synchronization in identical nonlinear systems, Physical Review E, Vol.49, No.6, 994, pp [7] Kapitaniak, T., hua, L. O. and G.-Q. Zhong, Experimental synchronization of chaos using continuous control, International Journal of Bifurcation and haos, Vol.4, No., 994, pp [8] Murali, K., Lakshmanan, M. and hua, L. O., ontrolling and synchronization of chaos in the simplest dissipative nonautonomous circuit, International Journal of Bifurcation and haos, Vol.5, No., 995, pp [9] aroll, T. L. and Pecora L. M., Synchronizing nonautonomous chaotic circuits, IEEE Transactions ircuits Systems-II, Vol.40, 993, pp [0] Kyprianidis, I. M., Stouboulos, I. N., Haralabidis, P. and Bountis T., Antimonotonicity and chaotic dynamics in a fourth-order autonomous nonlinear electric circuit, International Journal of Bifurcation and haos, Vol.0, No.8, 000, pp [] Kyprianidis I. M. and Stouboulos I. N., Synchronization of two resistively coupled nonautonomous and hyperchaotic oscillators, haos Solit. Fract., Vol.7, 003, pp References: [] Pecora, L. M. and arroll, T. L., Synchronization in chaotic systems, Physical Review Letters, Vol. 64, No.8, 990, pp [] Wu,. W., Synchronization in oupled haotic ircuits and Systems, World Scientific, 00. [3] uomo, K. M. and Oppenheim, A. V., ircuit implementation of synchronized chaos with applications to communications, Physical Review Letters, Vol.7, No., 993, pp [4] Kocarev, L. and Parlitz, U., General approach for chaotic synchronization with applications to communications, Physical Review Letters, Vol.74, No.5, 995, pp [5] Kolumban, G., Kennedy, M. P. and hua, L. O., The role of synchronization in digital communications using chaos - part I: