In-Phase and Anti-Phase Synchronization of Switching Phenomena ...

2010 IEEE Workshop onNonlinear Circuit Networks, Tokushima, Dec 3-4, 2010In-Phase and Anti-Phase Synchronization ofSwitching Phenomena in Coupled Chaotic CircuitsTakuya NishimotoDept. Information Business,Shikoku UniversityEmail: s2137034@edu.shikoku-u.ac.jpYasuteru HosokawaDept. Media and Information System,Shikoku UniversityEmail: hosokawa@keiei.shikoku-u.ac.jpYoshifumi NishioDept. Electrical and Electronic Eng.,Tokushima UniversityEmail: nishio@ee.tokushima-u.ac.jpAbstract— Switching phenomena of attractors can be observedin coupled chaotic circuits. In this study, we confirmed that inphaseand anti-phase synchronizations of switching phenomena.The phenomena can be observed in asynchronous states. Therefore,it is very interesting phenomena.I. INTRODUCTIONMany kinds of complex phenomena are observed on largescalecoupled chaotic circuits. Investigations of these phenomenaare very important works in order to declare nonlinearphenomena in the natural world. Electric circuits are suitablefor models of large-scale coupled nonlinear systems. Becausegetting electric parts is easily, the price is inexpensive, experimenttime is very short, repeatability of experiments isvery high, an electric circuit is real physical system and soon. Therefore there are many studies of large-scale coupledcircuit systems. In these studies, synchronization phenomenaare attracted attentions.On the other hand, some chaotic circuits have coexistingattractors. In these circuits, switching phenomena of attractorsare observed by changing parameters. Normally, in the caseof synchronization states, switching of attractors are also synchronized.And in the case of asynchrization states, switchingof attractors are also asynchronized. However, there is thepossibility that switching of attractors are synchronized in thecase of asynchrization states.In this study, we investigate coupled chaotic circuits andconfirm in-phase and anti-phase synchronizations of switchingphenomena. Especially, the case that the number of circuit istwo is investigated.L1r nin1II. SYSTEM MODELCivnn2L2nCCn(a) Circuit schematicFig. 2. System model (N = 4)Fig. 3.f( id)V-Vrd(b)v-i characteristicsCoupled diodes modelnegative resistor and bi-directionally-coupled diodes. In thiscircuit, two attractors which are symmetrical about a originare observed by adjusting the value of the negative resistor.In this study, we investigate coupled chaotic circuits whichis coupled by resistors shown in Fig. 2. Firstly, a system equationis derived. Bi-directionally-coupled diodes are modeled asFig. 3. Then, we can derive a following function(∣ ∣∣∣v d = i d + V ∣ ∣∣∣ −r d∣ i d − V ∣)∣∣∣. (1)r didFig. 1.Circuit modelA circuit model [1] using in this study is shown in Fig. 1.This circuit consists of three memory elements, one linearThe others elements are modeled as linear elements. Eachcircuit number is defined as 1 ≤ n ≤ N. System equation- 15 -

TABLE ICOLOR DEFINITION OF FIG. 13 AND FIG. 14.Fig.Fig. 5Fig. 6Fig. 7Fig. 8Fig. 9Fig. 10Fig. 11Fig. 12StateRedGreenBlueLight blueGrayPinkLight greenYellow(a)(b)of switching phenomena are only observed. The area is around0.55 ≤ δ ≤ 0.63 and 0.46 ≤ α 2 ≤ 0.48. Namely, anti-phasesynchronization is not observed in the case of α 1 ≃ α 2 ora small value of coupling factor δ. In yellow areas as shownin Fig. 14, by decreasing α 2 , synchronization time becomeslong.IV. CONCLUSIONSIn this study, we have investigated coupled chaotic circuitsand have observed in-phase and anti-phase synchronizations ofswitching phenomena. In the case that the number of circuitis two, the relationship between parameters and phenomena isshown.REFERENCES[1] Y. Nishio, N. Inaba, S. Mori and T. Saito, “Rigorous Analyses of Windowsin a Symmetric Circuit,” IEEE Trans. Circuit and Systems., vol. 37, no. 4,pp. 473–487, 1990.[2] T. Nishimoto, Y. Hosokawa and Y.Nishio, “Anti-phase Synchronization ofSwitching Phenomena in Globally Coupled System of Chaotic Circuits,”IEICE Technical Report, pp. 1–4, 2010.[3] H. Sekiya, S. Mori and I. Sasase, “Synchronization of Self-SwitchingPhenomena on Full-Coupled Chaotic Oscillators,” IEICE Trans., vol. J83-A, no. 11, pp. 1264–1275, 2000.[4] T. Yamada, Y. Hosokawa and Y. Nishio, “Quasi Synchronization Phenomenain Coupled Chaotic Systems as a Ladder,” Proc. of IEEE InternationalWorkshop on Nonlinear Circuit Networks, pp. 68-71, 2008.Fig. 5. Anti-phase synchronization of switching phenomena. α n = 0.40,β 1 = 3.0, β 2 = 4.5, γ = 470.0 and δ = 0.31.(a)(b)Fig. 6. In-phase synchronization of switching phenomena. α n = 0.40,β 1 = 3.0, β 2 = 15.0, γ = 470.0 and δ = 0.33.(a)(b)Fig. 7. Asynchronization of switching phenomena. α n = 0.40, β 1 = 3.0,β 2 = 9.0, γ = 470.0 and δ = 0.10.Fig. 4. Coexisting attractors of a chaotic circuit as shown in Fig. 1. α 1 =0.40, β 1 = 3.0 and γ = 470.0.(a)(b)Fig. 8. Switching phenomena on one circuit only. α n = 0.40, β 1 = 3.0,β 2 = 12.0, γ = 470.0 and δ = 0.10.- 17 -

0.4δ(a)0.3(b)Fig. 9. Holding one attractor state in each circuit. α n = 0.40, β 1 = 3.0,β 2 = 7.5, γ = 470.0 and δ = 0.25.0.20.1(a)(b)03 6 9 12 15 18 21β2Fig. 10. Boundary area between Fig. 5 and Fig. 7. α n = 0.40, β 1 = 3.0,β 2 = 6.0, γ = 470.0 and δ = 0.22.Fig. 13. Relationship among parameter β 2 , δ and observed phenomena.α n = 0.40, β 1 = 3.0 and γ = 470.0.(a)0.7δ(b)0.6Fig. 11. Boundary area between Fig. 6 and Fig. 7. α n = 0.40, β 1 = 3.0,β 2 = 18.0, γ = 470.0 and δ = 0.21.0.50.4(a)(b)0.3α20.4 0.42 0.44 0.46 0.48Fig. 12. Anti-phase synchronization of switching phenomena with synchronizationphenomena. α 1 = 0.40, α 2 = 0.46, β n = 3.0, γ = 470.0 andδ = 0.52.Fig. 14. Relationship among parameter α 2 , δ and observed phenomena.α 1 = 0.40, β n = 3.0 and γ = 470.0.- 18 -