Synchronization of Self-Switching Phenomena in Chaotic Oscillators Coupled by One Resistor Hiroo Sekiya
Shinsaku Mori y
Abstract In the study of coupled chaotic oscillators, investigation of transition of the attractor's region is important problem to clarify spatiotemporal chaos. We observe self-switching phenomena in chaotic oscillators coupled by one resistor. In our system, synchronization of self-switching phenomena is observed for the rst time. The term \synchronization of self-switching phenomena" means that the all attractors of subcircuits switch the regions synchronously. In our system, two types of synchronization of self-switching phenomena can be found.
1 Introduction Recently, the spatiotemporal chaos attracts many researchers' attentions. A network of chaotic one-dimensional maps is investigated earnestly by Kaneko[1]-[3]. Especially, Coupled Map Lattice (CML) and Global Coupled Map (GCM) are well known as discrete time mathematical models and various phenomena were observed in these models. The studies of such systems are very important not only as models for nonlinear systems with many degrees of freedom but also for the clari cation of biological information processing and engineering applications. Actually, Aihara et al. proposed chaotic neural networks whose cell is a one dimensional chaotic map, and they have con rmed that such systems produce dynamical chaotic search for memorized patterns [4]. On the other hand, in electrical systems, spatiotemporal chaos can be investigated in the system of coupled chaotic oscillators [5]{[8]. The electrical system is one of real physical and continuous time models. In the chaotic oscillator which is included among double-scroll family[9], two asymmetric attractors located symmetrically with respect to the origin coexist in each subcircuit. Hence, each subcircuit can take two di erent states. Therefore, selfswitching phenomena can be observed in chaotic oscillators. The term \self-switching phenomena"
Iwao Sasase
in this study means that each chaotic solution of subcircuit switches the located regions by itself randomly. We can think the self-switching phenomena in the system of coupled oscillators as one of spatiotemporal chaos. In most of studies about spatiotemporal chaos of electrical systems, the chain or array structure of chaotic oscillators are investigated. They correspond to CML since each oscillator a ects the neighbor oscillators. On the other hand, the full-coupled systems correspond to GCM since each oscillator a ects the others one another. However, there are few studies about spatiotemporal chaos in full-coupled systems. Moreover, there is no study how self-switching in each oscillator a ects the other oscillators one another in full-coupled system. We recognize that the investigation of selfswitching phenomena in full-coupled system is important theme to clarify the spatiotemporal chaos of full-coupled systems in continuous time. In this study, we investigate the self-switching phenomena in N simple chaotic oscillators coupled by one resistor in case of N = 2; 3 and 4. The investigated system is one of the simplest full-coupled systems since only one coupling resistor is needed regardless of N . The chaotic oscillator used in this study is included among double-scroll family. By carrying out numerical calculations and circuit experiments, very interesting phenomena named synchronization of self-switching phenomena can be observed for the rst time. The term \synchronization of self-switching phenomena" means that the all attractors of subcircuits switch the located regions synchronously. In our system, two types of synchronization of self-switching phenomena can be found.
2 Circuit Model
The circuit model is shown in Fig. 1. In our system, N chaotic oscillators are coupled by one resistor. Each chaotic oscillator is one of the simple chaotic oscillators proposed by Shinriki et al[10] and it is one of double-scroll family. At rst, we approxi Dept. of Information and Computer Science, Keio University, 3-14-1 Hiyoshi, Kohoku, Yokohama, 223-8522 mate the i , v characteristics of the nonlinear resisJAPAN Tel: +81-45-563-1141 Ext. 3319, Fax: +81-45-563- tors consisting of diodes by the following function.
2773, Email:
[email protected] y Dept. of Electrical & Electronics Eng., Nippon Institute of Technology, Miyashiro, Saitama, JAPAN Email : s
[email protected]
idk = E1 vdk th
9
(1)
Figure 1: Circuit model. In the above equation, the letter k denotes the number of each oscillator and Eth expresses the threshold voltage of the diodes. By changing the variables and the parameters, v1k = Eth xk ; v2k = Ethyk ; p p ik = Eth C1=Lzk ; t = LC1 ; p \ " = d=d ; idk = Eth C1=Lidk p p = g L=C1; = C1=C2; = R C1=L; (2) the normalized equations are given as follows: 8 x_k = xk , idk > > > > > < y_k = fidk , zk g (3) N > X > > > > : z_k = (1 + !k )yk , zi i=1 (k = 1; 2 N ); = (xk , yk )9 and !k expresses the difwhere idk ference of real circuits' elements. At the following circuit experiments, we x the elements as follows: L = 52:3mH 0:1%; C1 = 0:0157F 0:6%; C2 = 0:0225F 2% and Eth = 0:6V . On the other hand, we x the parameters = 0:7 in the numerical calculations and !k = 0:002(k , 1). We investigate the phenomena generating in our system by changing the negative conductance of each oscillator g and the coupling resistor R in the circuit experiments, and in the numerical calculations. Figure 2 shows some example of chaotic attractors obtained from the chaotic oscillator in case of N = 1. The chaotic attractor coexists in the two regions which are symmetry with respect to the origin as shown in Fig. 2 (a). We call the region where an attractor is in Fig. 2 (a)(1) \L" and one in Fig. 2 (a)(2) \R" respectively. As the parameter varies further, double-scroll like attractor is observed via symmetry recovering crisis as shown in Fig. 2 (b). In the double-scroll like attractor, a chaotic solution switches between the region R and L randomly. In other words, a self-switching phenomenon between two regions occurs in a double-scroll like attractor.
Figure 2: Examples of attractors for N = 1 and
= 0 (circuit experiments). (a) Asymmetry chaotic attractor for = 0:39. (b) Double-scroll like attractor for = 0:46. (1) Attractor in the region L. (2) Attractor in the region R. Horizontal and Vertical: 0.5 V/div
Figure 3: Anti-region synchronization of selfswitching phenomena for N = 2 and = 0:48. (a) =0.25. (b) = 0:15. (c) = 0:1. (1) and (2) Numerical calculations. (1) Time waveforms of xi . (2) x1vs:x2 . (3) and (4) Circuit experiments. (3) Time waveforms of v1i Horizontal: 10ms/div, Vertical: 0.5 V/div. (4) v11vs:v12 . Horizontal and Vertical: 0.5 V/div.
3 Synchronization of Self-switching Phenomena In this section, self-switching phenomena are investigated in detail. We adjust in order that the system generates a self-switching phenomenon and choose as a parameter to vary.
3.1 In the case of two subcircuits
When N = 2, we observe synchronization of selfswitching phenomena in case of large as shown in Fig. 3(a). When one attractor travels from the re-
gion R (L) to L (R), the other switches the region from L (R) to R (L) synchronously. In other words, the solution changes between the states fR; Lg and fL; Rg randomly. For example, the state fR; Lg expresses that an attractor of subcircuit 1 is in the region R and one of subcircuit 2 is in the region L. We call this state \anti-region synchronization of self-switching phenomena". In this state, chaotic signals of each subcircuit are synchronized at antiphase since the system tends to minimize the current through the coupling resistor. As decreases, synchronization of self-switching phenomena cannot be observed as shown in Fig. 3(b). However, there is the part of synchronization of self-switching phenomena as shown in dashed-part in Fig. 3(b). As decreases further, each subcircuit switches asynchronously and chaotic signals of each subcircuit are also asynchronized as shown in Fig. 3(c). By paying attention to the switching interval, we con rm the switching interval is short when the system generates anti-region synchronization of selfswitching phenomena. While, when it generates asynchronization of self-switching phenomena, we observe that the switching interval is larger than that when synchronization of self-switching is generated.
3.2 In the case of three subcircuits In the case of N = 3, observed synchronization of self-switching phenomena is di erent from anti-region synchronization. Namely, in-region synchronization of self-switching phenomena can be observed in case of large as shown in Fig. 4(a). The term \in-region synchronization of self-switching phenomena" means that when one attractor switches from the region R (L) to L (R), the others switch from the region R (L) to L (R) synchronously. Namely, the solution switches between fR; R; Rg and fL; L; Lg randomly. In this state, chaotic signals of each subcircuit are asynchronized though anti-phase synchronization of chaos is observed in the case of N = 2. Note that self-switching is synchronized though chaotic signals are asynchronized. Namely, it can be said that synchronization of self-switching and synchronization of chaos are di erent phenomena. As decreases, in-region synchronization of self-switching phenomena bursts here and there as shown in Fig. 4(b). Compared with the phenomenon in Fig. 4(a), we recognize that switching interval in Fig. 4(b) is shorter. As decreases further, selfswitching phenomena of each subcircuit are asynchronized. In this state, the switching interval is shorter than that in Fig. 4 (a) and (b). The relationship between the switching interval and the amplitude of in the case of N = 3 is the opposite
Figure 4: In-region synchronization of selfswitching phenomena for N = 3 and = 0:48. (a) =0.2. (b) = 0:15. (c) = 0:1. (1), (2)and (3) Numerical calculations. (1) Time waveforms of xi . (2) x1 vs:x2 . (3) x1 vs:x3 . (4) and (5) Circuit experiments. (4) Time waveforms of v1i . Horizontal: (a) 200ms/div, (b) and (c) 20ms/div, Vertical: 0.5 V/div. (4) v11 vs:v12. Horizontal and Vertical: 0.5 V/div. of two subcircuit case. Let us note the horizontal axis' ranges of the experimental results in Fig. 4. They are 200ms/div in Fig. 4 (a) and 20ms/div in Fig. 4 (b) and (c). The switching interval in Fig. 4 (a) is much longer than that in Fig. 4 (b) and (c)
3.3 In the case of four subcircuits
When N = 4, in-region synchronization of selfswitching phenomena is observed like the case of N = 3 as shown in Fig. 5(a). Namely, the solution switches the states between fR; R; R; Rg and fL; L; L; Lg randomly. In this state, chaotic signals of each subcircuit are asynchronized. We expect that two pairs of anti-region synchronization of self-switching phenomena may be observed since two pairs of anti-phase synchronization of chaos is observed in the case of N = 4[8]. However, we can observe only in-region synchronization of self-
of synchronization of self-switching phenomena can be found. One is in-region synchronization which is that the all attractors switch the same regions synchronously. The other is anti-region synchronization which is that two attractors switch the opposite regions synchronously. We con rm the generation of in-region synchronization at N = 3 and 4, and anti-region synchronization at N = 2.
Acknowledgments The authors would like to thank Mr. Naoki Miyabayashi of Keio University for his many helpful comments related to this paper.
References Figure 5: In-region synchronization of selfswitching phenomena for N = 4 and = 0:46 (a) =0.17. (b) = 0:10. (1), (2), (3) and (4) Numerical calculations. (1) Time waveforms of xi . (2) x1 vs:x2 . (3) x1vs:x3 . (4) x1 vs:x4 . (5) and (6) Circuit experiments. (5) Time waveforms of v1i Horizontal: (a) 200ms/div, (b) and (c) 20ms/div, Vertical: 0.5 V/div. (6) v11vs:v12 . Horizontal and Vertical: 0.5 V/div. switching. As decreases, in-region synchronization cannot be observed as shown in Fig. 5(b) and switching interval is shorter than that in Fig. 5(a). We guess that the cause of generating synchronization of self-switching phenomena as follows. In two subcircuit case, there are 4 (=22) states by the combination of the region where each attractor is. As strength of coupling parameter is larger, the state fR; Rg and fL; Lg change from stable states to unstable ones. Therefore, the solution can be in the states only fR; Lg and fL; Rg and anti-region synchronization of self-switching phenomena occurs for large . Similarly, in the case of N = 3 and 4, there are N 2 states. As strength of coupling parameter is larger, 2N ,1 states change from stable states to unstable ones and only two states, namely, fR; R; Rg and fL; L; Lg in case of N = 3, and fR; R; R; Rg and fL; L; L; Lg in case of N = 4 remain stable states. Therefore, in-region synchronization can be observed.
4 Conclusion The self-switching phenomena in N simple chaotic oscillators coupled by one resistor in case of N = 2; 3 and 4 have been investigated by carrying out numerical calculations and circuit experiments. The synchronization of self-switching phenomena is observed for the rst time. In our system, two types
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