I. INTRODUCTION
Synchronization and the related bifurcation of chaotic sys- tems are good models to describe various higher-dimensional nonlinear phenomena in the field of natural science. In par- ticular, the breakdown of chaos synchronization has attracted many researchers' attentions and their mechanisms have been gradually made clear [1]-[8]. However, a lot of phenomena around chaos synchronization are still veiled as well as other nonlinear circuits problems. Hence, in order to understand and exploit such phenomena, it is important to discover them, to model them, and to investigate them. On the other hand, it is important to investigate synchroniza- tion phenomena of coupled circuits under a difficult situation for the circuits. For examples, there are some frustrations in the circuit systems such as the coupling structure, frequency errors and external stimuli. Setou et al. have observed interesting synchronization phenomena when van der Pol oscillators with different oscillation frequencies are coupled by a resistor as a star topology [9]. We have investigated a ring consisting of van der Pol oscillators with different oscillation frequencies. By using computer simulations, oscillation death, independent os- cillation and double-mode oscillation were observed [10][11]. We consider that if coupled circuits with more different char- acteristics are coupled, the coupled circuits feel the frustrations more strongly. In this study, we investigate synchronization phenomena observed in coupled chaotic circuits as a ring topology when the chaotic circuits generating chaotic and three-periodic at- tractors are arranged in the circuit system. By using computer simulations and circuit experiments, we observe a collision be- tween chaotic and periodic attractors when the ratio of chaotic attractor is larger than the periodic attractor. After the collision, the periodic attractor arranged in the next to the chaotic attractor produce the bursting part such as intermittency-like Yoshifumi Nishio Dept. of Electrical and Electronic Engineering, Tokushima University 2-1 Minami-Josanjima, Tokushima, Japan Email: nishio@ee.tokushima-u.ac.jp chaos. And, chaos degree of the chaotic attractor becomes weak by the effects of the adjacent periodic attractor. II.  Circuit Model In this study, we consider a ring of the chaotic circuits as shown in Fig. 1. In the circuit system, two adjacent chaotic circuits are coupled by a resistor R via an inductor of each chaotic circuit. By using this coupling method, two adjacent circuits tend to synchronize with anti-phase state. Fig. 1. Coupled chaotic circuits as a ring topology. First, the i − v characteristics of the diodes are approximated by two-segment piecewise-linear functions as
By changing the variables the normalized circuit equations are given as
where
In these equations, γ is the coupling strength, N is the number of coupled chaotic circuits and β denotes the bifur- cation parameter. By changing β , the obtained attractors are controlled. We define βc to generate the chaotic attractors, and βρ is defined to generate the three-periodic attractors. For the computer simulations, we calculate Eq. (3) using the fourth-order Runge-Kutta method with step size h = 0.005. We set the parameters of this circuit model as follows; α = 7.0, β c = 0.175, β p = 0.160 and γ = 0.01.
III.  S YNCHRONIZATION P HENOMENA
A. N=3
Table I summarizes the observed phenomena for coupling oscillator number N = 3 by changing the bifurcation parame- ter β c . In these tables, "P" denotes three-periodic attractor , "(P)" denotes three-periodic attractor with oscillation, "C" denotes chaotic attractor ,“I” is intermittency-like chaos and T = 1e + 05[ τ ]. The initial circuit arrangement of the chaotic and the three-periodic attractors is shown in the middle row in the tables. From these tables, we can see that the collisions are occurred between the chaotic and the three-periodic attractors. Namely, the three-periodic attractors are attracted by the bursting behavior form the adjacent chaotic circuits and three- periodic attractors behaves like the intermittency-like chaos near the three-periodic window.
Next, the phase difference between the adjacent circuits are summarized in Table. II. "Non-syn" in these tables means the asynchronization. The same attractors are synchronized at anti-phase state. However, the chaotic and the three-periodic attractors could not be synchronized.
As one example, the observed attractors and the phase differences by using the computer simulations (initial pattern: (P-C-C) in Table I) are shown in Fig. 2. We also confirm the similar attractors and the phase differences in circuit experiments as shown in Fig. 3. Figure 4 shows the time series of xr1 + xr1 when the solution of the first circuit across to the Poincaré section ( XR1 + XL1 < 0, Z1 = 0). We can see that the three-periodic
attractor changes to the intermittency-like chaos by influence of adjacent chaotic circuits. We also confirm that the timing of the bursting depends on the circuit parameters such as the bifurcation parameter and coupling strength.
B. N=10
Next, we change the number of coupling circuits to N = 10. The computer simulation results are summarized in Table III. The collisions are observed between the chaotic and the periodic attractors. For the case of β c = 0.175, the periodic attractors are affected by chaotic factors and are attracted to the intermittency-like chaos or the three-periodic attractor with oscillation. While, in the case of β c = 0.185, when the ratio of chaotic attractor is larger than 40%, four-periodic intermittency-like chaos is observed. Figure 5 shows the two band chaos attractor and four- periodic intermittency-like chaotic attractor observed from the computer simulations. The time wave form of the four-periodic intermittency-like chaos is shown in Fig. 6. In addition, Fig. 7 shows one examples of synchronization states when the chaotic and the periodic attractors are arranged by irregularly. The horizontal axis is the Poincaré time and the vertical axis shows the value of x Rk + x Lk when the solution of the first circuit crosses to the Poincaré section ( x R 1 + x L 1 < −1.2, z 1 = 0). Namely, if the circuit generates chaotic solution the curve of xrk + xlk oscillates. While if the circuit generates the periodic solution the curve becomes constant. From these results, when the periodic attractor is located between the chaotic attractors, the periodic attractors are attracted by the bursting behavior from the chaotic circuits and the periodic attractors behaves like the intermittency-like chaos.
IV.  Application for Large Scale Networks
Finally, we apply the proposed circuit model to the large- scale networks. Figure 8 shows two examples of synchroniza- tion states when the chaotic and the periodic attractors are arranged by irregularly. The horizontal axis is the Poincaé time and the vertical axis shows the value of xrk + xlk when the solution of the first circuit crosses to the Poincaré section ( XR1 + XL1 < −1.2, Z1 = 0). In this case, the parameter of the chaotic attractors is fixed with β c = 0.175 and β c = 0.185. We confirm the complex behavior which is that the periodic attractor changes to the intermittency-like chaos by affecting the chaotic attractor. The results of the statistical analysis for this large scale network is going to be introduced at the presentation.
V.  CONCLUSION
In this study, we have investigated synchronization phenom- ena observed in coupled chaotic circuits as a ring topology when two types of the bifurcation parameters to generate chaotic and three-periodic attractors were used. By using computer simulations and circuit experiments, we have ob- served the clustering phenomena of coupled chaotic circuits depending on the number and the position of the circuits generating chaotic attractors. As our future works, we develop an efficient method to ana- lyze the obtained synchronization phenomena of the proposed system. Considering the other types of chaotic circuits and the other coupling methods are also important subjects for us.