平成 27 年度電気関係学会四国支部連合大会 講演論文集 (2015 高知工科大学)
2015 SHIKOKU-SECTION JOINT CONVENTION RECORD OF THE INSTITUTES OF ELECTRICAL AND RELATED ENGINEERS (KOCHI)

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Synchronization and Chaos Propagation
betweem Chaotic and Periodic Attractors in Symmetric System
Shogo TAMADA

Kenta AGO Yoko UWATE(Tokushima University)

Yoshifumi NISHIO

1. Introduction

3. Simulation results

In this study, we investigate synchronization andchaos propagation in five coupled chaotic circuits invarious systems. We propose a ladder system thatthe central circuit generates chaotic attractor andthe other circuits generate three-periodic attractors.
Moreover, we compare the phase diﬀerence among
symmetric systems in the cases of adding the coupling
resistor from the ladder system.

Figure 2 shows some examples of the computer simulation results. Figure 2(a) shows the initial state when
all circuits are not connected. We observe the statethat the chaos are propagated to the only adjacent circuits from the central chaotic circuit in a range of thecoupling strength g (see Fig. 2(b)). By increasing thecoupling strength g, all three-periodic attractors areaﬀected from chaotic attractor and all circuits close
to the synchronous state(see Fig. 2(c)). Moreover,Fig. 3 shows the four symmetric systems considered byadding three edges to the ladder system of Fig. 1(b).

2. System model
Figure 1(a) shows the chaotic circuit. we propose
the ladder system model in Fig. 1(b). Each circuit
is coupled by the resistors which correspond to theedges in this system. The central circuit generates thechaotic attractor and the other circuits generate thethree-periodic attractors.

Figure 3: Four symmetric systems.

Cn(a) Chaotic circuit.

Table 1 shows the phase diﬀerences of ladder systemand the four symmetric systems in the case of g = 0.01
and each system pattern corresponds to Fig. 3. In Table 1, the phase diﬀerences are masured between theadjacent circuits and the phase diﬀerences shows theaverage in each entire system. From Table 1, the phase
diﬀerences in the four symmetric systems are smaller
than the ladder system. Additionally, the phase diﬀerences of 7-A and 7-C are smaller than 7-B and 7-D in
the four symmetric systems. This reason can be considered that the central chaotic circuit is connected toall circuits in the systems of 7-A and 7-C.

(b) Ladder system model.

Figure 1: System model.
In the proposed ladder system, the circuits are connected to only adjacent circuits by the resistors. The
normalized circuit equations of the system are given as
follows:
 x˙ = αx + znnn




()

δ
1
1y˙n = zn −y+− y−(1)
2
δδ
∑



g(zn − zm ), z˙n = −xn − βyn −Table 1: Phase diﬀerences in the symmetric systems(g = 0.01).
C1 -C2phase diﬀerence [◦ ]C2 -C3C3 -C4C4 -C5

13.08

16.91

23.41

12.38

16.45

7.14
6.65
7.51
6.77

12.12
22.48
11.09
19.73

7.49
6.66
7.88
6.79

6.51
6.46
7.96
7.65

8.32
10.57
8.61
10.24

m∈snedges

where α represents the chaos degree. n represents thecircuit number up to 5 in this study. Sn is the set ofcircuits which are directly connected to Cn . g represents the coupling strength corresponding the coupling
resistor R.

4(Ladder)
7-A
7-B7-C
7-D.

Average

4. Conclusions
In this study, we have researched about synchronization and chaos propagation of five coupled chaotic circuits in the proposed systems. We observed that thechaotic attractor of the central circuit propagates toall circuits by increasing the coupling strength in theladder system. Moreover, we compared the phase difference among the ladder system and the four symmetric systems. As a result, the entire system is easy
to reach a state of synchronization when the central
chaotic circuit is connected to many circuits.

Figure 2: Attractor in the laddr system.

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