平成23年度 電気関係学会四国支部連合大会

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Synchronization of Chaos Circuits
Coupled by LC Circuit
Naoya Kondo, Ayana Shimada, Yoko Uwate and Yoshifumi Nishio(Tokushima University)

1. Introduction
Many nonlinear dynamical systems in various ﬁelds
have been conﬁrmed to exhibit chaotic oscillation.
Synchronization of chaos observed in coupled chaoticsystems attract many researchers’ attentions, becausesuch a phenomenon includes many unsolved interesting
behavior and at the same time possesses possible futureengineering applications.
There have been many researches on chaos synchronizations. We also have investigated chaotic circuits coupled
by a resistor, an inductor, and a transmission line.
In this study, we investigate two simple chaotic circuits
coupled by LC circuit.

2. Circuit Model
The chaotic circuit used in this study was proposed byInaba [1]. We investigate two Inaba circuits combined likeFig. 1 by LC circuit.

In the circuit equations, the parameters α, β, and γ arebifurcation parameters of each Inaba circuit, and the parameters δ and ε express the coupling capacitor and inductor, respectively.

3. Simulation results
We ﬁxed the parameters α= 4.0, β= 0.1, and γ= 0.27,
and investigate the inﬂuence of the coupling parameters δand ε on synchronization. Figure 2 shows two examplesof computer simulation results. In the ﬁgures, the attractors of the left Inaba circuit, the attractor of the right
Inaba circuit, and the synchronization state between thetwo Inaba circuits from the left. The parameter δ seems tocontrol the synchronization ability. Namely, for smaller δ
the two circuits do not synchronize. For relatively large δthe circuits start to synchronize as shown in Fig. 2. However, for large δ, the synchronization breaks down and theattractors diverge. Moreover, the parameter ε seems tocontrol the synchronization states. Namely, for smaller εthe two circuits tend to synchronize at in-phase as shownin Fig. 2(a), while for larger ε they tend to synchronize atanti-phase as shown in Fig. 2(b).

Figure 1: Two Inaba circuits coupled by LC circuit.
(a)
We can derive the normalized state equations of theInaba circuit coupled by LC circuit as follows.

()
 dx
α11n
= zn −xn + − xn − dτ2αα dyn
= zn − δω2dτ
dzn= −x − βy + γz(1)nnndτdω1= δω2
dτ
 dω2 = β(y + y ) − εω
1dτ

2(b)

14. Conclusions(n = 1, 2.)
where variables and parameters are deﬁned by the following equations.
√√
C1
L1 C1
V xn , In =V y n , v n = V zn
in =L1
L2
√
i0 =

√α=r

Powered by TCPDF (www.tcpdf.org)(n = 1, 2.)
L1 C1
C1V ω 1 , v0 =V ω2L0C0C1L1,β =,γ = gL1L2√Figure 2: Synchronization of chaos. α=4.0, β=0.1,γ=0.27, δ=0.7. (a) In-phase synchronization for
ε=0.5. (b) Anti-phase synchronization for ε=1.0.

L1C1L1,δ =,ε =C1C0L0In this study, we investigated the synchronization of two
Inaba circuit coupled by LC circuit. By carrying out computer calculations and circuit experiments, we observed
two diﬀerent types of chaos synchronization and investigated the eﬀect of the coupling parameters.
References[1] N. Inaba and S. Mori, ”Chaotic Phenomena in a Circuit with a Diode due to the Change of the OScillation
Frequency,” Trans of IEICE, vol. E71, no. 9, pp. 842-849,
Sep. 1988.

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