平成20年度電気関係学会四国支部連合大会

375

17-6

Synchronization Patternsin Three Cross-Coupled Chaotic Circuits
Yumiko Uchitani
Yoshifumi Nishio(Tokushima University)
1. Introduction
Studies on synchronization phenomena of coupledchaotic circuits are extensively carried out in various ﬁelds.
In our past studies, we investigated the state transition phenomenon in two cross-coupled chaotic circuits.
This state transition phenomena can be observed aroundin-phase synchronization, anti-phase synchronization andquadrature-phase synchronization.
In this study, synchronization patterns generated inthree cross-coupled chaotic circuits are investigated. Computer simulations and circuit experiments show that thissystem produces several phase patterns.

2. Circuit ModelFigure 1 shows the circuit model. In the circuit, threechaotic circuits are cross-coupled via inductors and smallresistors.

-gi 24
R

L2-g

R

Ri 21

L2i 22 R

Ri 26

L2

C1

i 25

i 11C2L 1 -gL2

v 21

R i 23(3)(4)

All other combinations of phase states can be obtained
from these type by using the symmetry of the couplingstructure.

v 22C1(yk − zk > 1)(|yk − zk | ≤ 1)
(yk − zk < −1)

TYPE I : [0, 0, 0]
TYPE II : [0, π/2, π/2]
TYPE III : [0, π, π]
TYPE IV : [0, π/2, π]

L2v 12id10 y −z +1kk

The synchronization states can be expressed by thephase diﬀerences of y2 and y3 with respect to the reference
waveform y1 . For example, fully in-phase synchronization
can be written as [0, 0, 0]. Also, if y2 is synchronized toy1 with π/2 phase diﬀerence (quadrature-phase) and y3 issynchronized to y1 with π phase diﬀerence (anti-phase),
the state can be written as [0, π/2, π]. By using this notation, all possible combinations of the phase states can besummarized as follows;

v 32L2

 yk − z k − 1

3. Synchronization Patterns

i 13

id3
v 31

v 11f (yk − zk ) =L1

C2

C1

where f are nonlinear functions corresponding to the v − icharacteristics of the nonlinear resistors and are describedas follows.

2.5

id2

i 12C2L1

0-2.50(a)2.5 -2.50(b)2.5 -2.50

2.5(c)

2.0Figure 1: Circuit model.

y1 0-2.0
2.0

By using the following variables and the parameters,√√
L1 ik1L1 ik2vk1

x=,w=, yk =,kk

CVCVV

22√
√
z =

vk2, t=L C τ, α =C2, β=L1

1 2k
C2√ C1
√V



LLC
d1
12

 γ=g, δ =, ²=R, “·”=

C2L2

L1

y2 0-2.0
2.0

y3 0-2.001200

Figure 2: Synchronization states (TYPE IV). α = 2.5,G,

β = 4.0, γ = 0.1, δ = 0.0007, and ² = 0.0005. (a) Attractor on y1 − y2 plane. (b) Attractor on y2 − y3 plane.
(c) Attractor on y3 − y1 plane. (d) Time waveform.

dτ(1)
the normalized circuit equations are given as follows.

ẋk = zk ẏ1 = α{γy1 − w1 − w2 − βf (y1 − z1 )}ẏ2 = α{γy2 − w3 − w6 − βf (y2 − z2 )} ẏ3 = α{γy3 − w4 − w5 − βf (y3 − z3 )}ż1 = β f (y1 − z1 ) + w4 + w3 − x1ż2 = β f (y2 − z2 ) + w2 + w5 − x2ż3 = β f (y3 − z3 ) + w1 + w6 − x3ẇ1 = δ(y1 − z3 − ²w1 ), ẇ2 = δ(y1 − z2 − ²w2 )


ẇ

 3 = δ(y2 − z1 − ²w3 ), ẇ4 = δ(y3 − z1 − ²w4 )ẇ5 = δ(y3 − z2 − ²w5 ), ẇ6 = δ(y2 − z3 − ²w6 )(d)Figure 2 shows computer calculated results of one example of the all possible types of synchronization statesobserved from the circuit in Fig 1. Also, the circuit experimental results show the similar phenomenon to the computer calculated results.

4 Conclusions(2)

In this study, we have investigate the synchronization
patterns characterized by the synchronization states inthree cross-coupled chaotic circuits. We conﬁrmed thatseveral patterns could be observed by giving diﬀerent initial conditions to the circuits.