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時変抵抗で結合されたカオス回路で観測される
複雑位相パターンについて
上手

洋子†

西尾

芳文††

ルディストープ†† Institute of Neuroinformatics, University / ETH Zurich
CH-8057 Winterthurerstrasse 190, Zurich, Switzerland†† 徳島大学大学院ソシオテクノサイエンス研究部〒 770–8506 徳島市南常三島 2–1
E-mail: †{yu001,ruedi}@ini.phys.ethz.ch, ††nishio@ee.tokushima-u.ac.jp
あらまし

近年、カオス結合系で観測される位相同期現象についての調査が活発に行われている。このような結合系

は様々な種類の位相パターンを生み出すことが知られており、この位相パターンを連想記憶や学習過程のモデリングに応用できるのではないかと期待されている。本研究では、リング状に時変抵抗で結合したカオス回路で観測される同期現象について調査を行う。コンピュータシミュレーションの結果、van der Pol 発振器で観測された位相伝搬派とは異なる複雑位相パターンを観測することができた。さらに、観測された複雑位相パターンの評価方法として、時間
空間エントロピーを適用した。
キーワード

結合発振器，複雑位相パターン，時変抵抗

Complex Phase Pattern in Chaotic Circuitswith Time-Varying Resistors
Yoko UWATE† , Yoshifumi NISHIO†† , and Ruedi STOOP†† University / ETH Zurich CH-8057 Winterthurerstrasse 190, Zurich, Switzerland†† Tokushima University 2–1 Minami Josanjima, Tokushima, 770–8506 Japan
E-mail: †{yu001,ruedi}@ini.phys.ethz.ch, ††nishio@ee.tokushima-u.ac.jp
Abstract Recently, studies on phase synchronization phenomena of coupled chaotic oscillators are extensively
carried out by many researchers. Such oscillatory systems can produce some kinds of phase patterns, and they may
be utilized modeling for associative memory and learning process. In this study, we investigate the synchronization
phenomena in chaotic circuits coupled by time-varying resistor as a ring. By carrying out computer simulations, we
conﬁrm the complex phase pattern which cannot be observed in simple oscillatory systems coupled by a resistor.
Furthermore, we apply a space-time entropy to evaluate complex phase patterns obtained form coupled chaotic
circuits.
Key words coupled oscillators, complex phase pattern, time-varying resistor

1. Introduction
Recently, studies on phase synchronization phenomena of
coupled chaotic oscillators are extensively carried out byring and a two-dimensional array [10]. Yamauchi et al. havediscovered very interesting wave propagation phenomena ofphase states between two adjacent oscillators in an array ofvan der Pol oscillators coupled by inductors [11].

many researchers [1]- [9]. Such oscillatory systems can pro-On the other hand, there are some systems whose dissi-

duce some kinds of phase patterns, and they may be utilized

pation factors vary with time, for example, under the time-modeling for associative memory and learning process. Endovariation of the ambient temperature, an equation describing

et al. have reported details of theoretical analysis and circuit

an object moving in a space with some friction and an equa-

experiments about some coupled oscillators as a ladder, ation governing a circuit with a resistor whose temperature—1—

gation and clustering) as shown in Fig. 1.

x1+x2
x2+x3
x3+x4
x4+x5
x5+x6
x6+x7
x7+x8
x8+x9
x9+x10x10+x11x11+x12x12+x13x13+x14x14+x15x15+x1In this study, we investigate the complex phase patternwhen coupled van der Pol oscillator is changed to chaotic
circuit. First, the case of even number coupling, the coexisτ
(a) Wave extinction.

tence of in-phase and anti-phase states are observed. In contrast, the case of odd number coupling, we can conﬁrm thecoexistence between in-phase and n-phase states. Second,the coexistence area changing the bifurcation parameter of

x1+x2
x2+x3
x3+x4
x4+x5
x5+x6
x6+x7
x7+x8
x8+x9
x9+x10x10+x11x11+x12x12+x13x13+x14x14+x15x15+x1chaotic circuits is investigated. By carrying out computer
simulations, we conﬁrm the complex phase pattern whichcannot be observed in simple oscillatory systems coupled byτ(b) Random pattern.

a resistor. Next, we apply a space-time entropy to evaluate complex phase patterns obtained form coupled chaotic
circuits.

2. Coupled Oscillators Model

x1+x2
x2+x3
x3+x4
x4+x5
x5+x6
x6+x7
x7+x8
x8+x9
x9+x10x10+x11x11+x12x12+x13x13+x14x14+x15x15+x1In this study, we consider a ring of chaotic circuits as shownin Fig. 2. In this circuit adjacent two chaotic circuits are coupled by one time-varying resistor (TVR). We realize the TVRτby switching a positive and a negative resistors periodicallyas shown in Fig. 3.

(c) Wave propagation.
x1+x2
x2+x3
x3+x4
x4+x5
x5+x6
x6+x7
x7+x8
x8+x9
x9+x10x10+x11x11+x12x12+x13x13+x14x14+x15x15+x1

R(t)

2π

r
τ

ω tt

0

(d) Clusterling.
Fig. 1 Four types of synchronization phenomena observed froma ring of van der Pol oscillators coupled by time varying-r2πpresistors.

Fig. 3 Characteristics of the TVR.
2L1 ILk IRk

2L1

2L1 IL(k+1) IR(k+1) 2L1

TVR

C

vk

L2

-r

First, the i − v characteristics of the diode are approxi-

ik+1

ik

vd(ik)

TVR

C
-r

vk+1

L2

TVR

vd(ik+1)

mated by two-segment piecewise linear function as
vd (ik ) =1(rd ik + E− | rd ik − E |).
2(1)

By changing the variables and the parameters,√
Fig. 2 Coupled oscillators model.

coeﬃcient is sensitive such as thermistor. However, there

IRk =

C
ExRk ,
L1

vk = Ezk , t =√In our previous research, we have investigated the synchronization phenomena in van der Pol oscillators coupledby time-varying resistor as a ring [12], [13]. We realized thetime-varying resistor by switching a positive and a negative
resistor periodically. We conﬁrmed the various interesting

L1
α=
,
L2
ω= √

ILk =

C
ExLk ,
L1

√
ik =

C
Eyk ,
L1

L1 Cτ ,

are few discussion about coupled oscillators coupled by atime-varying resistor.

√

√
β=r

C
,
L1

√
γ=R

C
,
L1

√
δ = rd

C
,
L1

1
ωτ ,
L1 C

the normalized circuit equations of the ring of chaotic circuits
are given as

phenomena (wave extinction, randam pattern, wave propa-

—2—


dxRk
1


= {β(xRk + xLk + yk ) − zk − γ(xRk + xL(k+1) )}


dτ
2






dxLk
1


= {β(xRk + xLk + yk ) − zk − γ(xLk + xL(R+1) )}


dτ
2




dyk

(2)

= α{β(xRk + xLk + yk ) − zk − f (yk )}

dτ






dzk


= xRk + xLk + yk


dτ







(k = 1, 2, 3, · · · , N )(a) In-phase synchronization.

where
f (yk ) =1
(δyk + 1− | δyk − 1 |)
2

(3)

and
xLN = xL1 ,xR0 = xRN .

(4)

It should be noted that γ corresponds to the coupling(b) Anti-phase synchronization.

strength and that β corresponds to the bifurcation parameter of chaotic circuits. Eq. (2) is calculated by using the

Fig. 4 Computer simulated result for N = 14.

α = 7.0, β =0.094, δ = 50.0, ω = 1.924, γ = (0.2 or −0.2).

fourth-order Runge-Kutta method.

Up-per ﬁgures: xRk + xLk vs zk . Middle ﬁgures: xRk + xLk
vs xR(k+1) + xL(k+1) . Lower ﬁgures: τ vs xRk + xLk .

3. Synchronization Phenomena

k = 1, 2, 3, . . . , 14.

3. 1 Even Number Coupling: N = 14
Figure 4 shows the computer simulated result for the case

Attractorof N = 14. N denotes the number of coupled oscillators. We

Phase
difference

can see that the ring of chaotic circuits coupled by TVR are

xR1+xL1xR2+xL2xR3+xL3xR4+xL4xR5+xL5xR6+xL6xR7+xL7xR8+xL8xR9+xL9
xR10+xL10
xR11+xL11
xR12+xL12
xR13+xL13
xR14+xL14
xR15+xL15

synchronized at in-phase or at anti-phase.
3. 2 Odd Number Coupling: N = 15
Figure 5 shows the computer simulated result for the caseof N = 15. We also can see that the ring of chaotic oscillators

τcoupled by TVR are synchronized with in-phase (Fig. 5(a)).
And the adjacent circuts are almost synchronized with antiphase as shown in Fig. 5(b). Because, the boundary condition is the ring structure, the phase diﬀerence between theadjacent circuits is not around π. Namely, in this case 15phase synchronization are observed.
3. 3 Complex Phase Patterns
Here, we investigate the synchronization phenomena when(a) In-phase synchronization.
Attractor
Phase
differencexR1+xL1xR2+xL2xR3+xL3xR4+xL4xR5+xL5xR6+xL6xR7+xL7xR8+xL8xR9+xL9
xR10+xL10
xR11+xL11
xR12+xL12
xR13+xL13
xR14+xL14
xR15+xL15

τ

the strength of the coupling parameter γ is decreased. We(b) 15-phase synchronization.

ﬁnd complex phase patterns as shown in Figs. 6-7. Thesephenomena can be observed by switching of the phase states
between the in-phase and the anti-phase synchronization

Fig. 5 Computer simulated result for N = 15.

α = 7.0, β =0.094, δ = 50.0, ω = 1.924, γ = (0.2 or −0.2).

Up-

per ﬁgures: xRk + xLk vs zk . Middle ﬁgures: xRk + xLk

with two adjacent chaotic circuits, whereas the white region

vs xR(k+1) + xL(k+1) . Lower ﬁgures: τ vs xRk + xLk .

shows the in-phase synchronization. From these ﬁgures, thek = 1, 2, 3, . . . , 15.

wave of the anti-phase propagates without regular rule.
Figures 8-9 show the coupling strength of breakdown synchronization from coexistence to phase pattern phenomena
by changing the bifurcation parameter β of the chaotic cir-parameter β.
The examples of complex phase pattern are shown in
Figs. 10-11 when the parameter β and γ are changed.

cuit. We conﬁrm that the coupling strength of breakdownof synchronization becomes large by increasing bifurcation—3—

(xR1+xL1)+(xR2+xL2)(xR2+xL2)+(xR3+xL3)(xR3+xL3)+(xR4+xL4)(xR4+xL4)+(xR5+xL5)(xR5+xL5)+(xR6+xL6)(xR6+xL6)+(xR7+xL7)(xR7+xL7)+(xR8+xL8)(xR8+xL8)+(xR9+xL9)(xR9+xL9)+(xR10+xL10)(xR10+xL10)+(xR11+xL11)(xR11+xL11)+(xR12+xL12)(xR12+xL12)+(xR13+xL13)(xR13+xL13)+(xR14+xL14)(xR14+xL14)+(xR15+xL15)(xR15+xL15)+(xR1+xL1)

Fig. 6 Complex phase pattern for N = 14.

α = 7.0, β =0.094, δ = 50.0, ω = 1.924, γ = (0.166 or −0.166).
(xR1+xL1)+(xR2+xL2)(xR2+xL2)+(xR3+xL3)(xR3+xL3)+(xR4+xL4)(xR4+xL4)+(xR5+xL5)(xR5+xL5)+(xR6+xL6)(xR6+xL6)+(xR7+xL7)(xR7+xL7)+(xR8+xL8)(xR8+xL8)+(xR9+xL9)(xR9+xL9)+(xR10+xL10)(xR10+xL10)+(xR11+xL11)(xR11+xL11)+(xR12+xL12)(xR12+xL12)+(xR13+xL13)(xR13+xL13)+(xR14+xL14)(xR14+xL14)+(xR15+xL15)(xR15+xL15)+(xR1+xL1)(xR1+xL1)+(xR2+xL2)(xR2+xL2)+(xR3+xL3)(xR3+xL3)+(xR4+xL4)(xR4+xL4)+(xR5+xL5)(xR5+xL5)+(xR6+xL6)(xR6+xL6)+(xR7+xL7)(xR7+xL7)+(xR8+xL8)(xR8+xL8)+(xR9+xL9)(xR9+xL9)+(xR10+xL10)(xR10+xL10)+(xR11+xL11)(xR11+xL11)+(xR12+xL12)(xR12+xL12)+(xR13+xL13)(xR13+xL13)+(xR14+xL14)(xR14+xL14)+(xR15+xL15)(xR15+xL15)+(xR1+xL1)

α = 7.0, β =0.094, δ = 50.0, ω = 1.924, γ = (0.136 or −0.136).

0.095

β (bifurcation parameter)(a) β=0.065.

τ(b) β=0.075.

τFig. 7 Complex phase pattern for N = 15.

0.09

τ

Switching: phase pattern(in-phase & anti-phase)τ(c) β=0.085.

0.085(xR1+xL1)+(xR2+xL2)(xR2+xL2)+(xR3+xL3)(xR3+xL3)+(xR4+xL4)(xR4+xL4)+(xR5+xL5)(xR5+xL5)+(xR6+xL6)(xR6+xL6)+(xR7+xL7)(xR7+xL7)+(xR8+xL8)(xR8+xL8)+(xR9+xL9)(xR9+xL9)+(xR10+xL10)(xR10+xL10)+(xR11+xL11)(xR11+xL11)+(xR12+xL12)(xR12+xL12)+(xR13+xL13)(xR13+xL13)+(xR14+xL14)(xR14+xL14)+(xR15+xL15)(xR15+xL15)+(xR1+xL1)

0.08
0.075

Coexistence(in-phase & anti-phase)

0.07(xR1+xL1)+(xR2+xL2)(xR2+xL2)+(xR3+xL3)(xR3+xL3)+(xR4+xL4)(xR4+xL4)+(xR5+xL5)(xR5+xL5)+(xR6+xL6)(xR6+xL6)+(xR7+xL7)(xR7+xL7)+(xR8+xL8)(xR8+xL8)+(xR9+xL9)(xR9+xL9)+(xR10+xL10)(xR10+xL10)+(xR11+xL11)(xR11+xL11)+(xR12+xL12)(xR12+xL12)+(xR13+xL13)(xR13+xL13)+(xR14+xL14)(xR14+xL14)+(xR15+xL15)(xR15+xL15)+(xR1+xL1)

τ

0.065

(d) β=0.095.
0.06
0.13

0.135

0.14

0.145

0.15

0.155

0.16

0.165

0.17

γ (coupling strength)

Fig. 10 Examples of pahse propagation by changing β from
0.065 to 0.095.

Fig. 8 Synchronization types (N = 14). α = 7.0, δ = 50.0, ω =α = 7.0, δ = 50.0, ω = 1.924, γ =(0.132 or −0.132).

1.924.

S=−

β (bifurcation parameter)

0.095
0.09

t
ct
1 1 ∑ ∑ ∑ ci,m∑ t log2 ∑ i,mt
nT
c
c
m i,m
m i,m
i

Switching(in-phase & 15-phase)ctwhere ∑ i,mt

0.085

m

ci,m

t

(5)

m

denotes the probabilities and m counts

the two possible states ai ∈ {0, 1} of cell. This space-time

0.08

entropy is a measure of the order some cellular automata
0.075

conﬁguration at a number of cell n and a ﬁxed time T (=65).
Coexistence(in-phase & 15-phase)

0.07

We apply the space-time entropy to evaluation of complex phase patterns obtained from a ring of chaotic circuits.

0.065
0.06
0.13

0.135

0.14

0.145

0.15

0.155

0.16

0.165

0.17

0.175

γ (coupling strength)

Fig. 9 Synchronization types (N = 15). α = 7.0, δ = 50.0, ω =1.924.

First, obtained complex phase patterns are changed to discrete pattern data like cellular automata. In order to distinguish in-phase or anti-phase synchronization, the thresholdvalue th = 1.0 is introduced as shown in Fig. 12. When th islarger than 1.0, the synchronization state is anti-phase.
The discrete data patterns obtained from Figs. 10-11 are

4. Entropy
In order to capture the properties of cellular automata aspace-time entropy have been introduced [16]. The spacetime entropy of cellular automata with n cells is deﬁned asfollows:

shown in Figs. 13-14. “¥ (1)” and “¤ (0)” are corresponding
to the in-phase and the anti-phase synchronization, respectively. We caluculate the space-time entropy for these discrete pattern data and the value of entropy are described in
subcaption of each ﬁgures. From these results, all entropies
show high value which is larger than 0.9.
—4—

(XR1+XL1)+(XR2+XL2)(xR1+xL1)+(xR2+xL2)(xR2+xL2)+(xR3+xL3)(xR3+xL3)+(xR4+xL4)(xR4+xL4)+(xR5+xL5)(xR5+xL5)+(xR6+xL6)(xR6+xL6)+(xR7+xL7)(xR7+xL7)+(xR8+xL8)(xR8+xL8)+(xR9+xL9)(xR9+xL9)+(xR10+xL10)(xR10+xL10)+(xR11+xL11)(xR11+xL11)+(xR12+xL12)(xR12+xL12)+(xR13+xL13)(xR13+xL13)+(xR14+xL14)(xR14+xL14)+(xR15+xL15)(xR15+xL15)+(xR1+xL1)2

th=1.0

1
0
-1

τ

-2
0

20000

40000

60000

80000

100000120000τ(a) γ=0.13.

Fig. 12 Threshold for distinguishing in-phase or anti-phase syn-(xR1+xL1)+(xR2+xL2)(xR2+xL2)+(xR3+xL3)(xR3+xL3)+(xR4+xL4)(xR4+xL4)+(xR5+xL5)(xR5+xL5)+(xR6+xL6)(xR6+xL6)+(xR7+xL7)(xR7+xL7)+(xR8+xL8)(xR8+xL8)+(xR9+xL9)(xR9+xL9)+(xR10+xL10)(xR10+xL10)+(xR11+xL11)(xR11+xL11)+(xR12+xL12)(xR12+xL12)+(xR13+xL13)(xR13+xL13)+(xR14+xL14)(xR14+xL14)+(xR15+xL15)(xR15+xL15)+(xR1+xL1)

chronization.

τ(b) γ=0.14.
(xR1+xL1)+(xR2+xL2)(xR2+xL2)+(xR3+xL3)(xR3+xL3)+(xR4+xL4)(xR4+xL4)+(xR5+xL5)(xR5+xL5)+(xR6+xL6)(xR6+xL6)+(xR7+xL7)(xR7+xL7)+(xR8+xL8)(xR8+xL8)+(xR9+xL9)(xR9+xL9)+(xR10+xL10)(xR10+xL10)+(xR11+xL11)(xR11+xL11)+(xR12+xL12)(xR12+xL12)+(xR13+xL13)(xR13+xL13)+(xR14+xL14)(xR14+xL14)+(xR15+xL15)(xR15+xL15)+(xR1+xL1)(a) β=0.065, S=0.932.

τ(b) β=0.075, S=0.934.

(c) γ=0.15.
(xR1+xL1)+(xR2+xL2)(xR2+xL2)+(xR3+xL3)(xR3+xL3)+(xR4+xL4)(xR4+xL4)+(xR5+xL5)(xR5+xL5)+(xR6+xL6)(xR6+xL6)+(xR7+xL7)(xR7+xL7)+(xR8+xL8)(xR8+xL8)+(xR9+xL9)(xR9+xL9)+(xR10+xL10)(xR10+xL10)+(xR11+xL11)(xR11+xL11)+(xR12+xL12)(xR12+xL12)+(xR13+xL13)(xR13+xL13)+(xR14+xL14)(xR14+xL14)+(xR15+xL15)(xR15+xL15)+(xR1+xL1)(c) β=0.085, S=0.948.

τ(d) γ=0.16.

(d) β=0.095, S=1.000.

Fig. 11 Examples of pahse propagation by changing γ from 0.13

Fig. 13 Discrete data obtained from Fig 10.

to 0.16. α = 7.0, β=0.094, δ = 50.0, ω = 1.924.

For comparison, we also apply the space-time entropy tocomplex phase patterns (Fig. 1) obtained from a ring of van(a) γ=0.13, S=0.961.

der Pol oscillators. The discrete data of phase patterns andthe value of the space-time entropy are shown in Fig. 15.
From these results, we conﬁrm that the space-time entropyof phase patterns obtained from chaotic circuits is higher(b) γ=0.14, S=0.974.

value than the van der Pol oscillators.
Finally we caluculate the time evolution of average entropy
and the obtained results are shown in Figs. 16, 17. In thecase of the chaotic circuits (Fig. 16), the every average en-(c) γ=0.15, S=0.987

tropy show high value. While, in the case of the patterns
obtained from the van der Pol oscillators, each pattern converge the diﬀerent entropy value. We consider that it is one
possibility to classiﬁcate these complex patterns obtained the(d) γ=0.16, S=0.989.
Fig. 14 Discrete data obtained from Fig 11.

ring of van der Pol oscillators and chaotic circuits by using
space-time entropy.

5. Conclusions

when the coupling strength decreases, we observed complex
phase pattern which cannot be observed in simple oscillatory circuits coupled by resistors. Furthermore, we appled

In this study, we have investigated synchronization phenomena in chaotic circuits coupled by time varying resistors
as a ring. By computer simulations, ﬁrst we conﬁrmed the co-the space-time entropy to evaluate complex phase patterns.
The space-time entropy obtained from the coupled chaotic
oscillators showed high values have been observed.

existence of in-phase and anti-phase synchronizations. Next,—5—

(a) S=0.575.

(b) S=0.920.

(c) S=0.720(d) S=0.447.
Fig. 15 Discrete data obtained from Fig 1.
1.2(b) γ=0.14(c) γ=0.15

1(a) γ=0.13

S(t)

0.8

(d) γ=0.16

0.6

0.4

0.2

0

0

200

400

600

800

1000
t

1200

1400

1600

1800

2000

Fig. 16 Time evolution of average entropy for chaotic circuits.
1.2(b) Random pattern

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0.8
S(t)

(c) Wave propagation
0.6(d) Clusterling
0.4(a) Wave extinction

0.2

0

0

200

400

600

800

1000
t

1200

1400

1600

1800

2000

Fig. 17 Time evolution of average entropy for van der Pol oscillators.

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