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社団法人 電子情報通信学会THE INSTITUTE OF ELECTRONICS,INFORMATION AND COMMUNICATION ENGINEERS
信学技報TECHNICAL REPORT OF IEICE.
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 ZurichCH-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 Circuits
with 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 by
many researchers [1]- [9]. Such oscillatory systems can pro-
duce some kinds of phase patterns, and they may be utilized
modeling for associative memory and learning process. Endo
et al. have reported details of theoretical analysis and circuit
experiments about some coupled oscillators as a ladder, a
ring and a two-dimensional array [10]. Yamauchi et al. have
discovered very interesting wave propagation phenomena of
phase states between two adjacent oscillators in an array of
van der Pol oscillators coupled by inductors [11].
On the other hand, there are some systems whose dissi-
pation factors vary with time, for example, under the time-
variation of the ambient temperature, an equation describing
an object moving in a space with some friction and an equa-
tion governing a circuit with a resistor whose temperature
— 1 —
x1+x2x2+x3x3+x4x4+x5x5+x6x6+x7x7+x8x8+x9
x9+x10
x11+x12x12+x13x13+x14x14+x15
x15+x1
x10+x11
τ
(a) Wave extinction.
x1+x2x2+x3x3+x4x4+x5x5+x6x6+x7x7+x8x8+x9
x9+x10
x11+x12x12+x13x13+x14x14+x15
x15+x1
x10+x11
τ
(b) Random pattern.
x1+x2x2+x3x3+x4x4+x5x5+x6x6+x7x7+x8x8+x9
x9+x10
x11+x12x12+x13x13+x14x14+x15
x15+x1
x10+x11
τ
(c) Wave propagation.
x1+x2x2+x3x3+x4x4+x5x5+x6x6+x7x7+x8x8+x9
x9+x10
x11+x12x12+x13x13+x14x14+x15
x15+x1
x10+x11
τ
(d) Clusterling.
Fig. 1 Four types of synchronization phenomena observed from
a ring of van der Pol oscillators coupled by time varying
resistors.
2L12L1 2L1 2L1
TVR C
vd(ik)
TVR TVRL2
-r
vk
ILk IRk IL(k+1) IR(k+1)
ik
C
vd(ik+1)
L2
-r
vk+1
ik+1
Fig. 2 Coupled oscillators model.
coefficient is sensitive such as thermistor. However, there
are few discussion about coupled oscillators coupled by a
time-varying resistor.
In our previous research, we have investigated the syn-
chronization phenomena in van der Pol oscillators coupled
by time-varying resistor as a ring [12], [13]. We realized the
time-varying resistor by switching a positive and a negative
resistor periodically. We confirmed the various interesting
phenomena (wave extinction, randam pattern, wave propa-
gation and clustering) as shown in Fig. 1.
In this study, we investigate the complex phase pattern
when coupled van der Pol oscillator is changed to chaotic
circuit. First, the case of even number coupling, the coexis-
tence of in-phase and anti-phase states are observed. In con-
trast, the case of odd number coupling, we can confirm the
coexistence between in-phase and n-phase states. Second,
the coexistence area changing the bifurcation parameter of
chaotic circuits is investigated. By carrying out computer
simulations, we confirm the complex phase pattern which
cannot be observed in simple oscillatory systems coupled by
a resistor. Next, we apply a space-time entropy to evalu-
ate complex phase patterns obtained form coupled chaotic
circuits.
2. Coupled Oscillators Model
In this study, we consider a ring of chaotic circuits as shown
in Fig. 2. In this circuit adjacent two chaotic circuits are cou-
pled by one time-varying resistor (TVR). We realize the TVR
by switching a positive and a negative resistors periodically
as shown in Fig. 3.
2πp
2πR(t)
ωtt0
r
-r
Fig. 3 Characteristics of the TVR.
First, the i − v characteristics of the diode are approxi-
mated by two-segment piecewise linear function as
vd(ik) =1
2(rdik + E− | rdik − E |). (1)
By changing the variables and the parameters,
IRk =
√C
L1ExRk, ILk =
√C
L1ExLk, ik =
√C
L1Eyk,
vk = Ezk, t =√
L1Cτ,
α =L1
L2, β = r
√C
L1, γ = R
√C
L1, δ = rd
√C
L1,
ω =1√L1C
ωτ ,
the normalized circuit equations of the ring of chaotic circuits
are given as
— 2 —
dxRk
dτ=
1
2{β(xRk + xLk + yk) − zk − γ(xRk + xL(k+1))}
dxLk
dτ=
1
2{β(xRk + xLk + yk) − zk − γ(xLk + xL(R+1))}
dyk
dτ= α{β(xRk + xLk + yk) − zk − f(yk)}
dzk
dτ= xRk + xLk + yk
(k = 1, 2, 3, · · · , N)
(2)
where
f(yk) =1
2(δyk + 1− | δyk − 1 |) (3)
and
xLN = xL1, xR0 = xRN . (4)
It should be noted that γ corresponds to the coupling
strength and that β corresponds to the bifurcation param-
eter of chaotic circuits. Eq. (2) is calculated by using the
fourth-order Runge-Kutta method.
3. Synchronization Phenomena
3. 1 Even Number Coupling: N = 14
Figure 4 shows the computer simulated result for the case
of N = 14. N denotes the number of coupled oscillators. We
can see that the ring of chaotic circuits coupled by TVR are
synchronized at in-phase or at anti-phase.
3. 2 Odd Number Coupling: N = 15
Figure 5 shows the computer simulated result for the case
of N = 15. We also can see that the ring of chaotic oscillators
coupled by TVR are synchronized with in-phase (Fig. 5(a)).
And the adjacent circuts are almost synchronized with anti-
phase as shown in Fig. 5(b). Because, the boundary condi-
tion is the ring structure, the phase difference between the
adjacent circuits is not around π. Namely, in this case 15-
phase synchronization are observed.
3. 3 Complex Phase Patterns
Here, we investigate the synchronization phenomena when
the strength of the coupling parameter γ is decreased. We
find complex phase patterns as shown in Figs. 6-7. These
phenomena can be observed by switching of the phase states
between the in-phase and the anti-phase synchronization
with two adjacent chaotic circuits, whereas the white region
shows the in-phase synchronization. From these figures, the
wave of the anti-phase propagates without regular rule.
Figures 8-9 show the coupling strength of breakdown syn-
chronization from coexistence to phase pattern phenomena
by changing the bifurcation parameter β of the chaotic cir-
cuit. We confirm that the coupling strength of breakdown
of synchronization becomes large by increasing bifurcation
(a) In-phase synchronization.
(b) Anti-phase synchronization.
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). Up-
per figures: xRk + xLk vs zk. Middle figures: xRk + xLk
vs xR(k+1) + xL(k+1). Lower figures: τ vs xRk + xLk.
k = 1, 2, 3, . . . , 14.
Attractor
Phasedifference
xR1+xL1
xR2+xL2
xR3+xL3
xR4+xL4
xR5+xL5
xR6+xL6
xR7+xL7
xR8+xL8
xR9+xL9
xR10+xL10
xR11+xL11
xR12+xL12
xR13+xL13
xR14+xL14
τxR15+xL15
(a) In-phase synchronization.
Attractor
Phasedifference
xR1+xL1
xR2+xL2
xR3+xL3
xR4+xL4
xR5+xL5
xR6+xL6
xR7+xL7
xR8+xL8
xR9+xL9
xR10+xL10
xR11+xL11
xR12+xL12
xR13+xL13
xR14+xL14
τxR15+xL15
(b) 15-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
vs xR(k+1) + xL(k+1). Lower figures: τ vs xRk + xLk.
k = 1, 2, 3, . . . , 15.
parameter β.
The examples of complex phase pattern are shown in
Figs. 10-11 when the parameter β and γ are changed.
— 3 —
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)
τ(xR15+xL15)+(xR1+xL1)(xR14+xL14)+(xR15+xL15)
Fig. 7 Complex phase pattern for N = 15. α = 7.0, β =
0.094, δ = 50.0, ω = 1.924, γ = (0.136 or −0.136).
0.06
0.065
0.07
0.075
0.08
0.085
0.09
0.095
0.13 0.135 0.14 0.145 0.15 0.155 0.16 0.165 0.17
β (b
ifurc
atio
n pa
ram
eter
)
γ (coupling strength)
Coexistence (in-phase & anti-phase)
Switching: phase pattern (in-phase & anti-phase)
Fig. 8 Synchronization types (N = 14). α = 7.0, δ = 50.0, ω =
1.924.
0.06
0.065
0.07
0.075
0.08
0.085
0.09
0.095
0.13 0.135 0.14 0.145 0.15 0.155 0.16 0.165 0.17 0.175
β (b
ifurc
atio
n pa
ram
eter
)
γ (coupling strength)
Coexistence (in-phase & 15-phase)
Switching (in-phase & 15-phase)
Fig. 9 Synchronization types (N = 15). α = 7.0, δ = 50.0, ω =
1.924.
4. Entropy
In order to capture the properties of cellular automata a
space-time entropy have been introduced [16]. The space-
time entropy of cellular automata with n cells is defined as
follows:
(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)
τ(xR15+xL15)+(xR1+xL1)(xR14+xL14)+(xR15+xL15)
(a) β=0.065.
(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)
τ(xR15+xL15)+(xR1+xL1)(xR14+xL14)+(xR15+xL15)
(b) β=0.075.
(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)
τ(xR15+xL15)+(xR1+xL1)(xR14+xL14)+(xR15+xL15)
(c) β=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)
τ(xR15+xL15)+(xR1+xL1)(xR14+xL14)+(xR15+xL15)
(d) β=0.095.
Fig. 10 Examples of pahse propagation by changing β from
0.065 to 0.095. α = 7.0, δ = 50.0, ω = 1.924, γ =
(0.132 or −0.132).
S = − 1
n
1
T
∑i
∑t
∑m
cti,m∑
mct
i,m
log2
cti,m∑
mct
i,m
(5)
wherect
i,m∑m
cti,m
denotes the probabilities and m counts
the two possible states ai ∈ {0, 1} of cell. This space-time
entropy is a measure of the order some cellular automata
configuration at a number of cell n and a fixed time T (=65).
We apply the space-time entropy to evaluation of com-
plex phase patterns obtained from a ring of chaotic circuits.
First, obtained complex phase patterns are changed to dis-
crete pattern data like cellular automata. In order to distin-
guish in-phase or anti-phase synchronization, the threshold
value th = 1.0 is introduced as shown in Fig. 12. When th is
larger than 1.0, the synchronization state is anti-phase.
The discrete data patterns obtained from Figs. 10-11 are
shown in Figs. 13-14. “¥ (1)” and “¤ (0)” are corresponding
to the in-phase and the anti-phase synchronization, respec-
tively. We caluculate the space-time entropy for these dis-
crete 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)(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)
τ(xR15+xL15)+(xR1+xL1)(xR14+xL14)+(xR15+xL15)
(a) γ=0.13.
(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)
τ(xR15+xL15)+(xR1+xL1)(xR14+xL14)+(xR15+xL15)
(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)
τ(xR15+xL15)+(xR1+xL1)(xR14+xL14)+(xR15+xL15)
(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)
τ(xR15+xL15)+(xR1+xL1)(xR14+xL14)+(xR15+xL15)
(d) γ=0.16.
Fig. 11 Examples of pahse propagation by changing γ from 0.13
to 0.16. α = 7.0, β=0.094, δ = 50.0, ω = 1.924.
For comparison, we also apply the space-time entropy to
complex phase patterns (Fig. 1) obtained from a ring of van
der Pol oscillators. The discrete data of phase patterns and
the value of the space-time entropy are shown in Fig. 15.
From these results, we confirm that the space-time entropy
of phase patterns obtained from chaotic circuits is higher
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 the
case of the chaotic circuits (Fig. 16), the every average en-
tropy show high value. While, in the case of the patterns
obtained from the van der Pol oscillators, each pattern con-
verge the different entropy value. We consider that it is one
possibility to classificate these complex patterns obtained the
ring of van der Pol oscillators and chaotic circuits by using
space-time entropy.
5. Conclusions
In this study, we have investigated synchronization phe-
nomena in chaotic circuits coupled by time varying resistors
as a ring. By computer simulations, first we confirmed the co-
existence of in-phase and anti-phase synchronizations. Next,
-2
-1
0
1
2
0 20000 40000 60000 80000 100000 120000
(XR1+XL1)+(XR2+XL2)
τ
th=1.0
Fig. 12 Threshold for distinguishing in-phase or anti-phase syn-
chronization.
(a) β=0.065, S=0.932.
(b) β=0.075, S=0.934.
(c) β=0.085, S=0.948.
(d) β=0.095, S=1.000.
Fig. 13 Discrete data obtained from Fig 10.
(a) γ=0.13, S=0.961.
(b) γ=0.14, S=0.974.
(c) γ=0.15, S=0.987
(d) γ=0.16, S=0.989.
Fig. 14 Discrete data obtained from Fig 11.
when the coupling strength decreases, we observed complex
phase pattern which cannot be observed in simple oscilla-
tory circuits coupled by resistors. Furthermore, we appled
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.
— 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.
0
0.2
0.4
0.6
0.8
1
1.2
0 200 400 600 800 1000 1200 1400 1600 1800 2000
S(t
)
t
(a) γ=0.13
(b) γ=0.14
(d) γ=0.16
(c) γ=0.15
Fig. 16 Time evolution of average entropy for chaotic circuits.
0
0.2
0.4
0.6
0.8
1
1.2
0 200 400 600 800 1000 1200 1400 1600 1800 2000
S(t
)
t
(a) Wave extinction
(b) Random pattern
(c) Wave propagation
(d) Clusterling
Fig. 17 Time evolution of average entropy for van der Pol oscil-
lators.
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