complex phase pattern in chaotic circuits with...

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, ††[email protected]

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, ††[email protected]

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.


x9+x10

x11+x12x12+x13x13+x14x14+x15

x15+x1

x10+x11

τ

(b) Random pattern.


x9+x10

x11+x12x12+x13x13+x14x14+x15

x15+x1

x10+x11

τ

(c) Wave propagation.


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:




(a) β=0.065.




(b) β=0.075.




(c) β=0.085.




(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 —




(a) γ=0.13.




(b) γ=0.14.




(c) γ=0.15.




(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.


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.


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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— 6 —

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