Chaotic Switching of Phase States in Time-Varying
Coupled Circuits
Yoko Uwate and Yoshifumi Nishio
Dept. of Electrical and Electronic Engineering,
Tokushima University, 2-1 Minami-Josanjima, Tokushima, Japan
Phone: +81-88-656-7470, Fax: +81-88-656-7471,
Email: {uwate, nishio}@ee.tokushima-u.ac.jp
1. Introduction
Synchronization phenomena in complex systems are very interesting to describe various
higher-dimensional nonlinear phenomena in theﬁeld of natural science. Studies on synchronization phenomena of coupled oscillators are extensively carried out in various ﬁelds, physics [1]-[4],biology [5], [6] engineering [7]-[11] and so on. Because many researchers suggest that synchronization phenomena of coupled oscillators have somerelations to information processing in the brain.
We consider that it is very important to investigate the synchronization phenomena of coupledoscillators to realize a brain computer for the future engineering application.
On the other hand, there are some systems
whose dissipation 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 equation governing a circuit with a resistor whose temperature coeﬃcient is sensitive such as thermistor.
However, there are few discussions about coupledoscillators coupling by a time-varying resistor.
In our previous research, we have investigatedsynchronization phenomena in van der Pol oscillators coupled by a time-varying resistor [12]. We
realized the time-varying resistor by switching a
positive and a negative resistors periodically. Bychanging the duty ratio p, we conﬁrmed that thecharacteristics of the synchronization phenomenachanged as follows. First, for smaller p, the two
coupled oscillators are synchronized only in antiphase. Second, for intermediate p, the coexistenceof the in-phase and the anti-phase synchronizations can be observed. Finally, for larger p, only

the in-phase synchronization can be conﬁrmed.
We consider what complex synchronization phenomena can be observed, when chaotic circuits
are coupled by time-varying resistor.
In this study, two chaotic circuits coupled bya time-varying resistor are investigated. First,
the coexistence of in-phase and anti-phase synchronization are observed. Next, we conﬁrm
that interesting synchronization phenomena withswitching phase states between the two chaotic
circuits can be observed when the strength of coupling parameter R is decreased. Furthermore, thesojourn time of the in-phase and the anti-phase
are investigated. We conﬁrm that the sojourntime depends on the frequency and the strengthof the time-varying resistor.
2. Circuit Model
Figure 1 shows the circuit model, which is the
chaotic version of the circuit investigated in [12].
In the circuit, two identical chaotic circuits arecoupled by a time-varying resistor whose characteristics are shown in Fig. 2 [13]. In this study,
we consider the case that the function representing the variation of the TVR is the square wave
with the angular frequency ωt and the duty ratiop.
First, the i − v characteristics of the diodes areapproximated by two-segment piecewise-linear
functions as
vd (ik ) = 0.5(rd ik + E − |rd ik − E|).

(1)

By changing the variables and parameters,


Ik =

C
Exk ,
L1



ik =

C
Eyk ,
L1

vk = Ezk ,

L1

L1

I2

I1

circuits are ﬁxed as α = 7.0, β = 0.084, γ = 0.1,
ω = 1.924 and p = 0.5.

i1
v1

C

i2
v2

L2

L2

C

z1

TVR

-r

z2

x2

-r

x1

x2

(a)

Figure 1: Circuit model (TVR is a Time-Varying Resis-

x1

(b)

(c)

tor).

x1
R(t)

2π

x2

R

ωtt

0

Figure 3: In-phase synchronization (computer simulation-R

results). (a) 1st circuit attractor (x1 vs z1 ). (b) 2nd circuit attractor (x2 vs z2 ). (c) Phase diﬀerence (x1 vs x2 ).
(d) Time wave form (τ vs x1 and x2 ).

2πpFigure 2: Characteristics of the TVR.


τ

(d)

L1
t = L1 Cτ , α =
,
L

 2
C
C
, δ = rd
,
γ=R
L1
L1



C
, (2)
L1
1
ωt ,
ω=√
L1 C
β=r

z1

(a)

⎧
dxk
⎪
⎪
= β(xk + yk ) − zk ± γ(x1 + x2 )
⎪
⎪
⎪
⎪
⎨ dτ

x2
(b)

x1
(c)

x1

(3)
x2
(d)

where the sign of the coupling term changes according to the value of the time-varying resistor.
The normalized characteristics of the diodes are
given asf (yk ) = 0.5 (δyk + 1 − |δyk − 1|).

x2

x1

the normalized circuit equations are given asdyk= αβ(xk + yk ) − zk − f (yk )
⎪
dτ
⎪
⎪
⎪
⎪
dzk
⎪
⎩
(k = 1, 2)= xk + yk
dτ

z2

τ

Figure 4: Anti-phase synchronization (computer simula-tion results). (a) 1st circuit attractor (x1 vs z1 ). (b) 2nd
circuit attractor (x2 vs z2 ). (c) Phase diﬀerence (x1 vs x2 ).
(d) Time wave form (τ vs x1 and x2 ).

(4)

3. Synchronization Phenomena
3.1. In-Phase and Anti-Phase Synchronization
We observed that the two coupled oscillators
are synchronized in in-phase and anti-phase as
shown in Figs. 3 and 4. These two synchronization states can be obtained by giving diﬀerent initial conditions. The parameters of the chaotic

We also conﬁrm that the two coupled chaotic
circuits are synchronized in in-phase or at antiphase in circuit experiments as shown in Figs. 5
and 6.
The one parameter bifurcation diagram of
x1 for in-phase and anti-phase synchronization
modes are shown in Fig. 7. Figure 8 show the one
parameter diagram of the phase diﬀerence. We
can conﬁrm the coexistence of in-phase and antiphase synchronizations for 0.055 < β < 0.090.

v1

v2

0

I2

-0.2
-0.4
-0.6

unstable

-0.8

I1

I2

(a)

-1

I1

-1.2

(c)

(b)

-1.4
-1.6
0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

β

I1

(a) Bifurcation diagram of x1 for in-phasesynchronization mode.

I2
-0.5
-0.6

(d)

-0.7
-0.8

Figure 5: In-phase synchronization (circuit experimental

-0.9
-1

results). (a) 1st circuit attractor (I1 vs v1 ). (b) 2nd circuit attractor (I2 vs v2 ). (c) Phase diﬀerence (I1 vs I2 ).
(d) Time wave form (t vs I1 and I2 ). L1 = 300mH, L2 =10mH, C = 33nF, r = 700Ω and R = 100Ω.

v1

v2

unstable

-1.1
-1.2
-1.3
-1.4
0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

β

(b) Bifurcation diagram of x1 for anti-phaseI2

synchronization mode.

Figure 7: One parameter bifurcation diagrams for α =
I1
(a)

I2
(b)

I1

7.0, γ = 0.1, ω = 1.924. Horizontal axis: β.

(c)

180

anti-phase

160

I1
I2
(d)

Figure 6: Anti-phase synchronization (circuit experimen-

tal results). (a) 1st circuit attractor (I1 vs v1 ). (b) 2nd
circuit attractor (I2 vs v2 ). (c) Phase diﬀerence (I1 vs I2 ).
(d) Time wave form (t vs I1 and I2 ). L1 = 300mH, L2 =10mH, C = 33nF, r = 700Ω and R = 100Ω.

3.2. Switching Phase States
In this section, we investigate the synchronization phenomena when the strength of the coupling
parameter γ is decreased. We can conﬁrm thatthe switching of the phase states between the inphase and the anti-phase is observed as shown in
Fig. 9. These switching phenomena could not be
conﬁrmed in the two van der Pol oscillators coupled by time-varying resistor. The chaotic circuits
coupled by time-varying resistor has possibility togenerate complex phenomena.
Next, we pay our attention to the sojourn timeof the in-phase state and the anti-phase state. We
carry out the 30 moving average of the phase difference between two coupled chaotic circuits todistinguish the in-phase state and the anti-phase

phase difference

140
120
100

coexistence

80
60
40
20

in-phase

0
0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

β

Figure 8: Phase diﬀerence.
state more correctly. The simulated result of themoving average of the phase diﬀerence is shownin Fig. 10. We deﬁne in-phase or anti-phase synchronization, by using the 30 moving average ofthe phase diﬀerence. Namely, when the phase difference is smaller or larger than 90 degrees, thesynchronization state of two chaotic circuits are
determined to in-phase or anti-phase state.
The frequency distribution of the sojourn timeof the synchronization states is investigated. Figure 11 shows the simulated results of the frequency distribution. From these ﬁgures, thefrequency distribution of the in-phase and theanti-phase is similar when the parameters of thechaotic circuits are set as follows: α = 7.0,
β = 0.084, γ = 0.095 and ω = 1.924.

frequency

phase difference [degree]

160
140
120
100

160

160

140

140

120

120

100

frequency

180

80

80
60

40

40

20
0

80

100

60

20

0

500

1000

1500

2000

0

0

500

sojourn time

60

1000

1500

2000

sojourn time(a) In-phase.

(b) Anti-phase.

Figure 11: Frequency distribution of sojourn time.

40
20

100000
0

200

400

600

800

1000

τ

1200

1400

1600

1800

2000

Figure 9: Switching phase states (γ = 0.095).
180

phase difference [degree]

160
140

antiphase120

average of sojourn time0In-phase

10000

1000

Anti-phase

100

100
90
80
60

10
1.905

in-phase

1.915

1.92

1.925

1.93

1.935

Figure 12: Average sojourn time in dependence on ω.

20
0

1.91

omega

40

0

200

400

600

800

1000

1200

1400

1600

1800

2000

τ

Figure 10: Moving average of the phase states.
Furthermore, the average sojourn time of thein-phase state and the anti-phase state when
the frequency ω of the time-varying resistor ischanged. The simulated result is shown in Fig. 12.
The horizontal axis is frequency ω and the verticalaxis is average sojourn time. The average sojourntime of the in-phase state increase by increasing
ω. On the other hand, the average sojourn timeof the anti-phase state is almost constant. Fromthese results, we can see that the anti-phase state
does not depends on the frequency ω of the timevarying resistor. When ω is smaller than 1.908,
two chaotic circuits coupled by time-varying resistor do not synchronous neither the in-phase state
nor the anti-phase state. And ω is larger than
1.930, the only in-phase state can be occurred.
Some example of the switching synchronization
state and the frequency distribution when ω are
set to 1.922 and 1.926 are shown in Figs. 13 and14, respectively. In the case of ω = 1.922, thesojourn time of the in-phase state is longer than
the anti-phase state (Fig. 13). In the case of ω =1.926, the sojourn time of the anti-phase state islonger than the in-phase state (Fig. 14).

Next, we investigate synchronization in dependence on the coupling strength γ. The average sojourn time of the in-phase state and the anti-phasestate is shown in Fig. 15. The average sojourntime of the in-phase state increase by increasing
γ. On the other hand, the average sojourn timeof the anti-phase state is almost constant. Fromthese results, we can see that the anti-phase state
does not depends on γ as well as frequency ω.
Finally, we investigate synchronization in dependence on the parameter mismatch. The parameter β of the other side chaotic circuit is ﬁxed
as β = 0.084. We change the parameter β2 ofthe one side chaotic circuit from β2 = 0.085 toβ2 = 0.078. The average sojourn time of the inphase state and the anti-phase state is shown in
Fig. 16. The average sojourn time of the both inphase state and anti-phase increase by increasing
parameter mismatch.
Some example of the switching synchronization
state and the frequency distribution when β2 are
set to 0.083 and 0.080 are shown in Figs. 17 and18, respectively. In the case of β2 = 0.083, thesojourn time of the both in-phase state and antiphase sate are short (Fig. 17). In the case of ω =0.080, the sojourn time of the both in-phase state
and anti-phase sate become to long (Fig. 18).

1800
180

1600

average of sojourn timephase difference [degree]

160
140
120
100
80
60
40

1400
1200

In-phase

1000
800

Anti-phase

600
400

20

200
0

0

200

400

600

800

1000

1200

1400

1600

1800

2000

0
0.086

τ

0.088

0.09

0.092

400

350

350

300

300

250

250

150

100

100

50

500

1000

1500

0.1

4000
3500

50

0

0.098

200

150

0

0.096

Figure 15: Average sojourn time in dependence on γ.

0

2000

0

500

1000

1500

2000

sojourn timesojourn time

(b)

(c)

Figure 13: Switching phase state for ω = 1.922. (a) Moving average of switching synchronization state. (b) Frequency distribution of sojourn time (in-phase). (c) Frequency distribution of sojourn time (in-phase).

average of sojourn time

frequency

frequency

400

200

0.094

R

(a)

3000
2500

In-phase
2000
1500
1000
500
0
0.077

Anti-phase

0.078

0.079

0.08

0.081

0.082

0.083

0.084

0.085

0.086

β2
180

Figure 16: Average sojourn time in dependence on parameter mismatch.

phase difference [degree]

160
140
120
100

4. Conclusions

80
60
40
20
0

0

200

400

600

800

1000

1200

1400

1600

1800

2000

1500

2000

τ

100

100

80

80

frequency

frequency

(a)

60

40

20

0

60

40

20

0

500

1000

1500

sojourn time

(b)

2000

2500

0

0

500

1000

2500

sojourn time

In this study, we have investigated phase diﬀerences in two chaotic circuits coupled by a timevarying resistor when the frequency of the timevarying resistor is changed. By carrying out computer calculations, we can conﬁrmed that thesynchronization phenomena of the two coupledchaotic circuits is depend on the frequency of
the time-varying resistor. Further, we investigated the sojourn time of the synchronization
state when the frequency ω and coupling strength
γ are changed. The average sojourn time of theanti-phase has no inﬂuence for changing ω and γ.

(c)Figure 14: Switching phase state for ω = 1.926. (a) Moving average of switching synchronization state. (b) Frequency distribution of sojourn time (in-phase). (c) Frequency distribution of sojourn time (in-phase).

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0

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60

20

0

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1000

τ

1500

2000

0

2500

0

500

1000

(b)

τ

1500

2000

2500

(c)

Figure 17: Switching phase state for β2 = 0.083.
(a) Moving average of switching synchronization state.
(b) Frequency distribution of sojourn time (in-phase).
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160
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120

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(a)
100

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0

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180

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1000

(b)

τ

1500

2000

2500

0

0

500

1000

τ

1500

2000

2500

(c)

Figure 18: Switching phase state for β2 = 0.080.
(a) Moving average of switching synchronization state.
(b) Frequency distribution of sojourn time (in-phase).
(c) Frequency distribution of sojourn time (in-phase).

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