Switching Phase States of Chaotic Circuits
Coupled by Time-Varying Resistor
Yoko Uwate and Yoshifumi Nishio
Dept. of Electrical and Electronics Engineering, Tokushima University
2-1 Minami-Josanjima, Tokushima, Japan
Email: {uwate, nishio}@ee.tokushima-u.ac.jp
Abstract— In this study, two chaotic circuits coupled by atime-varying resistor are investigated. We assume that the timevarying resistor is realized by switching a positive and a negative
resistors alternately. By carrying out circuit experiments andcomputer simulations, we confirm that interesting switchingphase states between in-phase and anti-phase is observed.

Furthermore, the sojourn time of the in-phase and the antiphase are investigated. We confirm that the sojourn timedepends on the frequency and the strength of the time-varying
resistor.

I. I NTRODUCTION

II. C IRCUIT M ODEL

Synchronization phenomena in complex systems are very
interesting to describe various higher-dimensional nonlinear
phenomena in the field 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 some relations to information processing in the brain. We
consider that it is very important to investigate the synchronization phenomena of coupled oscillators 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 timevariation of the ambient temperature, an equation describing
an object moving in a space with some friction and anequation governing a circuit with a resistor whose temperature
coefficient is sensitive such as thermistor. However, there arefew discussions about coupled oscillators coupling by a timevarying resistor.
In our previous research, we have investigated synchronization phenomena in van der Pol oscillators coupled by atime-varying resistor. We realized the time-varying resistor byswitching a positive and a negative resistors periodically. Bychanging the duty ratio p, we confirmed that the characteristicsof the synchronization phenomena changed as follows. First,
for smaller p, the two coupled oscillators are synchronized
only in anti-phase. 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 coupledby time-varying resistor.
In this study, two chaotic circuits coupled by a time-varying
resistor are investigated. First, the coexistence of in-phase andanti-phase synchronization are observed. Next, we confirm
that interesting synchronization phenomena with switchingphase states between the two chaotic circuits can be observed
when the strength of coupling parameter R is decreased.
1-4244-0921-7/07 $25.00 © 2007 IEEE.

Figure 1 shows the circuit model, which is the chaotic
version of the circuit investigated in [12]. In the circuit, two
identical chaotic circuits are coupled by a time-varying resistor
whose characteristics are shown in Fig. 2 [13].
L1

L1

I2

I1

i1
v1

C

i2
L2-r

TVR

v2

L2

C-r

Fig. 1.

Circuit model (TVR is a Time-Varying Resistor).

R(t)2πRωtt0-RFig. 2.

2πpCharacteristics of the TVR.

First, the i−v characteristics of the diodes are approximatedby two-segment piecewise-linear functions as
vd (ik ) = 0.5(rd ik + E − |rd ik − E|).
By changing the variables and parameters,CC
Exk , ik =Eyk , vk = Ezk ,Ik =L1
L1
CL1, β=r,t = L1 Cτ , α =LL
21CC
1γ=R, δ = rd, ω=√ωt ,L1
L1
L1 C

209

(1)

(2)

v1

the normalized circuit equations are given as⎧dxk⎪⎪= β(xk + yk ) − zk ± γ(x1 + x2 )⎪⎪⎪⎨ dτdyk(3)= αβ(xk + yk ) − zk − f (yk )⎪dτ⎪⎪⎪⎪⎩ dzk = xk + yk(k = 1, 2)dτ
where the sign of the coupling term changes according to thevalue of the time-varying resistor. The normalized characteristics of the diodes are given asf (yk ) = 0.5 (δyk + 1 − |δyk − 1|).
(4)
III. S YNCHRONIZATION P HENOMENA
A. 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 chaoticcircuits are fixed as α = 7.0, β = 0.084, γ = 0.1 andω = 1.924.
z1z2

v2

I1

I2(a)

I1
I2(d)
Fig. 5. In-phase synchronization (circuit experimental 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.

v1v2x2x1

I2

I1x2x1(b)I1(c)(b)(a)(a)

I2I2(b)

I1(c)

I1(c)

I2x1(d)Fig. 6. Anti-phase synchronization (circuit experimental results). (a) 1stcircuit 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.

x2τ

(d)

Fig. 3. In-phase synchronization (computer simulation 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 ).

z1

z2

x2

x1(a)

x2(b)

B. Switching Phase Statesx1(c)x1x2(d)The one parameter bifurcation diagram of x 1 for in-phaseand anti-phase synchronization modes are shown in Fig. 7.
Figure 8 show the one parameter diagram of the phasedifference. We can confirm the coexistence of in-phase andanti-phase synchronizations for 0.055 < β < 0.090.

τFig. 4. Anti-phase synchronization (computer simulation results). (a) 1stcircuit 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 ).

We also confirm that the two coupled chaotic circuits aresynchronized in in-phase or at anti-phase in circuit experiments as shown in Figs. 5 and 6.

In this section, we investigate the synchronization phenomena when the strength of the coupling parameter γ isdecreased. We can confirm that the switching of the phasestates between the in-phase and the anti-phase is observed asshown in Fig. 9. These switching phenomena could not be
confirmed in the two van der Pol oscillators coupled by timevarying resistor. The chaotic circuits coupled by time-varying
resistor has possibility to generate complex phenomena.
Next, we pay our attention to the sojourn time of the inphase state and the anti-phase state. We carry out the 30
moving average of the phase difference between two coupledchaotic circuits to distinguish the in-phase state and the antiphase state more correctly. The simulated result of the moving
average of the phase difference is shown in Fig. 10. We

2101800-0.2

160

-0.6

phase difference [degree]-0.4
unstable-0.8-1-1.2
-1.4
-1.6
0.02

0.03

0.04

0.05

0.06

0.07

β0.08

0.09

0.1

(a) Bifurcation diagram of x1 for in-phase synchronization mode.
-0.5

140
120
100
80
60
40
20

-0.6

0-0.7
-0.8

0400

unstable

600

800

1000

1200

τ

1400

1600

1800

2000

Switching phase states (γ = 0.095).

Fig. 9.

-0.9-1200-1.1

180-1.2-1.3

160
0.03

0.04

0.05

0.06

0.07

β0.08

0.09

phase difference [degree]

-1.4
0.02

0.1(b) Bifurcation diagram of x1 for anti-phase synchronization mode.
Fig. 7. One parameter bifurcation diagrams for α = 7.0, γ = 0.1, ω =1.924. Horizontal axis: β.
180anti-phase

120
100
90
80
60

in-phase

40

140

20

120

00200

400

600

800

100

coexistence

80

Fig. 10.

60
40

00.02

in-phase
0.03

0.04

0.05

0.06

0.07

0.08

0.09

frequency

200.1βFig. 8.

160

160

140

140

120

120

100

1008060

1600

1800

2000

80

40200500

1000

sojourn time

define in-phase or anti-phase synchronization, by using the30 moving average of the phase difference. Namely, when
the phase difference is smaller or larger than 90 degrees, thesynchronization state of two chaotic circuits are determined toin-phase or anti-phase state.
The frequency distribution of the sojourn time of thesynchronization states is investigated. Figure 11 shows thesimulated results of the frequency distribution. From thesefigures, the frequency distribution of the in-phase and the antiphase is similar when the parameters of the chaotic circuits are
set as follows: α = 7.0, β = 0.084, γ = 0.095 and ω = 1.924.
Furthermore, the average sojourn time of the in-phase state
and the anti-phase state when the frequency ω of the timevarying resistor is changed. The simulated result is shown inFig. 12. The horizontal axis is frequency ω and the verticalaxis is average sojourn time. The average sojourn time of thein-phase state increase by increasing ω. On the other hand, theaverage sojourn time of the anti-phase state is almost constant.
From these results, we can see that the anti-phase state does
not depends on the frequency ω of the time-varying resistor.
When ω is smaller than 1.908, two chaotic circuits coupled bytime-varying resistor do not synchronous neither the in-phasestate nor the anti-phase state. And ω is larger than 1.930, the1400

60

2001200τMoving average of the phase states.

40

Phase difference.

1000

frequency

phase difference

160

antiphase

140

1500

2000

00500

1000

1500

2000

sojourn time(a) In-phase.
(b) Anti-phase.
Fig. 11. Frequency distribution of sojourn time.

only in-phase state can be occurred.
Some example of the switching synchronization state andthe frequency distribution when ω are set to 1.922 and 1.926
are shown in Figs. 13 and 14, respectively. In the case ofω = 1.922, the sojourn 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 is longer than the inphase state (Fig. 14).
Finally, we investigate synchronization in dependence on thecoupling strength γ. The average sojourn time of the in-phasestate and the anti-phase state is shown in Fig. 15. The averagesojourn time of the in-phase state increase by increasing γ.
On the other hand, the average sojourn time of the anti-phasestate is almost constant. From these results, we can see that theanti-phase state does not depends on γ as well as frequency
ω.
IV. C ONCLUSIONSIn this study, we have investigated phase differences in two
chaotic circuits coupled by a time-varying resistor when the211180100000

phase difference [degree]

average of sojourn time160

In-phase

10000

1000

Anti-phase

140
120
100
80
60
40
20
0

100

0

200

400

600

800

1000

1200

τ

1400

1600

1800

2000

(a) Moving average of switching synchronization state.
1.91

Fig. 12.

Average sojourn time in dependence on ω.

1.915

1.92

1.925

1.93

100

100

80

80

1.935

frequency

omega

180

frequency

10
1.905

60

40

60

40

phase difference [degree]

160
20

20

140
0

120

0

60
40
20
200

400

600

800

1000

1200

τ

1400

1600

1800

2000

350

350

300

300

250

250

200
150
100

200
150
100

50

50

500

2000

2500

00500

1000

1500

2000

2500

sojourn time1600

average of sojourn time400

frequency

frequency

40001500

18000(a)

01000

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

800500sojourn time

100

1000

1500

2000

00500

1000

1500

1200

In-phase

1000
800

Anti-phase

600
400

2000

sojourn timesojourn time

1400200(b)(c)
Fig. 13. Switching phase state for ω = 1.922. (a) Moving average ofswitching synchronization state. (b) Frequency distribution of sojourn time
(in-phase). (c) Frequency distribution of sojourn time (in-phase).

frequency of the time-varying resistor is changed. By carryingout computer calculations, we can confirmed that the synchronization phenomena of the two coupled chaotic circuits isdepend 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 the anti-phase has no influence
for changing ω and γ.
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0.088

0.09

0.092

0.094

0.096

0.098

0.1

RFig. 15.

Average sojourn time in dependence on γ.

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