The 15th IEEE International Workshop on Nonlinear Dynamics of Electronic Systems
NDES'07, Tokushima, Japan, July 23-26, 2007

Effect of Parameter Mismatch in Chaotic Circuits
Coupled by a 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, we investigate the effect of parametermismatch in chaotic circuits coupled by a time-varying resistor.
By computer simulations, we can observe very interesting synchronization phenomenon which means that the sojourn times ofin-phase and anti-phase states become longer when the parametermismatch increases.

I. I NTRODUCTION
Synchronization phenomena in complex systems are important 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, biology, engineering and so on [1][4]. Because many researchers suggest that synchronization
phenomena of coupled oscillators have some relations toinformation processing in the brain. We consider that it isvery important to investigate the synchronization phenomenaof coupled oscillators to realize a brain computer for the futureengineering 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 oscillators coupled by a time-varying
resistor.
In our previous study, two chaotic circuits coupled by atime-varying resistor were investigated [5]. First, the coexistence of in-phase and anti-phase synchronization was observed. Next, we have confirmed that interesting synchronization phenomena with switching phase states between the two
chaotic circuits can be observed when the strength of coupling
parameter R is decreased. Furthermore, the sojourn times of
the in-phase and the anti-phase have been investigated. We
confirmed that the sojourn time depends on the frequency andthe strength of the time-varying resistor.
In this study, we investigate the effect of parameter mismatch in chaotic circuits coupled by a time-varying resistor.
Namely, the two chaotic circuits with different characteristics
are coupled. We give the parameter mismatch of β to thetwo chaotic circuits. We consider that it is very important toobserve the difference of synchronization phenomena between
symmetric and asymmetric systems. By computer simulations,
we can observe very interesting synchronization phenomenon

which means that the sojourn times of in-phase and anti-phasestates become longer when the parameter mismatch increases.
II. C IRCUIT M ODEL
Figure 1 shows the circuit model, which is the chaotic
version of the circuit investigated in [6]. In the circuit, twoidentical chaotic circuits are coupled by a time-varying resistor
whose characteristics are shown in Fig. 2 [7].
L1

L1

I2

I1

i1
v1

Ci2
v2

L2

L2

C

TVR-r-rFig. 1.

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

R(t)

2πR

ωtt0-R
2πpFig. 2.

Characteristics 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,
rrC
CIk =Exk , ik =Eyk , vk = Ezk ,
L1
L1rp
C
L1t = L1 Cτ , α =, β=r,
L2
L1rr
1
C
C
γ=R, δ = rd, ω=√
ωt ,
L1
L1
L1 C- 176 -(1)(2)the normalized circuit equations are given asdxk= β(xk + yk ) − zk ± γ(x1 + x2 ) dτdyk= αβ(xk + yk ) − zk − f (yk )dτ dzk = xk + yk
(k = 1, 2)dτ(3)The one parameter bifurcation diagram of x1 for in-phaseand anti-phase synchronization modes are shown in Fig. 5.
Figure 6 shows the one parameter diagram of the phasedifference. We can confirm the coexistence of in-phase andanti-phase synchronizations for 0.055 < β < 0.090.
0-0.2

where the sign of the coupling term changes according to thevalue of the time-varying resistor. The normalized characteristics of the diodes are given as-0.4

f (yk ) = 0.5 (δyk + 1 − |δyk − 1|).

-1.4

-0.6

-1
-1.2

(4)

-1.6
0.02

III. S YNCHRONIZATION P HENOMENA-0.5

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
circuits are fixed as α = 7.0, β = 0.084, γ = 0.1, δ = 100,and ω = 1.924.

-0.7

z20.06

0.07

β0.08

0.09

0.1

-0.8
-0.9
-1

unstable-1.1
-1.2
-1.3
-1.4
0.02

0.03

0.04

0.05

0.06

0.07

β0.08

0.09

0.1

180x1

(b)

0.05

(b) Bifurcation diagram of x1 for anti-phase synchronization mode.
Fig. 5. One parameter bifurcation diagrams for α = 7.0, γ = 0.1, ω =1.924. Horizontal axis: β.

x2

(a)

0.04

-0.6x2x1

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

A. In-Phase and Anti-Phase Synchronizationz1

unstable-0.8

anti-phase160(c)
phase difference

140x1x2τ(d)

z2x2x1(a)x1(c)

x1x2(d)coexistence

8060

20

in-phase

00.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

βFig. 6.

Phase difference.

B. Switching Phase Statesx2(b)10040Fig. 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 ).

z1120

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 ).

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. 7. These switching phenomena could not beconfirmed in the two van der Pol oscillators coupled by timevarying resistor. The chaotic circuits coupled by time-varying
resistor have 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 method of the phase difference between two
coupled chaotic circuits to distinguish the in-phase state andthe anti-phase state more correctly. The simulated result of themoving average of the phase difference is shown in Fig. 8.

- 177 -180

constant. From these results, we can see that the anti-phasestate does not depend on the frequency ω of the time-varying
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 if in the case
ω is larger than 1.930, only the in-phase state can occur.

phase difference [degree]

160
140
120
1008060IV. E FFECT OF PARAMETER M ISMATCH

402000200400Fig. 7.

600

800

1000

τ

1200

1400

1600

1800

2000

Switching phase states (γ = 0.095).

We define in-phase or anti-phase synchronization, by using
the 30 moving average of the phase difference. Namely, when
the phase difference is smaller or larger than 90 degrees, thesynchronization state of the two chaotic circuits is determined
to in-phase or anti-phase state.
180

phase difference [degree]

160
140

antiphase

120
100
90
80
60in-phase

40
20
0

200

400

Fig. 8.

600

800

1000

τ

1200

1400

1600

1800

2000

1000

Moving average of the phase states.

900

average of sojourn time0

In this study, we investigate synchronization in dependence
on the parameter mismatch. Namely, the two chaotic circuitswith different characteristics are coupled. We give the parameter mismatch of β to the two chaotic circuits. The parameterβ of one chaotic circuit is fixed as β = 0.084. We change theparameter β2 of the other chaotic circuit from β2 = 0.084 toβ2 = 0.078. The average sojourn times of the in-phase state
and the anti-phase states are shown in Fig. 10. The averagesojourn times of the in-phase state increase by increasing
parameter mismatch. This synchronization phenomena is very
interesting. This is because, generally we assume that if theparameter mismatch becomes large, the average sojourn timedecreases. While, the average sojourn time of the anti-phasedecreases by increasing parameter mismatch.
Some examples of the switching synchronization state andthe frequency distribution when β2 are set to 0.083 and 0.080
are shown in Figs. 11 and 12, respectively. In the case of
β2 = 0.083, the sojourn times of both in-phase state and antiphase state are short (Fig. 11). In the case of β2 = 0.080, thesojourn times of in-phase state becomes longer and anti-phasestate becomes shorter (Fig. 12).

average of sojourn time

100000In-phase

10000

1000

Anti-phase

800

In-phase

700
600
500400300200Anti-phase

10000.077

Fig. 10.
101.905

1.91

1.915

1.92

1.925

1.93

0.079

0.08

0.081

0.082

0.083

0.084

0.085

No-mismatch

Average sojourn time in dependence on parameter mismatch β.

1.935

omega

Fig. 9.

0.078

β2100

V. OTHER PARAMETERS M ISMATCH

Average sojourn time in dependence on ω.

Furthermore, we investigate the average sojourn time of thein-phase state and the anti-phase state when the frequency ωof the time-varying resistor is changed. The simulated resultis shown in Fig. 9. The horizontal axis is frequency ω and thevertical axis is average sojourn time. The average sojourn timeof the in-phase state increases by increasing ω. On the otherhand, the average sojourn time of the anti-phase state is almost

Finally, we investigate synchronization in dependence onother parameters mismatch α and δ.
The parameter α of one chaotic circuit is fixed as α = 7.0.
We change the parameter α2 of the other chaotic circuit from
α2 = 6.5 to α2 = 8.0. The simulated result of average sojourntime is shown in Fig. 13. From this figure, we can confirm thatthe average sojourn time of in-phase and anti-phase is almost
similar when the parameter α2 is larger than 7.0. While, the- 178 -

180

3500

average of sojourn time160

phase difference [degree]

140
120
100
80
60
40
2000

2000

4000

6000

8000

τ

10000

0.4

0.35

0.35

probability

0.2
0.15

Fig. 13.

0.1

0.1
0.05
500

1000

1500

2000

02500

Anti-phase

6.6

6.8

77.2

7.4

7.6

7.8

8

Average sojourn time in dependence on parameter mismatch α.

0.20.1501000

No-mismatch

0.30.250.0501500

α211001000
0500

sojourn time

1000

1500

2000

average of sojourn time

probability

0.4In-phase

2000

06.4
0.45

0.3

2500

500

(a)
0.45

0.25

3000

2500

sojourn time

(b)

(c)

Fig. 11. Switching phase state for β2 = 0.083. (a) An example of switching
synchronization state. (b) Frequency distribution of sojourn time (in-phase).
(c) Frequency distribution of sojourn time (anti-phase).
180
160

900
800
700

In-phase

600

Anti-phase

500
400
300
200

phase difference [degree]

140

100
80

120
100

Fig. 14.

80

105

110

115

120

No-mismatch

Average sojourn time in dependence on parameter mismatch δ.

VI. C ONCLUSIONS
02000

4000

6000

8000

10000

In this study, we have investigated phase differences in two
chaotic circuits coupled by a time-varying resistor when theparameter of the two chaotic circuits are changed. By carryingout computer calculations, we have observed very interesting
synchronization phenomenon. Namely the sojourn times of inphase and anti-phase states become longer when the parametermismatch increases.

τ0.45

0.45

0.4

0.4

0.35

0.35

0.3

probability

probability

10040(a)0.250.2
0.15

0.3
0.25
0.2
0.15

0.1

0.1

0.05

0.05
500

95

600090δ220085

1000

1500

sojourn time

(b)

2000

2500

00500

1000

1500

2000

R EFERENCES

2500

sojourn time(c)Fig. 12. Switching phase state for β2 = 0.080. (a) An example of switching
synchronization state. (b) Frequency distribution of sojourn time (in-phase).
(c) Frequency distribution of sojourn time (anti-phase).

average sojourn time of both the in-phase and the anti-phaseincreases when the α2 is smaller than 7.0.
Next, we investigate synchronization in dependence on theparameter mismatch δ. The parameter δ of one chaotic circuit
is fixed as δ = 100. We change δ2 of the other chaotic circuitfrom δ2 = 80 to δ2 = 120. Figure 14 shows the simulated
results of average sojourn time. When δ2 is larger than 100,
the average sojourn time of the anti-phase is longer than
the in-phase state. While, when δ2 is smaller than 100, theaverage sojourn time of the in-phase increase and the antiphase decreases.

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