Noise-Induced Breakdown of Stochastic Resonant
Behavior of van der Pol Oscillators
Coupled by Time-Varying Resistor
Yoko Uwate

Yoshifumi Nishio

Ruedi Stoop

Institute of Neuroinfomatics,
University / ETH Zurich,
Winterthurerstrasse 190,
CH-8057 Zurich, Switzerland,
Email: yu001@ini.phys.ethz.ch

Dept. of Electrical and Electronic Eng.,
Tokushima University
2-1 Minami-Josanjima, Tokushima, Japan
Email: nishio@ee.tokushima-u.ac.jp

Institute of Neuroinfomatics,
University / ETH Zurich,
Winterthurerstrasse 190,
CH-8057 Zurich, Switzerland,
Email: ruedi@ini.phys.ethz.ch

Abstract— We explore the behavior of two van der Pol oscillators coupled by a stochastically time-varying resistor. We observe
switching between in-phase and anti-phase synchronization andanalyze the statistics of the switching, with and without additional
noise. We find stochastic resonant behavior of synchronizationin both regimes. Correlating the two noise sources quickly
destroys the stochastic resonance phenomenon in the anti-phasesynchronization regime. In the in-phase regime, noise correlation
first contributes to the synchronization before removing theresonance by taking over the synchronization by its own means.

may be of technical importance as it may open avenues fornovel strategies of parallel information processing.
2π

0.1
γ

0
2πλ
-0.1
ωτ

Fig. 1.

Characteristics of the TVR.

I. I NTRODUCTION
1

Coupled oscillatory circuits provide simple models for describing high-dimensional nonlinear phenomena occurring in
our everyday world. Synchronization, in particular, is one of
the most important features that can be described and exploredwith the help of oscillators, because, upon their coupling,strongly correlated rhythms among the oscillators emerge,
called synchronized states. Synchronization phenomena havebeen extensively reported in physical [1]-[4], biological [5],[6]and electrical [7],[8] systems.
Synchronization in networks of chaotic cells with stochastically switched couplings (’blinking networks’) have been
first reported in the context of the coupling among smallworld networks [9], where the authors sketched some potential
applications of this model for explorations in neuroscience.
In a previous research, we applied the blinking coupling
concept for the coupling among van der Pol oscillators. The
oscillators were coupled by means of a time-varying resistor,
implemented by periodically switching between a positive anda negative resistor [10], as shown in Fig. 1. Using computer
simulations, we confirmed the coexistence [11] of the in-phase(see. Fig. 2) and the anti-phase (see. Fig. 3) synchronizations,which depend upon the provided initial conditions [12]. This
coexistence phenomenon of synchronization is interesting,since in resistor-coupled systems of two oscillators, even if
multiple synchronization states exist, normally only one synchronization state becomes stable. The observed phenomenon

978-1-4244-3828-0/09/$25.00 ©2009 IEEE

0.5

x1 0

-0.5
-1
1
0.5

x2 0
-0.5
-1

2000

4000

Fig. 2.

6000

8000

10000

12000

τ

In-phase synchronization.

1
0.5

x1 0

-0.5
-1
1
0.5

x2 0
-0.5
-1

2000

4000

Fig. 3.

6000

8000

10000

12000

τ

Anti-phase synchronization.

In this contribution, we analyze the behavior obtained froma new type of time-varying resistor, where the state of thetime-varying resistor is determined from an event probability,
a setting that we propose to call “Stochastically Time-Varying
Resistor (STVR) coupling”. In this setting, switching between

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the in- phase and the anti-phase synchronized solutions isobserved, similar to findings related with other systems [13]. In
order to obtain a more detailed picture of the phenomenon, thesojourn time into the in-phase vs. the anti-phase synchronized
regime is investigated, as a function of the parameters of
STVR. In order to mimic the real world behavior, we add
noise to the capacitors involved in each of the circuits. We findthat the in-phase and anti-phase regime sojourn times depend
inversely upon the noise strength, when identical noise ξ1 = ξ2
are used for the two circuits. This discrepancy is graduallydestroyed if the noise in the the two oscillators is made evermore independent.
II. C IRCUIT M ODEL
We study a circuit of two (up to later added noise) identicalvan der Pol oscillators, coupled by a stochastically timevarying resistor (STVR), see Fig. 4. The characteristics of
the STVR is shown in Fig. 5. In the time interval [kπ/ωt ,(k + 1)π/ωt ], R(t) is piecewise constant, taking one of thevalues {r, −r}. The event probabilities p+ and p− for tasking
the values r or −r, respectively, satisfy the equation
p+ + p− = 1.0.

L

I1

I2

(1)

L
iR2

iR1
v1

STVR

C

OCS-2

OCS-1
Fig. 4.

v2

C

Circuit Model (STVR is a Stochastically Time-Varying Resistor).

R(t)

r
ω tt

0

By a change of variables and parameters



√
g1
g1
C
xk , iRk =vk =yk , t = LCτ ,
g3
3g3 L


L
C
1
, γ=r
, ω= √ωt ,ε = g1C
L
LC
The normalized circuit equations obtain their form⎧dxk⎪
= εxk (1 − xk 2 ) − yk
⎪
⎪
⎪
dτ
⎨
2

⎪
dyk
⎪
⎪
=
x
±
γ(τ
)
yj
⎪
k
⎩ dτ

(3)
(k = 1, 2 .)j=1

In Eq. 3, ε denotes to the nonlinearity of van der Pol oscillatorand γ(τ ) is the characteristics of STVR.
III. S YNCHRONIZATION P HENOMENAFor the following computer simulations, we fix the circuit
system parameters at ε = 2.0, γ = 0.1 and ω = 1.5.
For observing the in-phase and the anti-phase synchronizedsolutions as shown in Figs. 6, 7, the value of the couplingstrength γ is the least critical, whereas a strong oscillator
nonlinearity ε is needed and ω should be chosen, under theother given parameter values, from the interval (1.44, 1.58).
Figure 6 shows the discrete data x1 − x2 between two vander Pol oscillators at τ = kπ/ω. From this figure we cansee that the switching synchronization between the in-phaseand the anti-phase state is observed. The simulation resultof switching phase difference is shown in Fig. 7. We obtain
the phase difference when the solution of the 1st oscillator
crosses the Poincaré map defined as x1 < 0, y1 = 0. It is
worthwhile mentioning that for a periodically switched TVR,
the switching between the two regimes can generally not be
observed. For characterizing the switching by means of thesojourn time, we calculated a moving average of 30 steps ofthe phase difference, which was sufficient for distinguishing
between the in-phase state and the anti-phase state as shown in
Fig. 8. We attribute the behavior to the in-phase or to the antiphase synchronized state, respectively, if the averaged phasedifference is smaller or larger than 90 degrees. The resultof the statistics of the distribution among in-phase and antiphase sojourn time is shown in Fig. 9 for switching probabilityp+ = p− = 0.5. As can be expected, the difference between
distributions associated with the two regimes increases if theswitching probabilities are changed, see Figs. 10, 11.

-r

x1-x2
0

π

2π

4
3
2
1
0
-1
-2
-3
-4

3π 4π 5π 6π 7π 8π 9π ...

Fig. 5.

Characteristics of STVR.

The van der Pol oscillator consists of an active element(nonlinear resistor) characterized by a simple symmetric cubic
nonlinearity of the form
iRk = −g1 vk + g3 vk3 ,

g1 , g3 > 0,

k = 1, 2.

(2)

0

500

1000

1500

2000

2500

3000

k
Fig. 6. Difference x1 − x2 of the two coupled van der Pol oscillators with
STVR with parameter p+ = 0.5, p− = 0.5.

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160

phase difference [degree]

140

400

400

350

350

300

300

frequency

frequency

180

250
200
150
100

120

0

200
150
100

50

100

250

50

0

200

400

600

800

1000

1200

0

1400

0

200

400

sojourn time

80

600

800

1000

1200

1400

sojourn time(a) In-phase.

(b) Anti-phase.

Fig. 11. Frequency distribution of sojourn time for (p+ = 0.52, p− = 0.48).

60
40
20
0
200

400

600

800

1000

1200

1400

1600

1800

2000
n

Fig. 7. Switching phase states (horizontal axis: Poincaré point n, vertical axis:
phase difference between two van der Pol oscillators, p+ = 0.5, p− = 0.5).

180
160

phase difference [degree]

140

Anti-phase

120
100
90
80

In-phase

60

The collected results obtained by changing the STVR probability p+ are shown in Fig. 12, where we use the averaged
sojourn time to characterize the preference for one of the two
regimes. When the probability of STVR p+ is 0.5, we haveabout equal preference. By increasing p+ , the preference forthe anti-phase regime increases and decreases for the in-phaseregime. The switching phase state phenomena can be observedfor 0.42 ≤ p+ ≤ 0.58. If p+ > 0.58, in-phase synchronization
breaks down and only the anti-phase synchronization can beobserved. For the choice p+ < 0.42, in contrast, only the inphase synchronization can be observed. These findings canbe collected by displaying the percentage of the in-phasestate in dependence p+ . Based on simulations comprising
500,000 samples for each data point, we obtained the result
as displayed in Fig. 13, showing a striking resemblance withthe sigmoid function centered around p+ = 0.5.

40
10000

0
200

600

800

1000

1200

400

400

350

350

300

300

250
200
150
100

2000
n

200

In-phase stateAnti-phase state
1000

100

10
0.42

0.44

0.46

0.48

150

0.5

0.52

0.54

0.56

0.58

p+

Fig. 12.

50

0

200

400

600

800

1000

1200

0

1400

0

200

400

1200

300

250
200
150

100

200
150
100

50

50

600

800

sojourn time(a) In-phase.

1000

1200

1400

90

250

100

400

Average of sojourn time in dependence on p+ .

1400

ratio of in-phase state [%]

350

300

frequency

350

Fig. 10.

1000(b) Anti-phase.

400

200

800

Frequency distribution of sojourn time (p+ = 0.5, p− = 0.5).

400

0

600

sojourn time(a) In-phase.
Fig. 9.

frequency

1800

250

sojourn time

0

1600

100

50
0

1400

Moving average of the phase states (p+ = 0.5, p− = 0.5).

frequency

frequency

Fig. 8.

400

average of sojourn time

20

0

80
70
60
50
40
30
20
10

0

200

400

600

800

1000

1200

1400

sojourn time

0
0.42

0.44

0.46

0.48

0.5

0.52

0.54

0.56

p+

(b) Anti-phase.

Frequency distribution of sojourn time (p+ = 0.48, p− = 0.52).

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Fig. 13.

Ratio of in-phase synchronization state.

0.58

130

IV. I NFLUENCE OF N OISE

100%(ξ1=ξ2, fully shared)
99%
95%
90%
80%
60%
40%
20%
0% (fully independent)

average of sojourn time120

In realistic systems, noise is omnipresent. Therefore, it isimportant to study the effect of noise on our findings. Whenextended by noise contributions to the voltage of the twocapacitances, the normalized circuit equations with noise aredescribed by⎧dxk⎪= εxk (1 − xk 2 ) − yk + Dξk
⎪
⎪
⎪
⎨ dτ
⎪
dyk
⎪
⎪
⎪
⎩ dτ = xk ± γ(τ )

2

j=1

average of sojourn time

140
120
100
100%(ξ1=ξ2, fully shared)
99%
95%
90%
80%
60%
40%
20%
0% (fully independent)

Fig. 14.

1

2

70

Fig. 15.

In-phase state

0

80

50

(k = 1, 2 )

yj

160

40

90

Anti-phase state

60

180

60

100

(4)

where ξk denotes the Gaussian white noise and D is thenoise intensity. The two complementary behaviors are now
investigated at the switching probability p+ = 0.5, where
we are also interested in the influence that a correlation
between the two noise contributions would have. As a functionof the percentage of shared vs. independently chosen noise
contributions, we obtain the results as displayed in Figs.14,
15.
For independent noise contributions, we observe a stochastic
resonance phenomenon for in-phase and anti-phase synchronization regimes. Namely, the length of the average sojourntime becomes longer by adding weak noise. This happens at
considerable, but not dominant, noise strengths, in the case thatthe noise contributions at the capacitances are independent. As
a function of the percentage of noise contributions randomly
chosen vs. shared among the capacitances, the stochastic
resonance phenomenon increases in the in-phase regime. The
dependence clearly indicates that the sharing acts as if an additional connection among the two oscillators were established.
For fully correlated (shared) noise sources, upon increasing
the strength of the noise, this ’connection’ first supports andthen overwhelms the stochastic resonance phenomenon. In theanti-phase synchronization regime, the effect is contrary: theconnection naturally works against the stochastic resonancephenomenon.

80

110

D

3

4

5

Average sojourn time of in-phase by changing noise intensity.

0

1

2

D

3

4

5

Average sojourn time of anti-phase by changing noise intensity.

V. C ONCLUSION
In this study, we have investigated the switching phasestate of two van der Pol oscillators coupled by STVR. We
have analyzed the statistics of the switching, with and without
additional noise. The stochastic resonant behavior of synchronization was observed when the strength of the adding noise
is small. We have confirmed that correlating the two noise
sources quickly destroys the stochastic resonance phenomenon
in the anti-phase synchronization regime.
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