SWITCHING PHASE STATES IN TWO VAN DER POL OSCILLATORSCOUPLED BY STOCHASTICALLY TIME-VARYING RESISTOR
Yoko Uwate† , Yoshifumi Nishio‡ and Ruedi Stoop††

Institute of Neuroinfomatics, University / ETH Zurich, Winterthurerstrasse 190,
CH-8057 Zurich, Switzerland,
Email: {yu001, ruedi}@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

Abstract— We explore the synchronization andswitching behavior of a system of two identical vander Pol oscillators coupled by a stochastically timevarying resistor. Triggered by the time-varying resistor, the system of oscillators switches between
synchronized and anti-synchronized behavior. We
find that the preference of the synchronized/antisynchronized state is determined by the ratio of theprobabilities of the two resistor states. The length ofthe phases of maintained resistor states, however, has
not a decisive role in the process, since the switchingis triggered, on average, with very short latency.
I. I NTRODUCTION
Coupled oscillatory circuits provide simple models
for describing high-dimensional nonlinear phenomena occurring in the everyday world. Synchronization, in particular, is one of the most important features that can be described and explored with the helpof oscillators, because, upon their coupling, strongly
correlated rhythms among the oscillators emerge,
called synchronized states. Synchronization phenomena have been extensively reported in physical [1]-[4],biological [5],[6] and electrical [7],[8] systems.
Synchronization in networks of chaotic cells withstochastically switched couplings (’blinking networks’) appear have been first reported in the context of the coupling among small-world networks [9].
There, 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 ofa time-varying resistor, implemented by periodicallyswitching between a positive and a negative resistor.
Using computer simulations, we confirmed the coexistence of the in-phase and the anti-phase synchro-nizations, which are realized according to the provided initial conditions [12]. This coexistence phenomenon of synchronization is interesting, since inresistor-coupled systems of two oscillators, even if
multiple synchronization states exist, normally only
one synchronization state becomes stable. The observed phenomenon may be of technical importance,
as it may open avenues for novel strategies of parallel information processing. We also have proposeda new type of time-varying resistor, where the stateof the time-varying resistor is determined from an
event probability, a setting that we call “Stochastically Time-Varying Resistor (STVR) coupling”. In
this setting, switching between the in-phase and theanti-phase synchronized solutions has been observed,
similar to findings related with other systems [13].
In this contribution, we investigate how the latencyof the system switch depends on the switching properties of the STVR. We observe that the ratio of theprobabilities between the two states of the resister
governs the switching of the system.
II. C IRCUIT M ODEL [14]We study the circuit of two identical van der Pol
oscillators, coupled by a stochastically time-varying
resistor (STVR), see Fig. 1. The characteristics ofthe STVR are shown in Fig. 2. In the time interval[kπ/ωt , (k + 1)π/ωt ], R(t) is piecewise constant,taking one of the values {r, −r}. The event probabilities p+ and p− for taking the values r or −r (theSTVR states), respectively, satisfy the equation
p+ + p− = 1.

(1)

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

g1 , g3 > 0,

k = 1, 2. (2)

50

LL

I2

I1

iR2iR1v1
C

STVRv2COCS-2

OCS-1

Fig. 1. Circuit Model (STVR is a Stochastically TimeVarying Resistor).
R(t)rsimulation result of the differences of the phase of
the two oscillators of the switching system is shownin Fig. 4, where the phase difference is measured at
the solutions crossing of the Poincaré section x1 <0, y1 = 0. It is worthwhile mentioning that for a periodically switched TVR, the switching between thetwo regimes can generally not be observed. For characterizing the switching by means of the sojourn time,
we calculated a moving average of 30 steps of thephase difference, which was sufficient for distinguishing between the in-phase state and the anti-phase state(Fig. 5). We attribute the behavior to the in-phase or to
the anti-phase synchronized state, respectively, if theaveraged phase difference is smaller or larger than 90
degrees. Examples, for the probabilities p+ = 0.48
and p+ = 0.52, respectively, are shown in Fig. 6Anti-phase1ω tt

00.5x1,x2-rIn-phase

0-0.5-10π2π

3π 4π 5π 6π 7π 8π 9π ...

0.1

Fig. 2. Characteristics of STVR.

STVR

0-0.1

By means of a change of variables and parametersg1xk , iRk =g3

ε = g1sr

L, γ=r
C

g1
3g3ss

√
Cyk , t = LCτ ,L

70000

75000

80000

85000

90000

95000

τ

Fig. 3. Example of the switching phenomenon (p+ = 0.5,p− = 0.5).
180

1C, ω=√ωt ,L
LC

160
140

we obtain the the circuit equations
dxk
= εxk (1 − xk 2 ) − ykdτ
2Xdyk=x±γ(τ)yjk
 dτ(3)(k = 1, 2).

j=1

In Eq. 3, ε embodies to the nonlinearity of van der Poloscillator and γ(τ ) is the characteristics of the STVR.
III. S YNCHRONIZATION P HENOMENAFor the following computer simulations, we fix thecircuit system parameters at ε = 2.0, γ = 0.1 andω = 1.5. For observing the coexistence between
the in-phase and the anti-phase synchronized solutions, the value of the coupling strength γ is the least
critical, whereas a strong oscillator nonlinearity ε isneeded and ω should be chosen -at the chosen parameter values- from the interval (1.44, 1.58). The

phase difference [degree]

vk =

r

65000

120100806040200200400600

8001000

1200

1400

1600

1800

2000
nFig. 4. Switching phase states (horizontal axis: Poincaré
point n; p+ = 0.5, p− = 0.5).

The collected results obtained in function of theprobability p+ are shown in Fig. 7, where we use theaveraged sojourn time to characterize the preference
for one of the two regimes. When the probabilityof the STVR is p+ =0.5, we have about equal preference. By increasing p+ , the preference for the antiphase regime increases and decreases for the in-phaseregime. The switching phase state phenomenon canbe observed for 0.42 ≤ p+ ≤ 0.58. If p+ > 0.58, the51

180
160

phase difference [degree]

140

Anti-phase

120
100
90
80
60

In-phase

40
20
0200

400

600

800

1000

1200

1400

1600

1800

2000
n

180

180

160

160

140

140
phase difference [degree]

120
100
80
60
40

350

100

ave: 624

80

300
60
40

200120250

200500

1000
τ

1500

2000

00500

1000
τ1500

2000

(a) p+ = 0.48.
(b) p+ = 0.52.
Fig. 6. Moving average of the phase states.

probability

phase difference [degree]

Fig. 5. Moving average of the phase states (p+ = 0.5,p− = 0.5).

of positive/negative of the STVR before the switchingphase states in Fig. 8. The result is based on simulations of 100 averages of 500000 Poincaré sections.
The graph has a peak around 250τ and, upon increasing τ , the latency probability decreases gradually. The
average of latency of the positive/negative part afterswitching is 624τ , as is shown in Fig. 8.
The distribution of the latency of the positive/negative part of the original STVR is shown inFig. 9. This distribution is similar to the distributionof Fig. 8. From Fig. 9, we conclude that the latency ofthe positive/negative part of the STVR has no strong
relation with the switching behavior, because Fig. 8shows just the characteristics of the original STVR.

200150100
50in-phase synchronization breaks down and only theanti-phase synchronization remains. For the choicep+ < 0.42, in contrast, only the in-phase synchronization can be observed. These findings are collected bydisplaying the percentage of the in-phase state in dependence p+ in Fig. 7.

00500

1000

1500

2000

τ

2500

3000

3500

4000

Fig. 8. Distribution of the latency of the positive/negative
part of the STVR before switching phase states.
400000ave: 619

350000In-phase stateAnti-phase state

300000probability

average of sojourn time

10000

1000

250000200000150000100000

100

50000
010
0.42

0.44

0.46

0.48

0.5

0.52

0.54

0.56

0500

1000

1500

2000

τ

2500

3000

3500

4000

0.58

p+Fig. 7. Average of sojourn time in dependence on p+ (average time is 100, during the solution passes 500000
times to the Poincaré section).

IV. S WITCHING M ECHANISM
In order to gain more insight, we investigate therelationship between the characteristics of the STVR
and the timing of switching phase states. We first
assume that the length of the positive/negative partof the STVR influences the switching behavior. For
the following computer simulations, we fix the STVR
probability at p+ = p− = 0.5. We show the latency

Fig. 9. Distribution of the latency of the positive/negative
part of the original STVR.

In order to better characterize the switching, we define two types of switching phenomena as follows.
[Switch-1]: The synchronization state switches fromin-phase to anti-phase.
[Switch-2]: The synchronization state switches fromanti-phase to in-phase.
We calculate the ratio of the summed duration of
the positive state vs. the summed duration in thenegative state of the STVR, between switching phasestates. The probability distributions of Switch-1 andSwitch-2, respectively, are shown in Fig. 10. In52

the case of Switch-1, the distribution shifts to right(Fig. 10(a)), whereas, in the case of Switch-2, thedistribution shifts to left seen from the 50 percent(Fig. 10(b)). From these results, we see that the ratio of the positive vs. the negative part of the STVR
governs the switching phase states.

300

50%

probability

250

R EFERENCES

150[1] L.L. Bonilla, C.J. Perez Vicente and R. Spigler, “Timeperiodic phases in populations of nonlinearly coupled oscillators with bimodal frequency distributions,” Physica D: Nonlinear Phenomena, vol.113, no.1, pp.79-97, Feb. 1998.
[2] J.A. Sherratt, “Invading wave fronts and their oscillatory
wakes are linked by a modulated traveling phase resetting
wave,” Physica D: Nonlinear Phenomena, vol.117, no.1-4,pp.145-166, June 1998.
[3] G. Abramson, V.M. Kenkre and A.R. Bishop, “Analytic solutions for nonlinear waves in coupled reacting systems,” Physica A: Statistical Mechanics and its Applications, vol.305,no.3-4, pp.427-436, Mar. 2002.
[4] I. Belykh, M. Hasler, M. Lauret and H. Nijmeijer, “Synchronization and graph topology,” Int. J. Bifurcation and Chaos,vol.15, no.11, pp.3423-3433, Nov. 2005.
[5] C.M. Gray, “Synchronous oscillations in neural systems:mechanisms and functions,” J. Computational Neuroscience,
vol.1, pp.11-38, 1994.
[6] R. Stoop and C. Wagner, “Neocortex’s architecture optimizes
computation, information transfer and synchronizability, atgiven total connection length,” International Journal of Bifurcation and Chaos, vol.17, no.7, pp.2257-2279, 2007.
[7] T. Suezaki and S. Mori, “Mutual synchronization of two oscillators,” Trans. IECE, vol.48, no.9, pp.1551-1557, Sep. 1965.
[8] H.B. Fotsina and J. Daafouza, “Adaptive synchronization ofuncertain chaotic colpitts oscillators based on parameter identification” Physics Letters A, vol.339, pp.304-315, May. 2005.
[9] I. Belykh, V. Belykh and M. Hasler, “Blinking model andsynchronization in small-world networks with a time-varying
coupling,” Physica D, vol.195, no.1-2, pp.188-206, 2004.
[10] Y. Nishio and S. Mori, “Chaotic phenomena in nonlinear
circuits with time-varying resistors,” IEICE Trans. Fundamentals, vol.E76-A, no.3, pp.467-475, Mar. 1993.
[11] C. Huygens, J. Scavants XI, 79 (1665); XII, 86 (1665). For
a recent review of synchronization processes see S. Strogatz,
Sync: The Emerging Science of Spontaneous Order (Hyperion, New York, 2003).
[12] Y. Uwate and Y. Nishio, “Synchronization phenomena invan der Pol oscillators coupled by a time-varying resistor,” International Journal of Bifurcation and Chaos, vol.17, no.10,
pp.3565-3569, 2007.
[13] Y. Uwate and Y. Nishio, “Switching phase states of chaotic
circuits coupled by time-varying resistor,” Proceedings of ISCAS’07, pp. 1783-1786, May 2007.
[14] Y. Uwate, Y. Nishio and R. Stoop, “Noise-Induced breakdown of stochastic resonant behavior of van der Pol oscillators
coupled by time-varying resistor,” Proceedings of ISCAS’09,
pp. 1887-1890, May 2009.

50001020304050607080901008090100

ratio of positive part (STVR) [%](a) Switch-1 (in - anti).
300

50%
250

probability

In conclusion, the switching behavior of the system seems to be strongly controlled by the dynamical
properties of the oscillators. From the STVR, only theratio between the residence times of the states seems
important, whereas other properties seems to be ofmuch lesser importance. It is our impression that thesephenomena are the consequences of the dynamics of
the involved oscillators, rather than of the STVR.

200100200150100500010203040506070

ratio of positive part (STVR) [%](b) Switch-2 (anti - in).
Fig. 10. Probability distribution (ratio of positive part ofthe STVR).

Finally, we investigate the switching time as a function of the nonlinearity of the oscillator. The simulation results obtained for 500000 Poincaré sections areshown in Fig. 11. We observe that the switching frequency increases with the strength of the nonlinearityof the oscillator. This may be seen as an unexpected
result, since, with increased nonlinearity, oscillators
tend to be stronger synchronized.
3300
3200
3100

switching time

V. C ONCLUSIONS

3000
2900
2800
2700
2600
2500
2400
0.5

1

1.5

ε2Fig. 11. Switching time.

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