Synchronization in Three Coupled van der Pol Oscillatorswith Different Memristor Coupling Strength
Yukinojo Kotani, Yoko Uwate and Yoshifumi Nishio
Dept. of Electrical and Electronic Eng., Tokushima University
2-1 Minami-Josanjima, Tokushima 770–8506, Japan
E-mail: {kotani, uwate, nishio}@ee.tokushima-u.ac.jp

Abstract
In this study, we propose three coupled van der Pol oscillatorswith different memristor couplings strength. We investigate
synchronization state changing the one memristor couplingstrength. Compared with different resistor coupling strength,
amplitude death as new synchronization state is obtained. Inaddition, we research synchronization in dependence on theone memristor parameter to analyze the relative phase difference, the amplitude and the power consumption.
1. Introduction
Human brain is composed of over one hundred billion neurons and synapses. Information transmission in the humanbrain involves the transmission of electrical signals between
neurons through synapses. At this time, neurons are synchronized with each neuron. In electrical engineering, humanbrain is modeled by coupled oscillators, and synchronization
phenomena are investigated in various coupled oscillators fordevelopment of brain science [1], [2].
Many researchers investigated synchronization phenomena on coupled oscillators via resistors [3], inductors [4] andtime-varying resistors [5]. These circuit elements are characterized by a current. Recent years, a memristor has been
focused on as the synapse because resistance value of thememristor is characterized by a charge [6], [7]. In general,the charge is defined as the integral of the current. In humanbrain, the weight of the synapse is changed during transmitting information. Hence the memristor has ability of realizingthe synapse of human brain. Research on synchronization
phenomena on coupled oscillators with memristor synapses
provides the realization of human brain behaviors.
In previous study, three-phase synchronization state andin-phase and anti-phase synchronization state have been observed in three coupled van der Pol oscillators with different resistor coupling strength [2]. Also, our research groupproposed three coupled van der Pol oscillators with memristor coupling strength, and confirmed three synchronization
types: three-phase synchronization state, anti-phase synchro-nization and amplitude death [8]. However, the characteristics of all memristors are same. Synchronization phenomena
on coupled van der Pol oscillators with different memristor
coupling strength have not been studied.
In this study, we propose three coupled van der Pol oscillators with different memristor coupling strength. First, weinvestigate synchronization phenomena changing the characteristics of the one memristor to do numerical simulations.
Second, we analyze synchronization in dependence on theone memristor parameter by calculating the relative phase difference and the amplitude of oscillators. Finally, we research
the power consumption of our proposed circuit model.
2. Proposed Model
2.1 Memristor Model
We use the charge-controlled memristor model [6]. Figure
1 (a) shows a schematic model and the Charge q - flux φ curvecharacteristics of the memristor is shown in Fig. 1 (b).

11(a)

(b)

Figure 1: Memristor model. (a): Schematic model, (b):Charge q - flux φ characteristic curve.
Resistance value of the memristor M (q) is called memristance. In this model, M (q) is defined as the gradient of thepiecewise linear function φ(q) as Eqs. (1) and (2).

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φ(q) = bq + 0.5(a − b)(|q + 1| − |q − 1|).
(a (|q| < 1)dφ(q)M (q) ==dqb (|q| > 1)

(1)
(2)

where a and b are the parameters of two memristors between
second oscillator and the other oscillators. Similarly, c andFigure 2 shows the three coupled van der Pol oscillatorsd are the parameters of the memristor between first oscillator
with different memristor coupling strength.
and third oscillator. Next, these circuit equation Eq. (4) need
to be normalized in order to investigate synchronization phenomena calculating by Runge-Kutta method. We change thevariable and parameters as follows.
2.2 Coupled Circuit Modelsr
g1
g1 CCxk , iR,k = iL,k =yL,k , qk = zk , γ =vk =g3
g3 L
Lr
r
r
√
L
g1C,ζ = C,η = r.
t = LCτ, ε = g1
C
g3
L
(5)
r

Figure 2: Three coupled van der Pol oscillators with different
memristor coupling strength.
Where r is the tiny resistor to avoid L-loop in coupled oscillators as a ring structure. The iG,k − vG,k characteristic
curve of the nonlinear resistor is approximated by the following third-power polynomial equation.
3iG,k = −g1 vG,k + g3 vG,k(g1 , g3 > 0, k = 1, 2, 3). (3)In addition, the q − φ characteristic curve of the one memristor between first oscillator and third oscillator is different
from the other memristors.
The circuit equation are given as Eq. (4) from Kirchhoff’s
circuit laws.
dvk
C= −iG,k − iR,k − iL,kdtdiR,k= vk − riR,k − M (q)(iR,k + iL,k+1 )
2L


dt

diL,k
2L= vk − riL,k − M (q)(iR,k−1 + iL,k )dtdqk= iR,k + iL,k+1
dt((4)

a(|q|<1)dφ(q)kk M (q ) ==kdqkb (|qk | > 1)


(

c (|q3 | < 1)
dφ(q3 )M (q3 ) ==dq3
d (|q3 | > 1)
φ(qk ) = bqk + 0.5(a − b)(|qk + 1| − |qk − 1|)
φ(q3 ) = dq3 + 0.5(c − d)(|q3 + 1| − |q3 − 1|).

Hence, the normalized circuit equation of the proposed model
are given as follows.
dxk= ε(1 − x2k )xk − yR,k − yL,k
dτ



dyR,k1= (xk − ηyR,k − M (z)(yR,k + yL,k+1 ))
2
 dτ



dyL,k1= (xk − ηyL,k − M (z)(yR,k−1 + yL,k ))dτ2dzk
= ζ(yR,k + yL,k+1 )dt((6)a (|zk | < 1)dφ(zk )M (zk ) ==
dzkb (|zk | > 1)


(

c (|z3 | < 1)
dφ(z3 )=M (z3 ) =dzd (|z3 | > 1)
3





φ(zk ) = bzk + 0.5(a − b)(|zk + 1| − |zk − 1|)
φ(z3 ) = dz3 + 0.5(c − d)(|z3 + 1| − |z3 − 1|).

The instantaneous power consumption of the memristor p
is defind as Eq. (7).
p = M (q)i2 .

(7)

The average power consumption P is obtained by integrating
the instantaneous power consumption p.
Z t
P =M (q)i2 dτ (t0 ≦ t).
(8)
t0

Therefore, the total average power consumption of our proposed model Pall can be calculated by Eq. (9).
Pall =33

k=1

k=1

1 X
1 X
Pk,k+1 =M (zk )(iRk + iLk+1 )2 . (9)TTwhere T is a long period, iL,4 = iL,1 and Pk,k+1 is the average power consumption of the memristor coupling betweenk-th oscillator and (k + 1)-th oscillator.

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Figure 4 shows the relationship between |∆θk | and d.

3. Results
For the computer simulations, the normalized circuit equation Eq. (6) are calculated by Runge-Kutta method with step
size h = 0.01. The parameters and variables are setting forτ = 20, 000, ε = 0.1, γ = 1.0, ζ = 0.1, η = 0.001, a = 0.1,
b = 10, c = 0.1, x1 = 1.0, x2 = 1.1, x3 = 1.2, y1 = 2.0,y2 = 2.1, y3 = 2.2, z1 = 1.0, z2 = 1.1 and z3 = 1.2.

Δθ3

Δθ3
Δθ2

Δθ2
Δθ1

3.1 Synchronization PhenomenaThis study analyzes synchronization phenomena by changing the one memristor parameter d. Figure 3 shows the threetypes of the time-series of x.

Δθ1(a)(b)

Figure 4: Relationship between |∆θk | and d. (a): 0 ≦ d ≦ 1,(b): 1 ≦ d ≦ 10.
x1 x3 x2

(a)x1 x3x2

x1 x3 x2(b)(c)

In Fig. 4, second oscillator anti-synchronized with first oscillator and third oscillator synchronized with first oscillator inthe region 0.0 ≦ d ≦ 0.15. In the region 0.2 ≦ d ≦ 2, thirdoscillator anti-synchronized with first oscillator and |∆θ2 |converges to almost 150°. In the region 2.5 ≦ d ≦ 10, threeoscillators synchronized with each oscillator.

Figure 3: Time-series of x (a): In-phase and anti-phase synchronization state (d = 0.01), (b): Amplitude death (s =In order to analyze the behavior of second oscillator, we0.5), (c): Three-phase synchronization state (d = 10).
investigate the relationship between the amplitude of threeoscillators ρk and the one memristor parameter d. Figure 5
In Fig. 3, in-phase and anti-phase synchronization occurs shows the relationship between ρk and d.
when d = 0.01. Amplitude death occurs when d = 0.5.
Three-phase synchronization occurs when d = 10. Therefore we confirm that three synchronization states: three-phase
ρ1 ρ3
ρ1 ρ3
synchronization state, in-phase and anti-phase synchronization state and amplitude death are obtained by changing d.

ρ2
3.2 Synchronization in dependence on dIn this subsection, we analyze three synchronization statesρ2in dependence on d in detail. We investigate the relationshipbetween the absolute value of the relative phase differences(b)(a)|∆θk | and the one memristor parameter d. The phase of firstoscillator as a reference is denoted θ1 and the relative phase
Figure 5: Relationship between ρk and d. (a): 0 ≦ d ≦ 1,
difference of k-th oscillator |∆θk | is defined as Eq. (10).
(b): 1 ≦ d ≦ 10.
yky1|∆θk | = |θk − θ1 | = | arctan− arctan |.
(10)
xkx1
In Fig. 5, ρ1 , ρ2 and ρ3 converge to almost 1.0 in the reHere, |∆θk | is calculated by Poincaré maps. The methods of gions 0 ≦ d ≦ 0.15 and 2.5 ≦ d ≦ 10. In the regions
0.2 ≦ d ≦ 2.5, ρ2 converges to almost zero. Therefore,
calculating |∆θk | are described as following steps.
first oscillator and third oscillator are anti-synchronized, and
1. A van der Pol oscillator with third-power has a stable then the amplitude of second oscillator is dead in the regions
limit cycle, so Poincaré section is defined as the plane 0.2 ≦ d ≦ 2.5.
that xk (τ ) > 0 and yk (τ ) = 0 in phase plane.
Here, we compare the simulation results of the relative
phase difference and the amplitude. In-phase and anti-phase
2. If x1 and y1 move from fourth quadrant to first quadrant
synchronization occur in the region 0.0 ≦ d ≦ 0.15. In the(x1 (τ − 1) > 0, y1 (τ − 1) < 0 and y1 (τ ) > 0), |∆θk |region 0.2 ≦ d ≦ 2, amplitude death occurs. In the region
is calculated by Eq. (10).
2.5 ≦ d ≦ 10, three-phase synchronization can be observed.

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3.3 Power Consumption

4. Conclusions

Generally, phase sift is caused by changes of the power
consumption of coupling parts [3, 8]. We investigate the relationship between the power consumption of the memristor
couplings P and the one memristor parameter d. Figure 6
shows the relationship between P and d.

We have proposed three coupled van der Pol oscillatorswith different memristor coupling strength. By changing
the charge-flux characteristic curve of the one memristor, wehave obtained three synchronization states: three-phase synchronization state, in-phase and anti-phase synchronization
state and amplitude death. Except for the case that the one
memristor coupling is larger than the other couplings, three
coupled van der Pol oscillators with different memristor coupling strength tends to synchronize to be minimum energy of
the whole circuit model when two memristor couplings arelarger than the one memristor coupling. For the future works,
we would like to design this circuit model and then we provide the fundamental evidences to do circuit experiments.

PallPallP1,2 P2,3 P3,1

P1,2 P2,3 P3,1

References(a)(b)

Figure 6: Relationship between P and d. (a): 0 ≦ d ≦ 1, (b):1 ≦ d ≦ 10.

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