Synchronizations in Three Coupled Oscillatorswith Memristor Synapses as Ring Structure
Yukinojo Kotani, Yoko Uwate and Yoshifumi Nishio
Dept. of Electrical Engineering
Tokushima University
2-1 Minami-Josanjima, Tokushima 770-8506, Japan
E-mail : {kotani, uwate, nishio}@ee.tokushima-u.ac.jp
Abstract— Synchronizations are obtained in coupled oscillators via
resistors. Recently, many researchers have focused on a memristor
synapse. Few studies have studied synchronizations in coupledoscillators of electronic circuit models with memristor synapses.
This study proposes the three coupled oscillators with memristor
synapses, and researches various synchronizations changing thecharacteristics of the memristor.
Keywords; Memristor; Synchronization; Oscillator

I.

INTRODUCTION

Synchronization is the interested phenomenon and studied in
the several fields. In electrical engineering, this phenomenon
was observed in the circuit model [1]. In previous study, varioussynchronizations were investigated in many kinds of coupledoscillators via resistors [2]. A memristor was proposed, thecharacteristics of the charge and the flux was defined as thepiecewise function by M. Itoh in 2008 [3]. It has dynamics for a
resistor because a charge is defined as an integral of a current.
The number of synchronization studies in coupled oscillators ofelectronic circuit models with memristor synapse was muchlittle, so we focus on a memristor synapse.

Figure 2. Three coupled oscillators with memristor synapses as ring.

We normalized the circuit equations of the coupled oscillators inFig. 2 for the computer simulations. The normalized circuit
equations describing first oscillators can be written by Eq. (1)𝑑𝑥1= 𝜀𝑥1 (1 − 𝑥1 2 ) − 𝑦1 − 𝑦6𝑑𝜏𝑑𝑦1 1= {𝑥1 − 𝑀(𝑧1 )𝛾(𝑦1 + 𝑦2 )}𝑑𝜏2𝑑𝑦6 1= {𝑥1 − 𝑀(𝑧3 )𝛾(𝑦5 + 𝑦6 )}𝑑𝜏2𝑑𝑧1){ 𝑑𝜏 = 𝜉(𝑦1 + 𝑦2This study proposed the three coupled oscillators withmemristor synapses and investigated various synchronizations
changing the characteristics of the memristor.
II.

CIRCUIT MODEL

Figure 1 shows a memristor model and the characteristics ofthe memristor. The characteristics of the memristor is defined as
the gradient of the piecewise function. Resistance value of thememristor is called memristance. This study proposed three
coupled oscillators with memristor synapses as shown in Fig. 2.

(a) Schematic symbol.

(b) Charge-flux characteristics.

Figure 1. Memristor model.

979-8-3503-2703-8/23/$31.00 ©2023 IEEE

(1)

where 𝑀 is refined as Eq. (2).
𝑀(𝑧1 ) =𝑑𝜑(𝑧1 )
𝑎 (|𝑧1 | < 1)={𝑏 (|𝑧1 | > 1)𝑑𝜏(2)

𝑧1 is correspond to 𝑞1 in change of the variables, so memristancedepends on 𝑧1 . 𝑀 is 𝑎 when the absolute value of 𝑧1 is smaller
than one. On the other hand, 𝑀 is 𝑏 when the absolute value of𝑧1 is larger than one. Similarly, normalized circuit equations
describing both second and third oscillators can be obtained.
In these simulations, the normalized circuit equations werecalculated by using Runge-Kutta method. The parameters were
set up to the nonlinearity 𝜀 = 0.1, the coupling strength 𝛾 = 1,
the coupling factor 𝜉 = 1, the scaling time 𝜏 = 20000 and step
size of Runge-Kutta method ℎ = 0.01. In addition, the variables
𝑥, 𝑦 and 𝑧 are correspond to the voltage 𝑣, the current 𝑖 and thecharge 𝑞 respectively.

251

ISOCC2023III.

RESULTS

A. Anti-phase and aynchronization
We chose the parameters a = 5.0 and b = 0.1 to strengthen
one memristance. Three oscillators oscillated in Fig. 4 (a) ~ (c).
First and second oscillators were anti-synchronized in Fig. 4 (d)and (g). On the other hand, third oscillator was asynchronizedwith the other oscillators in Fig. 4 (e) ~ (g). In Fig. 4 (h), 𝑀1 was𝑏, and 𝑀2 and 𝑀3 was a constantly. Coupling strength of thesynapse between first and second oscillators was much larger
than the other coupling parts. Therefore, first and second
oscillators were anti-synchronized. In addition, third oscillator
was not synchronized with first and second oscillators.

(a)

(b)

(a)

(b)

(c)

(d)

(e)

(f)

(c)

(g)
(d)

(e)

(g)

(f)

(h)

Figure 4. Anti-phase and asynchronization. (a) First oscillator attractor(𝑥1 vs 𝑦1 + 𝑦6 ). (b) Second oscillator attractor (𝑥2 vs 𝑦2 + 𝑦3 ). (c)Third oscillator attractor (𝑥3 vs 𝑦4 + 𝑦5 ). (d) Phase difference (𝑥1 vs𝑥2 ). (e) Phase difference (𝑥1 vs 𝑥3 ). (f) Phase difference (𝑥2 vs 𝑥3 ). (g)Time-series of 𝑥 . (h) Time-series of 𝑀 . 𝑎 = 5.0, 𝑏 = 0.1, initialvalues 𝑥1 = 1.0, 𝑥2 = 1.1, 𝑥3 = 1.2, 𝑦1 = 2.0, 𝑦2 = 2.1, 𝑦3 =2.2 , 𝑦4 = 2.3 , 𝑦5 = 2.4 , 𝑦6 = 2.5 , 𝑧1 = 1.0 , 𝑧2 = 1.1 , 𝑧3 =1.2.

B. Anti-phase and oscillation stop
We chose the parameters a = 0.1 and b = 10 to strengthen
two memristances. First oscillator stopped oscillating in Fig. 5(a). Second and third oscillators oscillated in Fig. 5 (b) and (c).
Second and third oscillators were anti-synchronized in Fig. 5 (f).
In Fig. 5 (h), 𝑀1 and 𝑀3 were 𝑏 , and 𝑀2 was 𝑎 constantly.
Coupling strength of the synapse between second and thirdoscillators was much smaller, so they were synchronized.
Besides, first oscillator stopped oscillating.

979-8-3503-2703-8/23/$31.00 ©2023 IEEE(h)Figure 5. Anti-phase and oscillation stop. (a) First oscillator attractor(𝑥1 vs 𝑦1 + 𝑦6 ). (b) Second oscillator attractor (𝑥2 vs 𝑦2 + 𝑦3 ). (c)Third oscillator attractor (𝑥3 vs 𝑦4 + 𝑦5 ). (d) Phase difference (𝑥1 vs𝑥2 ). (e) Phase difference (𝑥1 vs 𝑥3 ). (f) Phase difference (𝑥2 vs 𝑥3 ). (g)Time-series of 𝑥 . (h) Time-series of 𝑀 . 𝑎 = 0.1, 𝑏 = 10, initialvalues 𝑥1 = −1.0, 𝑥2 = −1.1, 𝑥3 = −1.2, 𝑦1 = 2.0, 𝑦2 = 2.1,
𝑦3 = 2.2 , 𝑦4 = 2.3 , 𝑦5 = 2.4 , 𝑦6 = 2.5 , 𝑧1 = 1.0 , 𝑧2 = 1.1 ,𝑧3 = 1.2.

IV. CONCLUSONSIn this study, we proposed three coupled oscillators withmemristor synapses and researched two types ofsynchronizations changing the characteristics of the memristor.
As a result, anti-phase and asynchronization were obtained. Also,anti-phase and oscillation stop occurred by giving synapses
dynamics.
REFERENCES
[1]

[2]

[3]

S. Moro, Y. Nishio and S. Mori, “Synchonization Phenomena inOscillators Coupled by One Resistor,” IEICE Trans. Fundamentals, vol.
E78-A, no. 2, pp. 244–253, February 1995.
Y. Uwate and Y. Nishio, “Synchonization in Three Coupled van der PolOscillators with Different Coupling Strength,” Proc. of Int. Conf.
NCSP’10, pp. 109–112, Mar 2010.
M. Itoh and L. O. Chua, “Memristor Oscillators,” Int. J. Bifucation andChaos, vol. 18, no. 11, pp. 3183-3206, Sep. 2008.

252

ISOCC2023