Synchronization of Two Memristor-Based Chaotic Circuits Coupled by Resistor
Tsubasa Tezuka† , Kana Kobayashi† , Yoko Uwate† , Yasuteru Hosokawa‡ , Yoshifumi Nishio††

Department of Electrical and Electronic Engineering, Tokushima University
2-1 Minami-Josanjima, Tokushima 770–8506, Japan
Email: {tezuka,kobayashi,uwate,nishio}@ee.tokushima-u.ac.jp‡Department of Media and Information Systems, Shikoku University
Furukawa, Ojin, Tokushima 771–1192, Japan
Email: hosokawa@keiei.shikoku-u.ac.jp

rived from Chua’s circuit by replacing Chua’s diode with amemristor.

AbstractMemristor is the fourth fundamental circuit element after theresistor, the capacitor and the inductor. The memristor has
attracted worldwide attention because it has a great potential. In this study, we investigate synchronization phenomenaof two memristor-based chaotic circuits coupled by a resistor. We confirm synchronization phenomena by changing thecoupling strength. Also, we investigate synchronization phenomena when changing the characteristic of the memristor.

In this study, we investigate synchronization phenomenaof two memristor-based chaotic circuits coupled by a resistor. We confirm synchronization phenomena by changing
coupling strength. Also, we investigate synchronization phenomena when changing the characteristic of the memristor.

2. System model

1. Introduction
Synchronization phenomena can be found in our life. Especially, synchronization phenomena have a closely related tonature world and animate beings. For example, we can confirm flashing firefly lights, croaking of a frog, bird call and soon. Synchronization phenomena have been studied in various
fields because it is widely applicable.
Chaos is an unpredictable phenomenon. Nishio-Inaba circuit is one of the simplest chaotic circuits [1]. We can confirm chaotic synchronization phenomena in cases where two
chaotic circuits are coupled. Chaotic synchronization phenomena have been studied extensively [2].
On the other hand, a memristor is the fourth fundamental
circuit element after the resistor, the capacitor and the inductor. Leon O. Chua proposed memristor in 1971 [3]. After
that, the memristor was called “missing” element for manyyears. However, the element was implemented by Strukov etal. [4]. The “memristor” is short for memory resistor. The
element has a relationship between charge and flux. M(q) ,which is the resistance value of the memristor is changing bypassing charge. M(q) is called memristance. The memristor
has attracted worldwide attention because it has a great potential. For example, the memristor is expected to apply tosuper-dense nonvolatile computer memory and neural computers. Also, memristor-based chaotic circuit is an interesting research topic because memristor is useful for designingnonlinear oscillators [5]. In [5], a nonlinear oscillator was de-Figure 1 shows a system model in this study. The systemmodel is derived from two Nishio-Inaba circuits by replacing
the diodes with a memristor. Each chaotic circuit consists oftwo inductors L1 and L2 , one capacitor C, one linear negative
resistor −r and one memristor M(q) . q is a charge that passthrough the memristor. The chaotic circuits are coupled by aresistor.
We assume that the memeristor is characterized by themonotone-increasing and the piecewise-linear nonlinearity[5]. a and b are the slope of the piecewise-linear function
where a > 0, b > 0. If a ̸= b, M(q) is discontinuous.

- 471 -Figure 1: System model.

The parameter δ is the coupling strength between the circuits.
The memristance M(w) is given as follows:

The circuit equations of this study are given as follows:di1 (1)= V (1) − M(q(1)) i1 (1),
 L1 (1)

dt




dv(1)
1


C(1)
= i2 (1) − i1 (1) − (V (1) − V (2)),
dtR
di (1)
 L2 (1) 2= −V (1) + r(1)i2 (1),

dt



dq(1)


= i1 (1),dtdi1 (2)
L1 (2)
dt

dv(2)

C(2)
dt



di2 (2)
L2 (2)

dt





dq(2)




dt

M(w(n)) =where M(w) is a when |w| < 1.0, and b when |w| > 1.0.
3. Simulation results1(V (1) − V (2)),
R= −V (2) + r(2)i2 (2),= i1 (2).
(1)

By changing the parameters:
x(n) = i1 (n), y(n) = V (n), z(n) = i2 (n),
w(n) = q(n),


1r(n)
1

, β(n) =, γ(n) =,α(n) =
L1 (n)
L2 (n)
L2 (n)
 δ = 1 , C(n) = 1.0,(n = 1, 2)
R

We set the parameters of the circuit as α(n) = 4.0, β(n) =1.0, γ(n) = 0.525, a = 0.2 and b = 10.0. Also, we set
the time step value of the Runge-Kutta method h is 0.0001.
We investigate synchronization phenomena by increasing thecoupling strength δ between the circuits.
Figure 2 shows the simulation result δ = 0.0000001. Figure 2 (a) shows a Lissajous figure. In the Lissajous figure,vertical axis and horizontal axis are V (2) and V (1). We candetermine synchronization or asynchronous when the Lissajous figure is diagonal or square. Figure 2 (b) shows a voltage difference. In the waveform y(1), y(2) shows a voltage
V (1) and V (2). We can determine synchronization or asynchronous when the amplitude of the waveform is high or low.
In case of δ = 0.0000001, two chaotic circuits are asynchronized.

(2)

the normalized circuit equations are given as follows: dx(1) = α(1)(y(1) + M(w(1)) )x(1),

dt




dy(1)


= z(1) − x(1) − δ(y(1) − y(2)),

dt





dz(1)


= −β(1)y(1) + γ(1)z(1),

dt





dw(1)


= x(1),
 dtdx(2)= α(2)(y(2) + M(w(2)) )x(2),

dt



 dy(2) dt = z(2) − x(2) + δ(y(1) − y(2)),
dz(2)
= −β(2)y(2) + γ(2)z(2),
dt


dw(2)

= x(2).
 dt(4)(n = 1, 2)= V (2) − M(q(2)) i1 (2),= i2 (2) − i1 (2) +dφ(w(n)),dw(n)

(a)

(b)

(3)

Figure 2: Simulation result. δ = 0.0000001, a = 0.2, b = 10.0.
(a)Lissajous figure. (b)Waveform.

Figure 3 shows the simulation result δ = 0.2, In case ofδ = 0.2, two chaotic circuits are synchronized than δ =0.0000001.
Figure 4 shows the simulation result δ = 0.6. In case ofδ = 0.6, two chaotic circuits are synchronized.

- 472 -

(a)

(a)

(b)

(b)

Figure 3: Simulation result. δ = 0.2, a = 0.2, b = 10.0.
(a)Lissajous figure. (b)Waveform.

Figure 4: Simulation result. δ = 0.6, a = 0.2, b = 10.0.

We investigate synchronization phenomena by changing
characteristic of memristor b. Figure 5 shows the simulation
result δ = 0.2, a = 0.2, b = 5.0. In case of b = 5.0, twochaotic circuits are synchronized than b = 10.0.
Figure 6 shows the simulation result δ = 0.2, a = 0.2,b = 100.0. In case of b = 100.0, two chaotic circuits are
asynchronized than b = 10.0.
We confirm that two chaotic circuits are syncrhonizedby changing coupling strength. Also, we confirm that thesynchronization phenomena are changed when changing thecharacteristic of the memristor b.

References

4. Conclusions
In this study, we have investigated the system model using
two memristor-based circuits that are coupled by a resistor.
As a result, two chaotic circuits are synchronized when increasing coupling strength. Also, we confirm that the chaoticcircuits are synchronized when decreasing the characteristicof the memristor b and asynchronized when increasing b.
In the future works, we will investigate the effect of thememristor on the chaotic circuit by comparing with two coupled original Nishio-Inaba circuits. Also, we investigate theeffect of when changing the characteristic of the memristor a.

(a)Lissajous figure. (b)Waveform.

[1] Y. Nishio, N. Inaba, S. Mori and T. Saito, “RigorousAnalyses of Windows in a Symmetric Circuit,” IEEE
Transactions on Circuits and Systems, vol. 37, no. 4,pp. 473-487, April 1990.
[2] Y. Z. Yin, “Synchronization of Chaos in a Modified
Chua’s Circuit Using Continuous Control,” International Journal of Bifurcation and Chaos, vol. 6, no. 11,
pp. 2101-2117, November 1996.
[3] L. O. Chua, “Memristor - the missing circuit element,”
IEEE Transactions on Circuit Theory, vol. 18, no. 5, pp.
507-519, September 1971.
[4] D. B. Strukov, G. S. Snider, D. R. Stewart andR. S. Williams “The missing memristor found,” Nature,
vol. 453, pp. 80-83, May 2008.
[5] M. Itoh and L. O. Chua, “Memristor Oscillators,” International Journal of Bifurcation and Chaos, vol. 18, no.
11, pp. 3183-3206, December 2008.

Acknowledgment
This work was partly supported by JSPS Grant-in-Aid for
Scientific Research 16K06357.

- 473 -(a)(b)
Figure 5: Simulation result. δ = 0.2, a = 0.2, b = 5.0.
(a)Lissajous figure. (b)Waveform.

(a)

(b)
Figure 6: Simulation result. δ = 0.2, a = 0.2, b = 100.0.
(a)Lissajous figure. (b)Waveform.

- 474 -