IEEE Workshop on Nonlinear Circuit Networks
December 8-9, 2023

The Behavior of Chaotic Circuit Using Memristor
in Series with Negative Resistance
Taishi Segawa, Yoko Uwate and Yoshifumi Nishio
Dept. of Electrical and Electronic Engineering, Tokushima University, Japan
2-1 Minami-Josanjima, Tokushima 770-8506, Japan
Email: {segawa, uwate, nishio}@ee.tokushima-u.ac.jp
Abstract—Combining chaos circuit and memristor may create
realistic brain circuit in the future. Toward this realization,
we would like to analyze the fundamental behavior of chaos
circuit with one memristor in series. Therefore, we introduce newcircuit equation and normalization parameter based on NishioInaba chaos circuit to do computer simulations. We confirmedthat memristor has a possibility to ease the strength of chaotic
property and make a periodic solution by observing attractor
and waveform. In addition, we also confirmed the amount ofmaximum and minimum memristance values can change thechaotic property of chaos circuit.

I. I NTRODUCTION
Memristor is a nonlinear electrical component that canchange this resistance value M (q) depending on the amountof charge passing through it and magnetic flux linkage. Dueto this property, memristor is considered as one of nonvolatile storage technologies [1], [2], [3]. This memristor has
attracted large attention as research topics and investigatedsome mathematical model like Hewlett − P ackard model(HP model) and piecewise linear approximation model. Thesemathematical models are used for research about synchronization phenomena of electrical circuit, application for neuralnetwork, and so on [2].
Chaos is a random state of disorder and confusion. Thisphenomena has investigated for a long time in electronicengineering [4]. The behavior of chaos circuit can be applied
to nonlinear phenomena in nature including human behavior.
In recently, the chaos circuit with memristor has attracted torealize a brain circuit that implements a neural network andalternative human brain in circuit simulator [5].
In this paper, we analyse the behavior of one series
memristor -based chaos circuit by observing attractor andwaveform. Also, we analyse how does the memristance value
M (q) affects the chaos circuit by changing the maximum andminimum memristance values.

Circuit model. (a) Original Nishio-Inaba chaos circuit.
(b) Proposed one series memristor-based chaos circuit.
Fig. 1.

A memristor in proposed circuit model is defined as HP
mathmatical model to analyses the behavior of this circuit
based on actual memristor value. The memristance M (q) ofHP model is given as follows:
RonM (q) = Rof f 1 − µv 2 q(1)
DWe approximate the I-V characteristic by the piecewiselinear function as follows:VVrdi2 +vd (i2 ) =− i2 −(2)2rdrdThen, the circuit equation of proposed model is describedas follows:

II. P ROPOSED C IRCUIT M ODEL
Figure 1 (a) shows the original Nishio-Inaba chaos circuit.
This chaos circuit is consisted by three memory elements,
one linear negative resistance and one nonlinear resistanceconsisting of two diodes. Fig. 1 (b) is proposed circuit
model in this study that is added one memristor in original
Nishio-Inaba circuit. One memristor M (q) is put into between
inductor L1 and linear negative resistance −r in series.

- 17 -di1= v + ri1 − M (q)i1L1dtdi2L2= v − vd (i2 )
dt
dv

C= −i1 − i2dt
dq

= i1
dt(3)

By changing values such thatri1 =√CV x;
L1

L1 CV y; v = V z;
L2q√t = L1 Ct; r LC1 = α;q = CV w;
L1
L2 = β;i2 =√rd

L1 C= γ;L2µv RDo2n CV = ξ;Rof fq(4)C
L1 = η;d” · ” = dτ(3) is normalized as
ẋ = z + αx − ηx(1 − ξw)
11γ

y−y+−ẏ=z−
2γγ(5)ż = −x − βy
 ẇ = x
In this paper, the values of two inductors L1 = 603.2mH, L2 = 206.5 mH, capacitor C = 0.0069 µF, nonlinear
resistor rd = 1.46 kΩ, and some memristance elements µv =10−14 m2 /Vs, D = 10 nm are fixed. When the value of one
linear negative resistance r is variable to analyse the nonlinear
property of proposed circuit model, memristance elements
values Ron and Rof f are fixed. On the other hand, when thevalues of memristance elements Ron and Rof f are variable
to analyse the influence of memristor for chaos circuit, one
linear negative resistance r is fixed.

The difference of attractor(In case of with memristor, Ron = 10 Ω, Rof f = 0.935 kΩ(µ = 0.100, ξ = 6.90×10−5 )(a) r = 4.38 kΩ · · ·α = 0.468, (b) r = 4.82 kΩ· · ·α = 0.516,(c) r = 5.07 kΩ· · ·α = 0.542, (d) r = 5.24 kΩ· · ·α = 0.560,(e) r = 5.76 kΩ· · ·α = 0.616.)
Fig. 2.

III. S IMULATION R ESULTS
A. Comparison of Proposed Circuit Model and Original
Nisho-Inaba Circuit.
This section shows comparison results of proposed circuit
model and standard Nishio-Inaba circuit model with fixed
memristance values. The difference of attractors are shownin Fig. 2 (a), (b), (c), (d) and (e). The value of linear negative
resistance −r differs under each patterns. In Fig. 2, left sides
are attractors of proposed model and right sides are original
Nishio-Inaba circuit attaractors. In addition, the difference ofwaveform under each cases are shown in Fig. 3 (a), (b), (c),(d) and (e). These attractors and waveforms comes from thecalculation results of Equation (5). Eq. (5) is calculated using
Runge-Kutta method with the step size h = 0.02 since τ = 0.
From Fig. 2, it can be seen that the attractor becomes
more complex as the value α increases. Compare (b), (c)and (d) with proposed model (left sides) and original model(right sides), the original model has a solution curve thatpasses near the origin, but the proposed model does not passnear the origin. Especially case (b), the solution curve arequite different. Original model has a chaos solution curve, but
proposed model has a 3 period solution. From this simulation
results, memristor has a possibility to ease the strength ofchaotic property.
In Fig. 3, the axis scales of all graphs are the same except forFig. 3 (e). The vertical axis scale for w of Fig. 3 (e) is 4 timeslarger than the others. From Fig. 3, a memristor remember theamount of charge that has passed through it. So, the graph ofw is linked to x that is equivalent to current i1 in Fig. 1 (b).
And also can be seen that the waveform with no memristor ischanged by adding memristor in series.
B. Influence of The Memristance Values for The Chaos Circuit
Here, the value of linear negative resistance is fixed andchanging minimum resistance value Ron and maximum resistance value Rof f to analyse influences of the memristance- 18 -

Attractor change by each Ron and Rof f values.
(α = 0.516)

Fig. 4.

The difference of waveform(Ron = 10 Ω, Rof f = 0.935 kΩ and the vertical axis scale for
w of (e) is 4 times larger than the others.
(µ = 0.100, ξ = 6.90×10−5 )(a) r = 4.38 kΩ · · ·α = 0.468, (b) r = 4.82 kΩ· · ·α = 0.516,(c) r = 5.07 kΩ· · ·α = 0.542, (d) r = 5.24 kΩ· · ·α = 0.560,(e) r = 5.76 kΩ· · ·α = 0.616.)
Fig. 3.

values for this chaos circuit. Fig. 4 shows attractors when thevalue of Ron and Rof f are gradually changed. Fig. 4 areusing α = 0.561 that value is shown as a 3 period solution
with memristor in Fig. 2 (b). The Ron and Rof f values ineach case are shown in Table I.
From Table I, the ratio of resistance Rof f /Ron = 100 is
set to be almost constant. From Fig. 4. it can be seen thatthe chaotic attractor gradually becomes a periodic solution as
the values of Ron and Rof f increases. At first case (a), theattractor is observed as chaos attractor. This chaotic attractor
becomes a 3 periodic solution attractor at case (e). After thatcase (e), this attarctor gradually becomes a almost 1 periodic
solution attractor at case (q). After case (q), the attractor
has a chaotic property a little bit again. From these results,
memristor has a possibility that it can ease the strength ofchaotic property in a chaos circuit. However, increasing thevalues of Ron and Rof f doesn’t ease the strength of chaotic
property definitely. Appropriate resistance values are necessaryto ease the chaotic property in this simulation condition.

- 19 -TABLE I
T HE Ron AND Rof f VALUES IN EACH CASE

(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
(l)
(m)
(n)
(o)
(p)
(q)
(r)

Ron Ω
56
7
8
9
10
10
11
12
13
14
15
16
17
18
1920
21Rof f kΩ
0.50
0.60
0.70
0.80
0.90
0.935
1.00
1.10
1.20
1.30
1.40
1.50
1.60
1.70
1.80
1.90
2.00
2.10

η
0.05350.06420.07490.0856
0.0963
0.100
0.107
0.118
0.128
0.139
0.1500.160
0.171
0.182
0.193
0.203
0.214
0.225

ξ
3.45×10−5
4.14×10−5
4.83×10−5
5.52×10−5
6.21×10−5
6.90×10−5
6.90×10−5
7.59×10−5
8.28×10−5
8.97×10−5
9.66×10−5
1.04×10−4
1.10×10−4
1.17×10−4
1.24×10−4
1.31×10−4
1.38×10−4
1.45×10−4

IV. CONCLUSIONSIn this study, we proposed new chaos circuit model that isbased Nishio-Inaba chaos circuit. One memristor is put into
between a inductor L1 and one linear negative resistance −rin series. The behavior of this circuit model is investigated as
seen attractor and waveform in each case that fixed the valueof memristance elements Ron and Rof f and fixed the valueof a linear negative resistance −r. As a result, we discoveredthat the proposed circuit has a possibility to ease chaotic
property and can became a periodic solution from chaotic
solution. In attractor comparison with memristor circuit andno memristor circuit, the circuit with memristor give more
periodic solution than no memristor one. Especially, in caseof the circuit with no memristor gives a chaos solution, thecircuit with memristor gives a 3 periodic solution expresses apossibility to ease chaotic property and can became a periodic
solution from chaotic solution. In addition, a proposed circuit
model can become a periodic solution from chaotic attractor
gradually as the values of memristor elements Ron and Rof fincreases. It mean as increase the influence of memristor tochaos circuit by changing the values of memristance elements,
we can adjustment the behavior of proposed circuit model aswell as parameter α.
For the future work, we would like to investigate the NishioInaba circuit where one memristor and one liner negative
resistance in parallel. We also would like to investigate thebehavior of chaos circuit with other memristor mathematicalmodel and other chaos circuit model.
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