IEEE Workshop on Nonlinear Circuit Networks
December 13-14, 2019

Comparison of Complexity of Chaos
in Many Degrees of Freedom Chaotic Circuits
Naoto Yonemoto, Katsuya Nakabai, Yoko Uwate and Yoshifumi Nishio
Dept. of Electrical and Electronic Engineering, Tokushima University
2-1 Minami-Josanjima, Tokushima 770–8506, Japan
Email: {yonex34, nakabai, uwate, nishio}@ee.tokushima-u.ac.jp

Abstract—This paper considers comparison of the complexityof chaos generated in many degrees of freedom chaotic circuits.
We increase the number of connected subcircuits from two to
three in order to produce more complex chaos. By means of thecircuit experiment and computer simulation, we show chaotic
attractors. From the results, we confirmed that more complex
chaos is generated in the each circuit.

v1
C1

L12

L11

vd (i12)-gI. I NTRODUCTION
Chaos has two major features. They are initial value sensitivity and long-term unpredictability. It is difficult to predict
long-term weather forecasts [1] due to initial values such astemperature, atmospheric pressure, and wind speed, etc.
Chaotic circuit is used to considers nonlinear phenomena
such as natural phenomena. It is faster and easier to experimentthan actual natural phenomena. Strictness analysis is difficult
for chaotic phenomena generated in high-dimensional systems [2]. Therefore, getting closer to high-dimensional chaosthat exists in nature leads to an effect that is useful for the realworld from a new perspective. For example, it may be possible
to improve the confidentiality of chaotic communications [3]or to be closer to human judgment in brain-type computers [4].
In this study, we investigate comparison of the complexity ofchaos generated in many degrees of freedom chaotic circuits.
In the previous study, the circuit consists of two Inaba’s circuits coupled by one linear negative resister has proposed [5].
At this time, the circuit is set a different parameter values,
especially natural frequencies for each circuit. Then we compare the two circuits in series with the three. We show chaotic
attractors generated in the circuit experiment and computer
simulation.

i12

i11

v2

i21

C2

i22

L22
L21

vd (i22)

Fig.1 Circuit model of two degrees of freedom chaoticcircuit.

II. S YSTEM M ODELIn the previous study, we use the circuit model of twodegrees of freedom chaotic circuit in Fig. 1. This circuit wasexpanded from the Inaba’s circuit to two degrees of freedom
chaotic circuit. In this circuit, the case where the naturalfrequency of the lower subcircuit was higher than the naturalfrequency of the upper subcircuit was investigated.

- 26 -The parameters are described as follows:√√dt = L11 C1 τ , ” · ” = dτ, α = g LC111 ,L11L1111
β1 = LL12 , β2 = L21 , β3 = L22 ,√L111
1γ=C,ε=C2rdC1 ,√v1 = Ex1 , i11 = E√
v2 = Ex4 , i21 = E

C1L11 x2 , i12 = E
C1
L11 x5 , i22 = E√√

C1
L11 x3 ,C1
L11 x6 .

The normalized circuit equations are described as follows:x˙1 = α(x1 + x4 ) − (x2 + x3 )x˙2 = x1 x˙3 = β1 (x1 − f (x3 ))(1)x˙4 = αγ(x1 + x4 ) − γ(x5 + x6 )x˙5 = β2 x4
 x˙6 = β3 (x4 − f (x6 )).

The parameters are described as follows:

The characteristic equation for the diode is described asfollows:
1f (x) = 2ε(x + ε − |x − ε|).

x˙1 = α(x1 + x4 + x7 ) − (x2 + x3 )x˙2 = x1x˙3 = β1 (x1 − f (x3 ))





x˙4 = αγ1 (x1 + x4 + x7 ) − γ1 (x5 + x6 )x˙5 = β2 x4
x˙6 = β3 (x4 − f (x6 ))x˙7 = αγ2 (x1 + x4 + x7 ) − γ2 (x8 + x9 )x˙8 = β4 x7
 x˙9 = β5 (x8 − f (x9 )).

(2)

III. P ROPOSED SYSTEM
In this study, we use the circuit model of three degrees offreedom chaotic circuit in Fig. 2. This circuit consists of two
degrees of freedom chaotic circuit and another Inaba’s circuit
connected in series in order to produce more complex chaos.

(3)

The characteristic equation for the diode is described asfollows:
1f (x) = 2ε(x + ε − |x − ε|).

(4)

IV. R ESULTS
A. Two degrees of freedom chaotic circuit
We show the experimental and computer simulation resultsof two degrees of freedom chaotic circuit in Fig. 1. Fig. 3 (a) is
the result of the top circuit from circuit experiment. Fig. 3 (b)is the result of the top circuit from computer simulation. Fig. 4(a) is the result of the bottom circuit from circuit experiment.
Fig. 4 (b) is the result of the bottom circuit from computersimulation.
Fig.2 Circuit model of three degrees of freedom chaoticcircuit.
The parameters are described as follows:√√dt = L11 C1 τ , ” · ” = dτ, α = g LC111 ,x2i11L11L11L11L1111
β1 = L, β2 = L
L21 , β3 = L22 , β4 = L31 , β5 = L32 ,12
√
L11C1
1
1
γ1 = C
C2 , γ2 = C3 , ε = rdC1 ,√v1 = Ex1 , i11 = E√
v2 = Ex4 , i21 = E√
v3 = Ex7 , i31 = E

C1L11 x2 , i12 = EC1
L11 x5 , i22 = EC1
L11 x8 , i32 = E√√√

v1

x1(b)

(a)

C1
L11 x3 ,Fig. 3 Chaotic attractors in the top circuit.

C1
L11 x6 ,x5

i21

C1L11 x9 .

v2x4(b)(a)Fig. 4 Chaotic attractors in the bottom circuit.

- 27 -

In the circuit experiment, the circuit parameters are chosen as C1 = 15[nF ], L11 = 300[mH], L12 = 30[mH],C2 = 7.5[nF ], L21 = 150[mH], and L22 = 15[mH]. In
the computer simulation, the circuit parameters are chosen asα = 0.3, β1 = 10.0, β2 = 2.0, β3 = 20.0, γ = 2.0, andε = 0.01.
As a result, the same attractors were observed qualitatively
in each circuit in circuit experiment and computer simulation.
B. Three degrees of freedom chaotic circuit
In this study, we change the number of Inaba’s circuit
connected from two to three. We show the experimental
and computer simulation results of three degrees of freedom
chaotic circuit. Fig. 5 (a) is the result of the top circuit fromcircuit experiment. Fig. 5 (b) is the result of the top circuitfrom computer simulation. Fig. 6 (a) is the result of the middle
circuit from circuit experiment. Fig. 6 (b) is the result of themiddle circuit from computer simulation. Fig. 7 (a) is the resultof the bottom circuit from circuit experiment. Fig. 7 (b) is theresult of the bottom circuit from computer simulation.

x2

i11

v1

x1

(a)

(b)
Fig. 5 Chaotic attractors in the top circuit.

In the circuit experiment, the circuit parameters are chosen
as C1 = 15[nF ], L11 = 300[mH], L12 = 30[mH], C2 =7.5[nF ], L21 = 150[mH], L22 = 15[mH], C3 = 5[nF ],L31 = 100[mH], and L32 = 10[mH]. In the computer
simulation, the circuit parameters are chosen as α = 0.3,
β1 = 10.0, β2 = 2.0, β3 = 20.0, β4 = 3.0, β5 = 30.0,
γ1 = 2.0, γ2 = 3.0, and ε = 0.01.
Comparing the results of two degrees and three, attractors
are similar in shape, and become more complex. The newly
connected circuit generates more complex chaotic attractor.
V. C ONCLUSION
In this study, we have investigated comparison of thecomplexity of chaos generated in multiple Inaba’s circuit inseries. We have increased the number of connected subcircuits
from two to three in order to produce more complex chaos.
We have evaluated the complexity using attractors obtained bycircuit experiment and computer simulation. First, results ofcircuit experiment and computer simulation were qualitatively
the same in the same circuit. Focusing on the chaotic attractor,
more complex chaos is generated in the new connected circuit.
When we compare the same circuits, it seems that attractors
become more complex.
As our future work, we will clearly evaluate the differencein complexity by using Poincare map. In addition, we will
research on various network types. We will investigate interactions between systems [6]. We regard three degrees of freedom
chaotic circuits as one system, and they will be coupledusing an inductor. First, we will investigate by connecting two
chaotic circuits and increasing the number of connections. At
this time, we will use a network topology such as ladders,rings, and stars to investigate basic networks [7].
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i21

v2x4

(a)

(b)
Fig. 6 Chaotic attractors in the middle circuit.

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x7(a)

(b)
Fig. 7 Chaotic attractors in the bottom circuit.

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