2020 RISP International Workshop on Nonlinear Circuits,Communications and Signal Processing (NCSP 2020)
Honolulu, Hawaii, USA, February 28 - March 2, 2020

Evaluation of Complexity of Chaos
in Many Degrees of Freedom Chaotic Circuits Using Poincare Map
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

2. System modelThis paper considers comparison of the complexity of 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 and Poincare maps. From the results, we confirmedthat more complex chaos is generated in the each circuit.

In the previous study, we use the circuit model of two degrees of freedom chaotic circuit in Fig. 1. This circuit is expanded from the Inaba’s circuit to two degrees of freedom
chaotic circuit. In this circuit, we investigate the case which
the natural frequency of the lower subcircuit is higher than
the natural frequency of the upper subcircuit.

1. Introduction
i12
Chaos has two major features. They are initial value sensii11
v1
tivity and long-term unpredictability. It is difficult to predict
C1
L12
long-term weather forecasts [1] due to initial values such as
L11temperature, atmospheric pressure, and wind speed, etc.
vd (i12)Chaotic circuit is used to considers nonlinear phenomena-gsuch as natural phenomena. It is faster and easier to exi22
i21
periment than actual natural phenomena. Strictness analv2
ysis is difficult for chaotic phenomena generated in highC2
L22
dimensional systems [2]. Therefore, getting closer to highLvd (i22)21
dimensional chaos that exists in nature leads to an effect thatis useful for the real world from a new perspective. For example, it may be possible to improve the confidentiality ofchaotic communications [3] or to be closer to human judgment in brain-type computers [4]. In addition, when we Fig.1 Circuit model of two degrees of freedom chaotic circuit.
use chaos for engineering applications, it is expected to be
The parameters are described as follows:
smaller type, faster speed, and larger scale [5].
In this study, we investigate comparison of the complex√√
ity of chaos generated in many degrees of freedom chaotic
dt = L11 C1 τ , ” · ” = dτ
, α = g LC111 ,circuits. In the previous study, it was proposed the circuit
consisting of two Inaba’s circuits coupled by one linear negL11L1111
ative resister [6]. At this time, the circuit is set a differentβ1 = LL12 , β2 = L21 , β3 = L22 ,parameter values, especially natural frequencies for each circuit. Then we compare the two circuits in series with the√C1three. We show chaotic attractors generated in the circuit exγ=C, ε = r1d LC111 ,2periment and computer simulation. In addition, we compare
the complexity with Poincare maps created by computer sim√
√ulation.
v1 = Ex1 , i11 = E LC111 x2 , i12 = E LC111 x3 ,√v2 = Ex4 , i21 = E393

C1L11 x5 , i22 = E√

C1
L11 x6 .

The normalized circuit equations are described as follows:

The parameters are described as follows:√√dt = L11 C1 τ , ” · ” = dτ
, α = g LC111 ,x˙1 = α(x1 + x4 ) − (x2 + x3 )x˙2 = x1
 x˙3 = β1 (x1 − f (x3 ))x˙4 = αγ(x1 + x4 ) − γ(x5 + x6 )
 x˙5 = β2 x4
 x˙6 = β3 (x4 − f (x6 )).

L11L11L11L1111
β1 = LL12 , β2 = L21 , β3 = L22 , β4 = L31 , β5 = L32 ,√L11C111
γ1 = CC2 , γ2 = C3 , ε = rdC1 ,(1)√v1 = Ex1 , i11 = E√
v2 = Ex4 , i21 = E

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

√v3 = Ex7 , i31 = E(2)C1L11 x2 , i12 = E
C1
L11 x5 , i22 = E
C1
L11 x8 , i32 = E

√
√√

C1
L11 x3 ,C1
L11 x6 ,C1
L11 x9 .

The normalized circuit equations are described as follows:
3. Proposed system
In this study, we use the circuit model of three degrees of
freedom chaotic circuit in Fig. 2. This circuit consists of twodegrees of freedom chaotic circuit and another Inaba’s circuit
connected in series in order to produce more complex chaos.

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 )).

(3)

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

(4)

4. Results
4.1. Two degrees of freedom chaotic circuit
We show the experimental and computer simulation results
Fig.2 Circuit model of three degrees of freedom chaotic cirof
two degrees of freedom chaotic circuit in Fig. 1. Figurecuit.
3 (a) is the result of the top circuit from circuit experiment.
Figure 3 (b) is the result of the top circuit from computer
simulation. Figure 4 (a) is the result of the bottom circuit fromcircuit experiment. Figure 4 (b) is the result of the bottom
circuit from computer simulation.

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x2

i11v1

v1x1(b)Fig. 5 Chaotic attractors in the top circuit.

Fig. 3 Chaotic attractors in the top circuit.

x5x5

i21v2x1(a)

(b)

(a)

i21x2

i11

v2x4x4(a)(a)(b)
Fig. 6 Chaotic attractors in the middle circuit.

(b)

Fig. 4 Chaotic attractors in the bottom 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], and L22 = 15[mH]. In
the computer simulation, the circuit parameters are chosen as
α = 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.

x8i31v3(a)4.2. 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 chaoticcircuit. Figure 5 (a) is the result of the top circuit from circuit experiment. Figure 5 (b) is the result of the top circuitfrom computer simulation. Figure 6 (a) is the result of themiddle circuit from circuit experiment. Figure 6 (b) is the result of the middle circuit from computer simulation. Figure 7(a) is the result of the bottom circuit from circuit experiment.
Figure 7 (b) is the result of the bottom circuit from computersimulation.

x7(b)Fig. 7 Chaotic attractors in the bottom 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.
4.3. Poincare map
We show simple Poincare maps in Figs. 8 - 10. We examine the behavior of the point when the periodic trajectory repeatedly passes through the cut surface in the computer simulation. Figure 8 (a) is the result of the top circuit in two
degrees of freedom chaotic circuit. Figure 8 (b) is the resultof the top circuit in three degrees of freedom chaotic circuit.
Figure 9 (a) is the result of the middle circuit in two degrees of395

freedom chaotic circuit. Figure 9 (b) is the result of the middle circuit in three degrees of freedom chaotic circuit. Figure
10 (b) is the result of the bottom circuit in three degrees of
freedom chaotic circuit.

x3

x3

x1

x1
(a)
(b)
Fig. 8 Poincare map in the top circuit.

x6x6chaos. We have evaluated the complexity using attractors obtained by circuit experiment and computer simulation, andPoincare maps obtained by computer simulation. Focusing on
the chaotic attractor, more complex chaos is generated in thenew connected circuit. However, when comparing the samecircuits, it seems that there is not much difference. Next,
when the Poincare map is observed, it can be seen that thecomplexity is high because the order is broken, especially inthe top circuit. From the results, we confirmed that the chaos
generated in some circuits are more complex.
As our future work, we will conduct rigorous analysis of
the chaos generated in chaotic circuits and research on various network types. In a rigorous analysis, we will show thebifurcation phenomenon in a diagram and aim to visualize theroute from the periodic solution to the chaotic solution [7]. In
research on various network types, we will investigate interactions between systems [8].
References[1] H. L. Tanaka, “Weather Forecasting and Chaos”, The
Journal of Reliability Engineering Association of Japan28 (7), 481-488, 2006.

x4x4(b)(a)Fig. 9 Poincare map in the middle circuit.

x9x7(b)Fig. 10 Poincare map in the bottom circuit.

[2] J. Liu, “High-Dimensional Chaotic System and its Circuit Simulation”, Telkomnika, Vol. 11, No. 8, pp.
4530 4538, August 2013.
[3] X. Zhang, M. Li, Y. Hong, M. Chen, Y. Shi, L. Huang,and X. Chen, “High-dimensional chaotic laser securecommunication system based on variable laser power”,
Proc. SPIE 11170, 14th National Conference on Laser
Technology and Optoelectronics (LTO 2019), 111703L(17 May 2019).
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[5] M. Sekikawa, T. Hamada, Y. Horio, and K. Aihara, “Integtated Circuits of Shinriki’s circuit and Inaba’s circuit”,
IEICE technical report 106 (136), 1-5, 2006-06-27.

In two degrees of freedom chaotic circuit, the circuit parameters are chosen as α = 0.3, β1 = 10.0, β2 = 2.0, [6] K. Suzuki, Y. Nishio, and S. Mori, “Twin Chaos - Simultaneous Oscillation of Chaos -”, The Institute of Electronβ3 = 20.0, γ1 = 2.0, and ε = 0.01. In three degrees ofics, Information and Communication Engineers (IEICE),
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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.
[7] M. Wada, Y. Nishio, and A. Ushida, “Analysis of BifurLooking at the attractor, the result was similar, but therecation Phenomena on Two Chaotic Circuits Coupled by
was a difference by creating Poincare map. Comparing thean Inductor ”, IEICE Trans. Fundamentals. Vol. E80-A.
figures, it can be seen that when the number of circuits isNo. 5 May 1997.
increased, the order is especially broken in the top circuit.
[8] S. Fujioka, Y. Yang, Y. Uwate, and Y. Nishio, “Asyn5. Conclusion
chronous Oscillations of Double-Mode and N-Phase in
In this study, we have investigated comparison of the coma Ring of Simultaneous Oscillators”, 2013 International
plexity of chaos generated in multiple Inaba’s circuit in seSymposium on Nonlinear Theory and its Applications
ries. We have increased the number of connected subcirNOLTA2013, Santa Fe, USA, September 8-11, 2013.
cuits from two to three in order to produce more complex

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