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

Effects of Higher Dimensions on Chaos
in Four Dimensional Hyperchaotic Systems
Tsuyoshi Isozaki, Takumi Nara, Yoko Uwate and Yoshifumi Nishio
Dept. of Electrical and Electronic Engineering, Tokushima University
2-1 Minami-Josanjima, Tokushima 770–8506, Japan
Email: {isozaki, nara, uwate, nishio}@ee.tokushima-u.ac.jp

Abstract—This paper considers the change of behavior withhigher dimensions in four dimensional hyperchaotic systems. In
particular, we focus on the shape and the complexity of chaotic
attractors by changing the number of elements in the circuit.
By means of the computer simulation and circuit experiment,
the changes in the shape of chaotic attractors are investigated.
Further, by means of simple poincaremap, the changes in thecomplexity of chaotic attractors are investigated. From the theseresults, it is shown that the system becomes more complex as thesystem becomes higher dimensions.

I. INTRODUCTION
Chaos is familiar phenomena in our daily lives. For
exmaple, natural phenomena and neurons which build thebrain and so on. From these viewpoints, the study of chaos has
attracted a great interest from various fields such as natural science, biology and engineering. Particularly in the engineering
field, confidential communications [1] are expected due to therandomness of chaos. In addition, chaos has attracted attention
and research as a complex behavior which obtained from a
simple low dimensional system. As the study progresses, it
has been shown that higher dimensions result in more complex
behaviors. In recent years, the studies on the behavior in highdimensional systems and analysis of complex networks [2]-[3]that interconnect systems are conducted.
We focus on the systems that generate hyperchaos. Hyperchaos is generated from high dimensional systems with aminimum dimension of four. Further, the behavior of hyperchaos is harder to predict more than general chaos because
it has at least two positive Lyapunov exponents. Therefore
hyperchaotic systems are expected to be more secure communications [4]. In this study, we use the four dimensional
hyperchaotic system proposed by Nishio et al. [5]. In thissystem, the results show that the inductor and capacitor valuesaffect the generated attractor. Therefore, we change dimensions by increasing the number of inductors and capacitors
and investigate the changes of dynamics.

Fig. 1. Chaotic circuit.

The circuit equations are given as follows:

dv1


C1= i1 − id

dt




dv2



 C2 dt = i2 − i1
 L di1 = v − v


121

dt





di2


 L2 dt = −v2 + ri2 .

(1)

By using the parameters and the variables:√
v1 = Ex1 , v2 = Ex2 , t = L1 C1 τ√L1
Ex3 , i2 =C1√

L1
1i1 =Ex4 , ε =C1G√
C1L1
L1 C1γC =, γL =, α=R.
C2L2L2√

C1L1The normalized circuit equations are given as follows:x˙1 = x3 − f (x1 )








 x˙2 = γC (x4 − x3 )
x˙3








 x˙4

=x2 − x1=−γL x2 + αx4 .

(2)II. EXTREAMELY SIMPLE HYPERCHAOS GENERATORSFigure 1 shows the chaotic circuit which generates hyperchaos [5]. This circuit consists of a negative resistor, twoinductors, two capacitors and one diode.

where the parameter f (x) is the equation for the diode anddescribed as follows:
1(3)f (x) = ε−1 (|x − 1| + x − 1) .
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We set the values of parameters. Computer simulationfor γC = 0.47, γL = 0.4, α = 0.16, ε = 0.01. Circuit experiment for L1 = 20mH, L2 = 50mH, C1 = 0.022µF, C2 =0.047µF .
III. SYSTEM MODELFigure 2 shows the chaotic circuit which one inductor andone capacitor are added to Fig. 1.

L2

L3 i3

L1i2iD

i1

IV. COMPUTER SIMULATION AND CIRCUIT EXPERIMENTFirstly, we compare the shape of attractors from the results
obtained from computer simulations and circuit experiments.
In this study, we measure the values of two capacitors close tonegative resistor and diode in the circuit. Figure 3(a) shows theattractors which obtained from the circuit of Fig. 1. Figure 3(b)shows the attractors which obtained from the circuit of Fig. 2.
Further the figures of left side show the results of circuit
experiment and the figures of right side show the results ofcomputer simulation in Fig. 3. In this results, we confirmedthat the shape of attractors did not change so much.

x2v2-rv3C3C2v2C1v1v1Fig. 2. Proposed model.

(a) Attractors obtained from the circuit of Fig. 1.

By using the parameters and the variables:√v1 = Ex1 , v2 = Ex2 , v3 = Ex3 , t = L1 C1 τ√√

L1Ex4 , i2 =C1

γC1 =C1C1L1L1, γC2 =, γL1 =, γL2 =C2C3L2L31L1 C1, ε=α=RL2G

√

L1
Ex6C1

v1x˙4x˙5 x˙6Fig. 3. Attractors by the computer simulation and circuit experiment.

V. SIMPLE POINCAREMAP= x4 − f (x1 )= γC1 (x5 − x4 )= γC2 (x6 − x5 )= x2 − x1x1(b) Attractors obtained from the circuit of Fig. 2.

C1.
L1

The normalized circuit equations are given as follows:x˙1
x˙2
 x˙3x3v3i1 =√L1Ex5 , i3 =C1

√

x1(4)

= γL1 (x3 − x2 )= −γL2 x3 + αx6 .

We set the values of parameters. Computer simulationfor γC1 = 0.45, γC2 = 0.21, γL1 = 0.5, γL2 = 0.4, α =0.06, ε = 0.01. Circuit experiment for L1 = 10mH, L2 =20mH, L3 = 50mH, C1 = 0.010µF, C2 = 0.022µF, C3 =0.047µF .

Secondly, we compare the complexity of attractors by using
simple poincaremap. The poincaremap is obtained by setting
a plane that crosses the trajectory of the attractor. Further
it is used to analyze the local properties of attractors byplotting points when the trajectory intersects the plane. Forexample, when the attractor has one periodic trajectory, itis expected that the trajectory always intersects the plane at
the same position. On the other hand, when the attractor has
multiple trajectories, it is expected that the trajectories intersect
the plane at various positions. That is, the attractor has acomplex trajectory and many points are plotted on the plane.
In this study, we evaluate the complexity of attractors fromtwo viewpoints using simple poincaremap. First, many points
are plotted extensively. Second, losing the structural order ofthe shape. In other words, the outline of the shape is not clear.
Figure 4(a) shows the attractors obtained from the circuit ofFig. 1. Figure 4(b) shows the attractors obtained from thecircuit of Fig. 2. Further the figures of left side show theattractors and the figure of right side show the attractors ofsimple poicaremaps in Fig. 4. In this results, we confirmed- 30 -that many points are plotted extensively in Fig. 4(b) and theoutline of the shape is clear in Fig. 4(a) but the outline of theshape is not clear in Fig. 4(b). Therefore, it is confirmed thatthe attractors which obtained from the circuits with increased
dimensions become complex.

[5] Y. Nishio, S. Mori and T. Saito “Extremely simple hyperchaos generators
including one diode”, Proc. IEEE international Symposium on Circuits
and Systems, vol. 6, pp. 2797-2800, May 1992.

x4x4x2x2(a) Attractors obtained from the circuit of Fig. 1.

x6x6x3x3(b) Attractors obtained from the circuit of Fig. 2.
Fig. 4. Attractors by the computer simulation.

VI. CONCLUSION
This study considers the change of behavior with higher
dimensions in four dimensional hyperchaotic systems. In particular, we focus on the shape and the complexity of chaoticattractors by changing the number of elements. By means ofthe computer simulation and circuit experiment, the change
in the shape of chaotic attractors are investigated. Further, bymeans of simple poincaremap, the change in the complexity ofchaotic attractors are investigated. Firstly, from the results ofthe computer simulation and circuit experiment, we confirmedthat the shape of attractors did not change so much. Secondly,
from the results of simple poincaremap, we confirmed that thecomplexity of attractors changed with increased dimensions.
R EFERENCES[1] Takami Matsuo, Member, Haruto Suemitsu, Non-member, Kazushi
Nakano, Member “A Design of a Nonlinear Decrypter for Chaotic SecureCommunications”, IEEJ Trans. EIS, vol. 123, No. 9, 2003
[2] Shuhei Hashimoto, Takahiro Chikazawa, Yoko Uwate and Yoshifumi
Nishio “Synchronization Phenomena due to Changes in Network Structure of Coupled Chaotic Circuit in Complex Networks”, Proceedingsof international Symposium on Nonlinear Theory and its Applications(NOLTA’ 17), pp. 522-525, Dec. 2017.
[3] Kenta Ago, Yoko Uwate and Yoshifumi Nishio, “Influence of Local
Bridge on a Complex Network of Coupled Chaotic Circuits”, Proceedingsof international Symposium on Nonlinear Theory and its Applications(NOLTA’ 14), pp. 731-734, Sep.2014.
[4] Xingyuan Wang, Mingjun Wang“A hyperchaos generated from Lorenzsystem”, Physica A: Statistical Mechanics and its Applications, vol. 387,issue. 14, pp. 3751-3758, 1 June 2008.

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