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

Investigating Dynamical Behavior with Higher Dimensions
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

2. Extreamely simple hyperchaos generators

This paper considers the change of dynamical behavior withhigher dimensions in four dimensional hyperchaotic systems.
In particular, we focus on the shape and the complexity ofchaotic attractors by changing the number of inductors andcapacitors in the circuit. By means of the computer simulation and circuit experiment, the changes in the shape ofchaotic attractors are investigated. Further, by means ofpoincaremap, the changes in the complexity of attractors areinvestigated. From the these results, it is shown that the system becomes more complex with the system becomes higherdimensions.

Figure 1 shows the chaotic circuit which generates hyperchaos [6]. This circuit consists of a negative resistor, two inductors, two capacitors and one diode. This circuit is a systemthat can generate hyperchaos very simple because nonlinear
element in the circuit is only one diode.

1. Introduction
Figure 1: Chaotic circuit.
Chaos is phenomena that exist close to us. For example,
natural phenomena and neurons which build the brain and soon. From these viewpoints, the study of chaos has attracted agreat attention from various fields such as natural science, bi- The circuit equations are given as follows:
ology and engineering. Particularly in the engineering field,
confidential communications [1] which using the randomnessof chaos, and a brain-type computer that incorporates braindv1
= i1 − idC1
information processing into a computer are expected and re
dt
searched. Further in these studies, analysis of high dimen
dv2



sional systems that combine many elements is important. In C2 dt = i2 − i1
recent years, the studies on the behavior in high dimensional
(1)
di1

systems and complex networks [2] - [3] that interconnect sysL1= v2 − v 1dttems are conducted.
di2
We focus on the systems that generate hyperchaos. Hy L2 dt = −v2 + ri2 .
perchaos is generated from high dimensional systems with aminimum dimension of four [4] - [5]. In this study, we use
the four dimensional hyperchaotic system proposed by Nishio
et al. [6]. In this system, the results show that the resonant where the v − i characteristics of the diode are approximatedfrequency governing the diode is related with generation of as follows:
hyperchaos. Therefore, we change dimensions by increasing
the number of inductors and capacitors and investigate the1(2)iD = G (|vk − E| + vk − E) .
changes of dynamics.
2389

By using the parameters and the variables:
v1 = Ex1 , v2 = Ex2 , t =√v1 = Ex1 , v2 = Ex2 , v3 = Ex3 , t =L1 C1 τ√L1
1Ex4 , ε =C1G
√
C1L1
L1 C1
γC =, γL =, α=R.
C2L2L2i1 =L1
Ex3 , i2 =C1√

By using the parameters and the variables:√√

C1L1i1 =γC1 =C1C1L1L1, γC2 =, γL1 =, γL2 =C2C3L2L3L1 C11, ε=α=RL2Gx˙3







 x˙4

=x3 − f (x1 )=γC (x4 − x3 )=x2 − x1=−γL x2 + αx4 .

√L1Ex6C1C1.
L1

The normalized circuit equations are given as follows:(3)x˙1x˙2 x˙3

where the parameter f (x) is the equation for the diode anddescribed as follows:
f (x) =

L1
Ex5 , i3 =C1√

L1
Ex4 , i2 =C1√

The normalized circuit equations are given as follows:
x˙1








 x˙2

√

√
L1 C1 τ1 −1ε (|x − 1| + x − 1) .
2x˙4







x˙5







 x˙6(4)=x4 − f (x1 )=γC1 (x5 − x4 )=γC2 (x6 − x5 )=x2 − x1=γL1 (x3 − x2 )=−γL2 x3 + αx6 .

(5)We set the values of parameters. Computer simulationfor γC = 0.47, γL = 0.4, α = 0.16, ε = 0.01. CirWe set the values of parameters. Computer simulation
cuit experiment for L1 = 20[mH], L2 = 50[mH], C1 =for γC1 = 0.45, γC2 = 0.21, γL1 = 0.5, γL2 = 0.4, α =0.022[µF ], C2 = 0.047[µF ].
0.06, ε = 0.01. Circuit experiment for L1 = 10[mH], L2 =At this time, in the circuit of Fig. 1, the value of inductors
20[mH], L3 = 50[mH], C1 = 0.010[µF ], C2 =and capacitors are set such that L2 > L1 , C2 > C1 .
0.022[µF ], C3 = 0.047[µF ].
At this time, in the circuit of Fig. 2, the value of inductors
and capacitors are set such that L3 > L2 > L1 , C3 > C2 >C1 .
3. System model
Figure 2 shows the chaotic circuit which one inductor andone capacitor are added to Fig. 1.

L2

L3 i3-r

C3v3

L1i2

C2v2

i1iD

C1v1

Figure 3 shows the chaotic circuit which two inductors andtwo capacitors are added to Fig. 1.

Figure 2: Proposed model 1.

Figure 3: Proposed model 2.

390

By using the parameters and the variables:from the circuit of Fig. 1. Figure 4(b) shows the attractors
which obtained from the circuit of Fig. 2. Figure 4(c) shows
the attractors which obtained from the circuit of Fig. 3. Further the figures of left side show the results of circuit experiment and the figures of right side show the results of computer simulation in Fig. 4. In this results, we confirmed thatthe shape of attractors does not change so much. That is, it isconfirmed that a slight increase in dimensions do not significantly affect the shape of the attractors.

v1 = Ex1 , v2 = Ex2 , v3 = Ex3 , v4 = Ex4
√

√

i1 =

L1
Ex5 , i2 =C1

L1Ex6C1√√√
L1
L1i3 =
Ex7 , i4 =
Ex8 , t = L1 C1 τ
C1C1γC1 =C1C1C1, γC2 =, γC3 =C2
C3C4v1

The normalized circuit equations are given as follows:x˙1x˙2
x˙3
 x˙4x˙5 x˙6



x˙7





 x˙8

x2v2

L1L1L1γL1 =, γL2 =, γL3 =L2L3L4
√
√
L1 C1
1 C1α=R, ε=.
L2G L1x1(a) Attractors obtained from the circuit of Fig. 1.

= x5 − f (x1 )x3v3= γC1 (x6 − x5 )= γC2 (x7 − x6 )= γC3 (x8 − x7 )= x2 − x1

v1x1(6)(b) Attractors obtained from the circuit of Fig. 2.
(Proposed model 1)= γL1 (x3 − x2 )= γL2 (x4 − x3 )v4= −γL3 x4 + αx8 .

x4

We set the values of parameters. Computer simulation
v1for γC1 = 0.47, γC2 = 0.21, γC3 = 0.1, γL1 = 0.5, γL2 =x10.25, γL3 = 0.1, α = 0.03, ε = 0.01. Circuit experimentfor L1 = 5[mH], L2 = 10[mH], L3 = 20[mH], L4 =(c) Attractors obtained from the circuit of Fig. 3.
50[mH], C1 = 0.0047[µF ], C2 = 0.010[µF ], C3 =(Proposed model 2)
0.022[µF ], C4 = 0.047[µF ].
At this time, in the circuit of Fig. 3, the value of inductors Figure 4: Attractors by the computer simulation and circuit experiand capacitors are set such that L4 > L3 > L2 > L1 , C4 > ment.
C3 > C2 > C1 .
4. Computer simulation and circuit experiment

5. Poincaremap

First, we compare the shape of attractors from the results
obtained from computer simulations and circuit experiments.
In this study, we measure the voltage values of two capacitors
close to negative resistor and diode in each circuit and show
to attractors. Figure 4(a) shows the attractors which obtained

Second, we compare the complexity of attractors by usingpoincaremap. The poincaremap is obtained by setting a planethat crosses the trajectory of the attractor. Further it is used
to analyze the local properties of attractors by plotting points
when the trajectory intersects the plane. For example, when391

the attractor has one periodic trajectory, it is expected that thetrajectory 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 atvarious positions. That is, the attractor has a complex trajectory and many points are plotted on the plane. Focusing on
this property, we evaluate the complexity of attractors using
poincaremap. Figure 5(a) shows the attractors obtained fromthe circuit of Fig. 1. Figure 5(b) shows the attractors obtained
from the circuit of Fig. 2. Figure 5(c) shows the attractors
obtained from the circuit of Fig. 3. Further the figures of left
side show the attractors and the figure of right side show theattractors of poincaremaps in Fig. 5. In this results, we confirmed that many points are plotted extensively in Fig. 5(b)and Fig. 5(c). Therefore, it is confirmed that the attractors
which obtained from the circuits with increased dimensions
become complex.

6. Conclusion
This study investigated the change of behavior with higher
dimensions in four dimensional hyperchaotic systems. In particular, we focus on the shape and the complexity of chaotic
attractors by changing the number of inductors and capacitors. By means of the computer simulation and circuit experiment, the change in the shape of attractors are investigated.
Further, by means of poincaremap, the change in the complexity of attractors are investigated. First, from the resultsof the computer simulation and circuit experiment, we confirmed that the shape of attractors did not change so much.
Next, from the results of poincaremap, we confirmed that thecomplexity of attractors changed with increased dimensions.

[1] Takami Matsuo, Member, Haruto Suemitsu, Nonmember, Kazushi Nakano, Member “A Design of aNonlinear Decrypter for Chaotic Secure Communications”, IEEJ Trans. EIS, vol. 123, No. 9, 2003.

x4x4[2] Shuhei Hashimoto, Takahiro Chikazawa, Yoko Uwateand Yoshifumi Nishio “Synchronization Phenomena
due to Changes in Network Structure of Coupled
Chaotic Circuit in Complex Networks”, Proceedings ofinternational Symposium on Nonlinear Theory and its
Applications (NOLTA’ 17), pp. 522-525, Dec. 2017.

x2

x2(a) Attractors obtained from the circuit of Fig. 1.

x6[3] Kenta Ago, Yoko Uwate and Yoshifumi Nishio, “Influence of Local Bridge on a Complex Network of Coupled Chaotic Circuits”, Proceedings of international
Symposium on Nonlinear Theory and its Applications(NOLTA’ 14), pp. 731-734, Sep.2014.

x6x3[4] Xingyuan Wang, Mingjun Wang “A hyperchaos generated from Lorenz system”, Physica A: Statistical
Mechanics and its Applications, vol. 387, issue. 14,
pp. 3751-3758, 1 June 2008.

x3(b) Attractors obtained from the circuit of Fig. 2.
(Proposed model 1)x8[5] Xia Huang, Zhao Zhao, Zhen Wang, Yuxia Li “Chaosand hyperchaos in fractional-order cellular neural networks”, Neurocomputing, vol. 94, pp. 13-21, 1 October 2012.

x8x4References[6] 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.

x4(c) Attractors obtained from the circuit of Fig. 3.
(Proposed model 2)

Figure 5: Attractors by the computer simulation.

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