EXTREMELY SIMPLE HYPERCHAOS GENERATORSINCLUDING ONE DIODE
Yoshifumi NISHIO t, Shinsaku MORI f, and Toshimichi SAITO $t Dept. of Electrical Engineering, Keio University
3 Dept. of Electrical Engineering, Hosei University
ABSTRACT:In this article, two extremely simple hyperchaos generators including one diode are analyzed. In the casethat we regard the diodes as ideal switches, the 2-dimensional
return map can be derived rigorously and its Lyapunov exponents can be calculated by using piecewise exact solutions. Especially, it is confirmed that the resonant frequencies governing
the diod are deeply related with the generation of hyperchaos.

the circuit dynamics are described byINTRODUCTION
In the study on autonomous chaotic circuits, threedimensional systems have reasonably well investigated, but
four-dimensional systems have only recently begun to investigated [I]-[5]. Higher-dimensional chaotic circuits include
various important problems, among them; physical meaningof chaos, classification of chaos and mathematical evidence
for chaos. In order to approach such problems, we consider
a hyperchaos generating circuit family which consist of four
linear memory elements] some positive and negative linear
resistors, and a diode. Hyperchaos is usually defined asa chaotic attractor with more than one positive Lyapunov
exponent. It cannot be observed from three-dimensional
autonomous circuits. Our circuit family are extremely simple hyperchaos generators] because nonlinear element in thecircuit is only one diode. In [3][4], we have proposed another h perchaos generators. However, the circuit analyzed
in [ 3 1 d are more complicated than our circuit family. Because the circuit in 31 includes a hysteresis resistor and thesolutions jump one ranch to the other one. Also, the circuit in [4] includes two diodes and hence it is three-region
system.
In t,his study we consider two ladder circuits in the family.
At first, me derive the 2-dimensional Poincar6 map by using the simplification of the diode. This sim lification havebeen proposed by Inaba and the authors [$ and its effectiveness has been confirmed for some circuits [4][5]. We canverify that these circuits generate hyperchaos by calculatingthe Lyapunov exponents of the Poincark map. Especially,
we can confirm that the resonant frequencies governing thediodes are deeply related with the generation of hyperchaos.
There are very few discussions related with such physicalexplanation of the generations of hyperchaos.

5,CIRCUIT MODEL

IFig. 1 shows two circuit models. These circuits consist of
only four memory elements, a linear negative resistor andone diode. In the circuits only the position of the diodediffers. At first, we approximated the v - i characteristicsof the diode byiD = 0.5G(lVk - El -t Vk - E)

Fig. 1 Circuit model.
(a) Circuit 1.

(1)

Using the following variables and parameters:

t=GT,v l = Exll(b) Circuit 2.

v 2 = Ex2,(c)- i characteristics of the diode.

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1. Circuit 1

2. Circuit 2I

1{l42r.3
24= 23-f(%1)= YC(Z4 - 5 3 )= 2 2 - 21
= - y L x 2 + cYx4

where=

$1

22 =
43 =
24=f ( x ) = O . ~ E - ' ( / X - 11( 6 ~+) (6b) : X I = 1(66) + ( 6 ~ ) : 2 3 = 0(7a) -+ (7b) : 2 2 = 1(7b) -+ ( 7 ~ ) : 2 4 - 2 3 = 0

23

YC(Z4 -x3-f(x2))
x2 - 21-YLx2+cYx4(3)+ x - 1)

(4Here, me regard the diodes as ideal switches; the diodebecomes E[V]voltage source if its branch voltage reaches
E[V]and it is opend if the branch current reaches zero. Itcorresponds t o E -+ 0 and (3) and (4)are simplified into

1. Circuit 1For x1 < 1For x1 = 1

2. Circuit 2For x2 < 1For x 2 = 1=2.3= 22 -21= -7LX2 + C Y 2 424

When (6b) switches into (sa), x1=1 and x3=0 are satisfied. Also when (7b) switches into (7a), 2 2 = 1 and x3-x4=0
are satisfied. Therefore, 2-dimensional Poincark ma T canbe derived using the exact solutions of (6) and (77 rigorously. Also, the Jacobi matrix DT of T can be derived.
Fig. 2 shows chaotic attractors of the 2-dimensional m a pping obtained from the Circuit 1 and corresponding trajectories. Note that Circuit 1 tends to exhibit chaos only for
L1C1<L2C2.Fig. 3 shows the Lyapunov exponents. 1stand 2nd Lya unov exponents, p l > p 2 , are calculating by
the following771

(5)

$1
22=

POINCARE MAP23Yc(24-23)(b){x.121324=

23=
1 - 21= -7L
ffx4+

where the transitional conditions are follows.
3x"

T00+x20-33+ Ex"t0I1I0-3-)XI

3: [0.5V/div]

vq : [0.5V/div]

Fig. 2 Chaotic attractors obtained from the Circuit 1 for LICl < L2C2.
( 7=~ 7c = 0.5 (L1 = 10mH, L2 = 20mH, Cl = 0.022pF, C2 = 0.044pF))(a) a = 0.13, (b) a = 0.18.

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Fig. 6 shows chaotic attractors of the 2-dimensional
mapping obtained from the Circuit 2 and corresponding
trajectories for LlCl>L&2. Fig. 7 shows the Lyapunov
exponents. In this case hyperchaos can be observed for0.161a10.26.
We carried out many numerical experiments for someandcombinations of "~r,and confirmed the following:
Circuit 1 generates hyperchaos only for LICl<L2C2.
Circuit 2 generates chaos for LICl<L2C2 and generates hyperchaos only for L ~ C I > L ~ C ~ .
The above results denote that the resonant frequency governing the diode is deeply related with the generation ofhyperchaos. Though this results may not be universal for
any types of the circuits, we consider that this study would
open the way to make clear the physical mechanism of thegeneration of hyperchaos.

pl>O implies chaos and p2>0 implies hyperchaos. For(u>O.19 the solution diverges to infinity. We can confirm
that the Circuit1 exhibits hyperchaos for 0.175aL0.18.
Fig. 4 shows chaotic attractors of the 2-dimensional mapping obtained from the Circuit 2 and corresponding trajectories for L1C1<L2C2. Fig. 5 shows the Lyapunov exponents. In this case chaotic att,ractors can be observed. However, Fig. 5 denotes that hyperchaos cannot be generat,ed,
because li?<O for any a.
.$+0.0= t + + ++t ,+ t '.

CO"S0.00.1

In this article, two extremely simple hyperchaos generators including one diode have been analyzed. Especially,
it is confirmed that the resonant frequencies governing thediod are deeply related with the generation of hyperchaos.
Our future research are to make detailed bifurcation diay a m and to investigate the relation between the resonant
requency governing the diode and the generation of hyperchaos for another circuit models.

+ a+ : 1st-exponent (pi)o : 2dim-exponent (pl+ p2)

Fig. 3 Lyapunov exponents obtained fromthe Circuit 1 for LICl < L2C2.

REFERENCES[l] E. N. Lorenz, "The Local Structure of a Chaotic Attractor
in Four Dimensions," Physica 13D, (1984) 90.

( 7=~7c = 0.5)
2.5

4x"
tx"

00-2.5
2.5, _ ___ T----r0+ X I 1r-----1-4041

I+ X I 4Iv2 : [0.5V/div]

Fig. 4 Chaotic attractors obtained from the Circuit 2 for LICl < L2Ca.
(TL = 1, TC = 0.3235 (Li= Lz = 10mH, C1 = 0.022pF, C2 = 0.068pF))(a) Q = 0.10, (b) Q = 0.17.
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[2] T. Matsumoto, L. 0. Chua and K . Kobayashi, "Hyper-[5] Y.

[3] T. Saito, "An Approach Toward Higher-Dimensional Hys-

[GI N. Inaba, T. Saito and S. Mori, "Chaotic Phenomena in a
Circuit with Negative Resistance and an Ideal Switch ofNishio, N. Inaba and S. Mori, "Chaos in a FourDimensional Autonomous Circuit with Two Diodes,"Trans. IEICE Japan, J74-A, 4, (1991) 681.

chaos: Laboratory Experiment and Numerical Confirmation," IEEE Trans., CAS-33, 11, (1986) 1143.

teresis Chaos Generators," IEEE Trans., CAS-37, 3, (1990)
399.
[4] T. Saito, "The Dead Zone Conductor Hyperchaos Gener-ator," Electronics and Communications in Japan, 73, 6,(1990) 58.
lSI-.-,~I

Diode," Trans. IEICE, E70, 8, (1987) 744.

[7] I. Shimada and T. Nagashima, "A Numerical Approach toErgodic Problem of Dissipative Dynamical Systems," Prog.

Theor. Phys., 61, (1979) 1605.

+I0.0++0-1.5

-1.5

0.1

0.0

+ : 1st-exponent

(p1)

. 0.1

0.0

--+a0.2-+a+ : 1st-exponent ( p l )+o : 2dim-exponent ( p 1 + p2)

o : 2dim-exponent (pl p2)

Fig. 7 Lyapunov exponents obtained fromthe Circuit 2 for LICl > L2C2.

Fig. 5 Lyapunov exponents obtained fromthe Circuit 2 for LICl < L2C2.

( 7=~ 1, 7c = 1.545)(7,5 = 1, 7c = 0.3235)

2.5

xm
T02.5 -II

41x"
Tx"
T00-2.50--+x1 2.5

I0-4-+XI

4

VI : [0.5V/div]
v2 : [0.5V/div]

Fig. 6 Chaotic attractors obtained from the Circuit 2 for LICl > L2C2.
( 7=~ 1, 7c = 1.545 ( L 1 = L.2 = 1 0 4 3 , C1 = 0.068pF, Cz = 0.044pF))(a) a = 0.14, (b) a = 0.26.
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