181

Twin Chaos- Simultaneous Asynchronous Oscillations of Chaos Katsunori Suzukit, I'oshifumi Nisliio$ and Shinsaku Mori]
Dept. of Electrical Engineering, Keio University
3-14-1 Hiyoshi, Koolioku, Yokoliatna, 223 Japan
Tel: +81-45-563-1141 Email: suzuki@niori.elec.keio.ac.jp$ Dept. of Electrical and Electronic Engineering, University of Tokushirna
2-1 Minami-Josanjirna, Tokushinia, 770 Japan
Email: iiisliio@ee.tokushitna-u.ac.jp

ABSTRACTIn this study, simultaneous asynchronous oscillatioiis ofchaos observed in a simple autononioiis circuit are iiivestigated by both of circuit experiments and coniputer
calculations. This circuit consists of two nonlinear subcircuits having different oscillation frequencies coupledby one linear negative resistor. I n the case that oscillation frequencies of two subcircuits are quite direrent,
both subcircuits exhibit asynclironous chaos siinultaiieously. Namely, the oscillation is alriiost divided into two
different kinds of chaos, one includes many lower frequency components and the other one includes tiiany
higher frequency components. We call this iiiteresting
phenomenon as twin chaos. Moreover, as the dinerent between oscillation frequencies of two subcircuits decreases, two difference kinds of chaos interfere mutually.
Finally, twin chaos disappears and chaos which is tlie
sarite as the case of oiie subcircuit appears dire to tiiutual leading-in.
1

INTRODUCTION

There have been many investigations of siiiiul(aneous
asynchronous oscillators with many degrees of freedom
since the subject had been introduced by van der Pol
who considered a lumped oscillator wit,li t,wo degrees
of freedom [1]-[5]. Schaffner lias considered simultaneous asynchronous multifrequencies oscillator with fifthpower negative nonlinear U - i characteristics [3]. IIe
sliowed that nonresonant two oscillations are possible insuch a system b u t are not self-starting. Disinan and Edison have investigated that self-biasing effects in class-C
oscillator can permit self-starting nonresonant two oscillations [4]. Saito e t a1 have considered simultaneous
asynchronous oscillations in negative conductance oscillator with nortliriear resoiiators [5].
On the other hand, in the study of aiitotioitious chaotic
circuits, three dimensional systems have reasonably wellinvestigated [6]-[9]. Inaba e l a1 have investigated ail autononious chaotic circuit constructed from three iiieniory eieinetits, one linear negative resistor and one diode[7], [a]. In the case that we regard the diode as an
ideal litiiiter [9], tlie switching operation of the diodechanges the oscillation frequency in the circuit considerably and this effect causes a sbretclting arid folding
rnechanisni. We have explained the physical ntechanisrnof the generation of chaos by chatiging oscillation frequency. Recently, four or higher diiriensional cliaotic
systems have begun t o be investigated (see Refs.[lO], [11]and references therein). Although a kind of siinultaneous asynchronous oscillator is a good example of Iiigherdiniensional systerns, studies on chaotic plierioinetia observed i t t such systems have not been reported as far as
we know.
In this study, simultaneous asynchronous oscillationsof chaos observed iri a siiiiple aut,oriorrtous circuit, basedoii one proposed by Inaha e l nl [7] are invest,iga,ted.
'lhe circuit niodel consist,s of two subcircui ts coiishuctedfroni lliree iiierriory eleiiiertt8sand one nodinear resistor
and oiily one negative coiiductance. Dy idealizing thediode [9], the circuit dynamics are simplified. 111 our system, two kinds of cliaos can be observed. One chaos includes many higher frequency components and the otherone includes niatiy lower frequency component's. I n thecase that our circuit exhibits asynchronous chaos siniultaneously, we obtain the spectral characteristics having
two crest,s oil lower frequency band and higher frequency
band. We confirm the existence of siniultaneous two
chaotic oscillations in this spectrum. We call this phenomeiioii as twirl chaos. Further, we carry out circuit
experinient,s in the case that the difference of oscillation
frequencies in both of subcircuits is small. Although theeffect of mutual leading-in occurs in the system, we canalso observe almost twill chaos. We believe tliat chaosgeiterated i t i our systeni is alniost divided iiito t.wo kindsof cliaos OH lower frequency band and higlter frequency

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182

h i d arid tliis result itidicates sotiie elfective suggestiotis
to make clear the chaotic plietionieiia in Iiigher ditiieiisional systems coupled saitie cltaotic circuit.^. 'I'he circuit,l i a s the broad spectrum arid the spectruni is cotit.rolletl
simply aiid arbitrarily by otily varying values of iiieiiiory
elettients. Therefore, we expect tliat, it is possil)le tliat.
this syst,etn applicates a w i s e generator.

Wltett t81iediode is 011, tlie diode is msunied t,o operateas F,'[\,'] coilstatit, voltage source. I3y usiiig tliis idealizalioti, circnib dytiaritics are divided irito sitiiplified fourpiecewise-littear clilferetitial equatiotis.

CI
ClItCUIT MODEL

2

Circuit tnodel is sliowti i t i P'ig. 1. 'I'liis t , w i i i cltaos geiierator is 1)asetl on the circuit. proposed by Itiaba e t n l [7]and consists of t w o subcircuits coupled i t i series a i i t l otie
tiegat,ive cotiductarice. 'l'lte suhcirciiit coiisists of threetiiemory elernetils and one diode.
At first, the i - u characteristics of tlie diode is approximated by following 2-segttientspiecewise-linear ftilic tioti.

C.

-g

DJ

r,

Fig. 1: (Jircuit i i i o d c ~ l . (a) Sutxircriit. ( t ) ) ' I ' w i i i cliaos
generalor.
All diodes iticluded i i i our circriit. are described by b;q. ( I )hecaiise of sariie tliresltold volt age E . llsiitg the following
variables atid paranielers,

3

EXP ERIM EN TAL RES U LT S
Si i i g le S U])c i r c U i t C &se

We cotisider sitigle S I It)circuit case. 'Hiis circuit, is pro
posed b y Itiaha c-1 nl [7] atid lliis circuit, exliibib a pe
riod doubliiig hfiircat.ioii roiit,e to cliaos. 'l'hey explaiiietl
the physical tiiecliatiisriis of the generatioti of chaos bycliangitig oscillat.ioti freque~icy[7], [8]. Figure. 2 sliows
the cliaotic a(,t,ract.orsaiicl its spect>ralcharitct,erist,icsoh

t.aitiet1by circuit experiiiierit,s. We ohserve spectral cltar
ac t e r is t ics i ri Ihe follow i ng co t id i liot is.
0 FF'I' emect.ive satiiple rate : 100.00 [kSa/s]
0 average : 256' l w i t i Cliaos(2)t.lie iiorirtalized equatiotis are tlescrihed as follows.
j:,

=

.(XI

t 2

=
=

21

i 3

i, =

where

+ 2.1)- (22+ E3)

PI ("1 - J[.r,,>

+

$5

=

cr-i(z1
p2.1

z(j

=

P 3 (21 -z 4 ) - Y("5

2E

"6)

l&,)

1

J(z)= -(x++ E- 12 - E l )(41

Here, in order to siuiplify t,lie aiialysis, we consider
the case t,hat the diode i t i t,lie circuit is nssunied t,o operate as aii ideal litnit,er [9]. This idealiza(iori correspotids
to the litnit rd -+ m, i.e. E --+ 0. Wlieti tlie diodeis off, the diode is ~ssurriedto o p e r a k as opeti circuit,.

Next., w e consider t,wiii cliaos g e i i e r a h r . 'I'liis circuit exIiit)its cliaos ori 1 ' 1 - i.5 plaiie via hriis. I~'igiire.:! sltows
t,he cl i aot ic d.t,rac t ors o t)ser vet1 i ii c i r c i t i 1, ex per i i i i ~i 1,s.
i
I t i Fig. :! (I)(a -I ) , cli;totic i t t trac:I.or iricliides tiiaiiy lower
Trr qit e ii e y COIt i poI1 P I I 1,s. I Iowe ver , 1.1i is c 1I aos i i I c I I i t l es i - ~few higlier frequeiicy coriipotteiits. O i i tlie otlier Iiaiitl, i t iFig. 3 ( I)(a-2Il cliaotfic aL(.ract,or iticlrides tiiatiy liiglierfrequeticy coiiipoiieiih. Figure. 4 sliows chaotic a(trach r s ot)served iii n (1 I iter icitl experi I I ieiit s. We o t)bniiietl
tlieiii by using 1Eunge-KiiLt.it iiiet,liod. 'l'lie iiuiiierical re^sults ttiatc11 tlie experittiertt,al ones well qualit,at,ively.
Figure. :! ( 2 ) sliows spechral characterist,ics o1)sPrvetl i i i
circuit experiiiietits wlien (,lie c i r c u i t exliil)it,s clt;tos. I i i
I'ig. 3 ( 2 ) ( a ) ,we cati observe cliaot>icspect,rirtti o t i Icwer
freq~ieiicyharid rroiii (1 [Ilz] t.o at)oiit 5 [kllz] i I i t d periodic otie oti liiglier rreclireiicy 1)aiitI around (1.1 [kllz].
'ltiis spect,rutri is siiriilar tto oiie added spedru tii i i iFig. 2 ( 2 ) ( a ) a.iid i i i Fig. 2 (2)(h). 'L'lie exist.ericeor ahlost,
periodic cliaos on liiglier Treq~ieticyI)aticl is verified arid
bot.li of (:linos oti lower frqiietiry hatid aiid liiglier frequeiicy t):trid are just, aI)otit, (,lie saiiie. A s (Y iii(.leiises aii(1Authorized licensed use limited to: Tokushima Univ. Downloaded on March 14,2010 at 06:49:40 EDT from IEEE Xplore. Restrictions apply.

183

reaches a = 1.027, the spectrum become to be chaotic
spectrum on the whole (see Fig. 3 (2)(b)). However, thespectrum has two crests around lower frequency bandand higher frequency band. On higher frequency band,the existence of almost periodic chaos which is not more
periodic than chaos in Fig. 3 (1) is verified. Therefore,
two chaotic oscillations having quite different frequency
components are simultaneously generated in the twin
cham generator.
Moreover, we consider the case the difference of the oscillation frequencies in two subcircuits is small. Circuit
experimental results for the case that oscillation frequen-cies of two subcircuits are similar are shown in Fig. 5.
By comparison with Fig. 3 (2), each chaotic attractor
is affected by higher frequency or lower frequency components. However, spectral characteristics is two crests
around lower frequency band and higher frequency band.
When two oscillation frequencies are closer, each chaosinterferes each other. Finally, twin chaos disappears andchaos which is the same as the case of one subcircuit appears due to mutual leading-in.

0.0

0.0

24.4

18.8

frcqiiciicy [kllx]

24.4

18.8

Irequeucy [kllx]

..l.-

(a)

(2)

.. .

0.0

24.4

48.8

Irmlueilcy [kl Iz]
frnlitciicy [kl Iz](b)Fig. 2: Experimental results for single subcircuit case.
(1) Chaotic attractor. (2) Spectral characteristics. (a) a= 0.758, PI = 10.0 (C, = 0.022 [pF], 1511 = 500 [mH], ,512
= 50 [mH]). (b) a = 0.774, p1 = 10.0 (Cl = 0.0010 [pF],
1511 = 10 [mH], LIZ = 1 [mH]).

(b)
Fig. 3: Experimental results for twin chaos generator. /31=10.0, p2 = 50.0, P3 = 500.0, 7 = 22.0 (C1= 0.022 [pF], L.11 = 500 [mH], L I Z = 50 [mH], Cz =0.0010 [pF], L21 = 10 [mH], L Z Z= 1 [mH]) (1) Chaotic
attractor. (2) Spectral characteristics. (a-1) Projection
onto v1 - i l l . (a-2) Projection onto u2 - izl. (a) a =0.745. (b) a = 1.027.

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184

1

5 __1----ti“
0in our system if it is possible that the interference isalmost igtiored. Therefore, We believe tliat chaos generated in our system is almost divided iiito N kinds ofcliaos and this results would give the eflective suggestions to make clear the nonlinear pheriomeria in higher
ditiierisional systems coupled same chaotic circuits.
REFERENCES

3[I] B. van der Pol, “An oscillation hysteresis in a triode
generator with 2-degrees of freedom,” Phil. Mag.,
Vol. 43, pp. 700-719, Apr. 1922.

Fig. 4: Numerical results for twin chaos geiierator. c* =0.591, pi = 10.0, /?a = 50.0, 03 = 500.0, 7 = 22.0

[2] D . van der Pol, “The nonlinear theory or electric
oscillatioiis,” Proc. I R E , Vol. 22, pp. 1051-1086,
Sept. 1934.

-8

-3

0
1’1

[3] J . S . Schaffner, “Simultaneous oscillations in oscillators,” IRE IPrans. Czrcuzt Theory., Vol. CT-1, pp. 28, June. 1954.

[4] M. I. Disman and W. A. Edson, “Simultaneous
asyncl~roirous oscilla~ions in class-C oscillators,”
Proc. IRE., Vo1.46, pp. 895-903, May. 1958.

- 65.(J

I

-

- -

IJ.0

.

-

I
IS x

I

24 4
rlrclrlc.llt[ki 1x1

Fig. 5: Spectral characteristics of twin cliaos for nnotlier
parameters. a = 0.701, p1 = 10.0, ,& = 10.0, ,&I =100.0, y = 10.0 CI = 0.010 [ { i t ? ]L11,
= 100 [mH], L I Z= 10 [111H], L z z == 10 [niH], Cz = 0.0010 [pF],
1 [mHI)
4

CONCLUSIONSIn this study, twin chaos generator have been inves1.igated. In the case that oscillation frequeiicies of two subcircuits are quite different, both subcircuits exhibit asyiiclironous chaos sirnultaneonsly. I n circuit experinieiits
we can observe spectral characteristics having two crests
on lower frequency band and higher frequency band. We
confirm that twin chaos is generated in our circuit,. As
a results, twin chaos is almost divided into t,wo different kinds of chaos, one includes inany lower frequency
components and the other one includes inany liiglier frequency components. Moreover as the difference between
oscillation frequencies of two subcircuits decreases, twodifferent kinds of chaos interfere mutually. Finally, twin
chaos disappears and chaos which is the saine as thecase of one subcircuit appears due to mutually leadingin. In order to obtain stable asynchronous N cliaos Iiaving different oscillation frequencies simultaneously, wemust choose quite different oscillatioii frequencies of N
subcircuits. Simultaneous asynchronous N chaos liaving different oscillation frequencies are generated sbably[5] ‘f. Saito, T. Takagi and K . Mano, ~ ‘ A ~ y ~ i ~ l i r o n o u ~Multimode Oscillations in Negative ConductanceOscillator with Nonlinear Resonators,” Traits.
IECE., Vol. 57, A No. 7, pp. 511-518, July. 1974.
(in J apauese).

[GI T. Matsumoto, L. 0. Chua and hl. Koinuro, T h e
Double Scroll,” IEEE ‘I’rans. Ctrcuzl Sysl., CAS-32,
NO. 8, pp 797-818, Aug. 1985.
[7] N . lnaba and S. Mori, “Chaotic Plienornena in a
Circuit with a Diode due to the Change of the Oscillation Frequency,” Trans. IEICE, Vol E71 , No. 9,
pp. 842-849, Sep. 1988.
[8] N . Inaba and T. Saito, “On Physical Mecliaiiisitis of
Chaos in a Third Order Autorioinous Circuit Familywill] a I)iode,” Trans. IElCE, Vol. J74-A1 No. 12,
pp. 1766-1773, Dec 1991. (in Japanese).
[9] N . limba, ‘f. Saito arid S. Mori, “Chaotic Plietio~iiena in a Circuit with a Negative Resistance and an
Ideal Switch of Diodes,” ’3%” IEICE., Vol E70,
NO. 8, pp. 744-754, Aug. 1987.

[lo] T. Mntsuinoto, L 0. Chua and I<. Kobayaslii, “Hyperchaos. Laboratory Experirrient and NuniericalCoi~firrrration,”IEKE ‘lians. Czrcuzl Syst., CAS-33,
No. 11, pp. 1143-1147, Nov. 1986
[ l l ] T. Saito, “An Approach Toward Iligher Uirnensiorial Hysterisis Chaos Generators,” 16EIC 7knns.
Gzrcutl Syst., CAS-37, No. 3, pp 399-409, March.
1990.

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