CHAOTIC
ITINERANCYPHENOMENA
ON COUPLED
WDOUBLESCROLLSCHAOTIC
Masahiro
Dept. of Electrical

WADA,

Yoshifumi

NISHIOCIRCUITS

and Akio USHIDA& Electronic Eng., Tokushima University, Tokushima
Tel: +81-88-656-7470,
Fax: +81-88-656-7471,
E-mail: wada@ee.tokushima-u.ac.iD

ABSTRACTaIn this study, chaotic itinerancy
phenomena on coupled chaotic circuits are investigated.
A piecewise linear element included in a chaotic circuit consists of4n - 1 segments.
In this circuit, we can confirm ndouble scrolls or various scrolls attractors.
The featuresof the obtained results are respected for various engineering systems such as chaos communication
systemswith robustness against various interferences.

G770, JAPANbL

NR

Figure1: Chua’s circuit

1. INTRODUCTION
Many types of coupled chaotic circuits have been researched as one of the most interesting
phenomena of
the nonlinear
circuits in the electricalfields.
Manychaotic circuits constructedby electrical elements arealso existing.
Since they had been investigatedin detail up to now, and many paper have reported someinteresting
chaotic phenomena[l-51.
Chua’s circuit isone of them, which is well known chaotic circuit as
we can observe a variety of chaotic behavior.
There
are some papers which treated the case of coupled two
Chua’s circuits[l,2]or many coupled Chua’s circuits[3].
Further, complicatedchaotic behavior
were found inseveral coupled systems. In large scale coupled chaotic
circuits we can see spatio-temporalchaotic phenomena
and chaotic itinerancy
phenomena.
On the other hand,
Suykens et al.[6] proposed a concept which they modify
the characteristicsof the nonlinear resistor included in
the Chua’s circuit due to generate n-double scrolls, andthey utilized it to the CNNs by one dimensional
arrayof the circuits[7].
By using such method, more complicated chaotic behavior can be generated in simpled
chaotic circuits. However, there seems to be no reporton n-double scrolls by circuit experiment.
Moreover,coupled systems of the n-double scrolls have not been
investigatedin detail.
Firstly, we consider a modified Chua’s circuit which
can generate n-double scrolls chaotic attractors.
Wecarry out both numerical simulationand circuit experiment, then we realize 2-double scrolls chaotic attractor
on electrical circuits.
Secondly, we consider the modified Chua’s circuits coupled by one resistor.
In this0-7803-5474-5/99/$10.00 1999 IEEENR

Figure2: Chua’s

diode with piecewise

linearsegments

study, chaotic itinerancy
phenomena on coupled modified Chua’s circuits are investigatedin detail.
Moreover we calculate autocorrelation
function of the modified Chua’s circuit. We consider that such complicatedchaotic waveforms are expected to be utilized for realizat ion of several chaotic applications
such a chaos
communication
system with robustness against various
interferences
including multi-useraccess.

2. CIRCUITMODELSIMULATION& CIRCUITAt first, we consider the Chua’s circuit which cangenerate n-double scrolls attractor.
The concept of thiscircuit was proposed by Suykens et al[6]. When a piecewise linear resistor incl uded in the circ uit is approximated by 4n - 1 segments, we can obtain n-double
scrolls attractor.
The circuit diagram is illustratedin Fig. 1. This
circuit consists of only three memory elements, one re-V-487

Figure 3: 2-double scrolls chaotic attractors.
a = 10.0,
13 = 0.265. m = [-1.15, -0.71, -1.75, -0.711, c = [l.O,2.2, 3.01.
Figure 5: Realizationof piecewise linear resistor byusing O.P.Amps.
OP Amp type is TL082CP,
RI = R2= 21.7[k0], R3 = 1.69[kfl], R4 = R5 = 237[Q], R6 =1.56[kSt], R7 = Rg = 5.06[kfl], Rg = 1.46[kS1], thresholdvalue of the diodes is about 3.O[V].

(a >(b)(a >Figure 4: 3-double scr 111schaotic attractors.
a = 10.0,,!3 = 0,265. (a) m = -1.15, -0.73, -1.35, -0.73, -1.30,-0 731, c = [l.O, 2.2, 3.5, 5.5, 8.01, (b) m = [-1.15, -0.78,-1:35, -0.78, -1.23, -0.781, c = [l-O, 2.5, 3.5, 5.0, 6.51.
sistor and one nonlinear element.
The nonlinear
element is defined as piecewise linear as shown in Fig.
2 and it is shaped symmetricallyfor the origin point.
If we choose adequate parameters,
we can confirm ndouble scrolls chaotic attractors.
The circuit dynamics of general Chua’s circuit areas follows:

i = WqY - x - f(x))
ti = P(x - y) + 2
i = -ywhere characteristicsof the nonlinear
circuit parameters
areresistor(1)
and the2n-1
f(X)= &2n-lX+ix(si-l-si)((X+CiI-/X-CiI)

(2)

i=lt

=WlJEg,

= Vx,

c‘l”

=-$

vc2 = Vy,iL

(3)where rni and ci are values of the slope of segments andbreakpoints,respectively.
V is any value, hereafter V isFigure 6: 2-double scrolls chaotic attractor
obtainedfrom the circuit experiment.
Cl=
4.45[nF], C2 =49.3[nF], L = 5.25[mH], l/G = l.O6[kfl].
(a) characteristic of Chua’s diode, v&N/div]
vs. &R[lOmA/div],(b) vcr[5V/div]
vs. vc&V/div],(c) v&N/div]
vs.
iL [5mA/div],(d) iL [5mA/div]
vs. vc2 [lV/div].

refixed at l.O[V]. W e sh ow some numerical calculation
sults obtained by using the Runge-Kutta
method withtime step size Ar = 0.001.
Fig.
3 shows 2-double
scrolls attractors.
We can see such chaotic attractors
when adequate parameters
are given. Similarly, we canobtain 3double
scrolls attractors
as shown in Fig. 4.
Thus we can obtain n-double scrolls attractors
easily.
In this study, we carried out circuit experimenton
We realized the nonlinear
resistor
electricalcircuits.
by using OP. Amps, some linear resistor and diodes asshown in Fig 5. Some experimental
results are shown inFig. 6. We can realize 2-double scrolls chaotic attractors. In circuit experiments,
it is difficult normally torealize a nonlinear resistor which has a lot of segments
by using only electrical elements.
However, if we will
be able to realize that, we will obtain more complicatedchaotic phenomena on real electrical circuits.

V-488

3.

CIRCUITSIMULATIONFORCHUA’SCIRCUITSCOUPLEDIn this section, we focus on complicatedchaotic phenomena on coupled modified Chua’s circuits.
Here weconsider coupled modified Chua’s circuits connected byone resistor as shown in Fig. 7. Our aim is to investigate chaotic itinerancy
phenomena on this simple
coupled chaotic circuits. This coupled modified Chua’s
circuits is very simple because the system is governedby six dimensional
piecewise linear equations.
We assume that two modified Chua’s circuits arecoupled by one resistor G,, and we defined a new following parameter of coupling:y = GxJLc(4)(G, = l/R,).

2

For computer calculations,in order to consider the difference of real circuit elements, the circuit equation of
coupled modified Chua’s circuits is written by using anadditionalparameter
Aw as follows:
x*k=d(Yk - Jtk- f (xk))!ik;I,-==

P(xk-{I- Yk) + xk + (-l)“?‘(yl+ (k - l)Aw}yl,-92)

to O.5[& and vertical corresponds to the value of autocorrelation
function.
We can confirm the value ofautocorrelation
function in the case of 3-double scrolls
converge faster than the double scrolls case. It means it
has robustness for various interferencesin the 3-double
scrolls case. Therefore,
we consider that it can makea communication
system with robustness for variousinterferencesby using n-double scrolls features.
4.

CONCLUSIONSIn this study, we investigatedchaotic itinerancy
phenomena on coupled chaotic circuits which have piecewise linear elements. We can confirm n-double scrolls
chaotic attractors on computer simulation or circuit experiments.
Chaotic itinerancy
phenomena on coupledchaotic circuits have been investigated.
The autocorrelation functions of both 3-double scrolls and double
scrolls were calculated.
We consider that such complex
chaotic waveforms with chaotic itinerancy
are expected
to be utilized for realizationof chaos communication
systems with robustness against various interferences
includingmulti-useraccess because of its quick decayof correlation
functions.

(5)

where, k = 1 and 2. Hereafter, Eq. (5) is calculated byusing the Runge-Kutta
method with time step size AT= 0.002 and Aw is 0.01.
We show some numerical simulation
results. Figs.
8 and 9 show chaotic phenomena of the 2-double scrolls
case and 3-double scrolls case, respectively.
From theleft of the attractors,
attractorof one modified Chua’s
circuit, attractorof the other modified Chua’s circuitand synchronization
state, respectively.
When we focus
on the synchronization
state of both attractors,
we canconfirm that the state of two variables of xl and x2 arewandering on the xl - 22 plane. Namely, chaotic itinerancy phenomena were confirmed in this system. We
can also confirm several interesting
chaotic phenomena
such as chaos synchronization,
bursting of chaos synchronizationand chaotic itinerancy
phenomena.
In order to clarify stabilityof synchronization
between bothcircuits, we investigate
the relation between existenceof extime and place.
Fig. 10 shows a probability
istence on the xl - x2 plane for 3-double scrolls case.
Vertical axis means time 7 where synchronization
state
exist in the correspondingarea. These figures indicatethat frequency of chaotic itinerancy
can be controlled
by tunning coupling parameter value.
We also investigate
autocorrelation
function of thismodified n-double scrolls Chua’s circuit.
In Fig. 11,
autocorrelation
results obtained from computer calculation are shown.
Fig.
11(a) and (b) are 3-double
scrolls case and double scrolls case, respectively.
Horizontal corresponds to time which time distance is set

Acknowledgment
This work was supported
Research Grant.
5.

by The Mazda

Foundation’s

REFERENCESand L.O. Chua, Int. J. of Bifurcation and Chaos, vol. 4, no. 2, pp. 477-482, 1994.

[l] T. Kapitaniak[2] V.C. Anishchenko,
T.E. Vadivasova,
V.V. Astakhov,O.V. Sosnovtseva,
C.W. Wu and L.O.
Chua, Int. J. of Bifurcation and Chaos, vol. 5, no.
6, pp. 1677-1699, 1995.
[3] A.P. Munuzuri,
V. Perez-Munuzuri,
M. GomezGesteira, L.O. Chua and V. Perez-Villar,
Int. J.
of Bifurcation and Chaos, vol. 5, no. 1, pp. 17-50,
1995.
[4] Y. Nishio and A. Ushida, IEEE Trans. on Circuitand Systems-I, vol. 42, no. 10, pp. 678-686, Oct.
1995.
[5] Y. Nishio and A. Ushida,1037-1040, 1997.

Proc. of ISCAS’97,pp.

Suyk ens and J. Vandewalle,
IEEE Trans.
on Circuit and Systems-I, vol. 40, no. 11, pp. 861-

[6] J.A.K.

867, Nov. 1993.
[7] J.A.K.
J. Vandewalle
Suyk ens, A.P. Muiiuzuri,and L.O. Chua, Proc. of NOLTA’97, pp. 205-208,
1997.

V-489

. ++
1ZNR
7G?iG

irdtc, vcl vc-2 c,Figuresistor

7: Modified

Chua’sLLcircuits

coupledby one re16000
14000
12000
lOOO0r

A

A

8000
6000
4000
2000

(a )

(b)

050000
45000
4oOO0
35000
30000
25000
20000
15000
1oOOo
5000
004
._
Figure 8: Z-double scrolls case. Chaotic itinerancy
phenomenon on coupled modified
Chua’s circuits which
NR has 7 segments. a = 10.0, p = 0.265, (a) y = 0.150(b) y = 0.300 (Rx e 1.26[kSl]),CRX N, 2.52[k0]),

10-1

Figure 10: Probabilityof existence on the q -Q planefor 3-double scrolls case. Total time [T]: 106. (a) y =0.10, (b) Y = 0.15, (c) y = 0.30.

(a >(b)(a )0II-0.02'--0.04

I0C-ItI’0‘1100200’

“1300

400

1’

500

600

700

800

900100E

J

I
300

I
400

I
500

I600

I
700

I
800

I900100E““““‘I0.02
0(b)-0.02

Figure 9: 3-double scrolls case. Chaotic itinerancy
phenomena on coupled modified Chua’s circuits which NB
has 11 segments. cy = 10.0, p = 0.265, (a) y = 0.100(c) yCR 21 3.78[kR]), (b) y = 0.150 (Rx ftl 252[kG]),= i.300 (Rx N 1.26[kR]).

-0.04
0Figurescrolls,

V-490

100200

11: Autocorrelation(b) d ou bl e scrolls.

function
Horizontal:

for (a) 3-double
time[ x 0.5~1.