CHAOTIC PHENOMENAIN AN AUTONOMOUS CIRCUIT
WITH NONLINEAR INDUCTOR
Yoshifumi Nishio t, Naohiko Inaba $ and Shinsaku Mori tt Department, of Electrical Engineering, Keio University
3-14-1 Hiyoshi, Kohoku-ku, Yokohama, 223 Japan$ Department of Information Science, Utsunomiya University
2753 Ishii-machi, Utsunomiya, 321 Japan

ABSTRACT: In this article an analytic investigation of thechaotic phenomena in an autonomous circuit which contains anonlinear inductor. The circuit dynamics are described by athree-dimensional piecewise-linear differential equation. In thecase that the nonlinear inductor is assumed to have ideal saturation 4 - i characteristic, the solution of the equation is constrained onto a two-dimensional plane piecewisely. Therefore,
one-dimensional Poincard map can be derived strictly and it isproved to be chaos in the sense of Li-Yorke. The justifiability of
the idealization of the nonlinear inductor is verified theoreticallyand experiment ally.

nomena very well. Especially the PoincarC map is proved to bechaos in the sense of Li-Yorke[S] with computer assistance.
The justifiability of the idealization of the nonlinear inductor
is verified theoretically and experimentally.
The equation discussed in this article is a new type of constrained equations applied t o chaos-generating circuits. The results of this article show that the analyzing method using theidealization of the characteristics of nonlinear elements is also
effect,ive for nonlinear inductors.
VINTRODUCTION
Recently many chaos-generating autonomous circuits havebeen proposed a.nd analyzed. Almost all of these circuits include
nonlinear resistors. There are very few discussions about chaos-.
generating autonomous circuits which do not contain nonlinear
resistors except for the model reported by Akiyama et al.[l].
They have analyzed an autonomous circuit with Josephson element. Though the equivalent circuit of the Josephson element
is usually represented by a linear capacitor, a nonlinear resistor
and a nonlinear inductor, they have verified that chaotic phenomena ca.n be observed in the case that the equivalent circuitof the Josephson element does not contain the nonlinear resistor[l]. However, the 4-2 characteristic of the nonlinear inductorin the equivalent circuit of the Josephson element is represented
by i = sin 4 and the rigorous analysis seems to be difficult,.
In the mean time, Saito and we have proposed the analyzing
method using the idealization of the v-i characteristics of diodes
and have verified that, it is effective to clarify the mechanism ofchaos for some circuits[2]-[4]. In the case that this idealizationis considered, three-dimensional systems are degenerated int,otwo-dimensional systems piecewisely and the phenomena canhe explained by using one-dimensional discrete map.
In this article an autonomous three-dimensional circuit withnonlinear inductor is analyzed. The circuit model is shown inFig. l(a) and it contains no nonlinear resistors. The 4-i characteristics of the nonlinear inductor is shown in Fig. l(b) and themeasured cha.racteristics is shown in Fig. l(c). Note that ournonlinear inductor has negative inductance in the center region
and that it is saturable inductor which is quite different fromt.he nonlinear inductor in the equivalent circuit of the Josephsonelement.
In order to analyze the chaotic phenomena observed from thecircuit, we use a degeneration technique which means that thenonlinear inductor in t,he circuit is assumed to exhibit ideal saturat,ion characteristic. In the case that this degeneration technique is used, the solution of the equation is constrained onto atwo-dimensional plane piecewisely. Therefore, one-dimensional
PoincarC map can be derived strict,ly and it can explain the phe--r

I

(b)

(c)

-4Fig. 1 Circuit. model.

CIRCUIT MODEL
At first, we represent the 4 - i characteristics of the nonlinear
inductor by the following %segment piecewise-linear function asshown in Fig. lib).
(1)

Then the governing equation of the circuit is described by the f d lowing piecewise-linear three-dimensional differential equation.

dvCdt

=io

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wlirre 4, 11 and zo denote the magnetic flux in the nonlinear
iritluctoi, the voltage across C and the current through Lo, re-Since E is very small, A, is negative and large and E: is close by
the boundary z = f l . Therefore, it is considered that trajectories in D* oscillate around Pi flatten onto E; rapidly and hit
the boundary 2 = f l near Lbf.

~pectivrly Note that the inductance L2 in the saturate regionof t he nonlinear inductor is extremely small.
Via t h e iescaling,X

Tt lir no1 malizetl equation IS represented as follows.

= -bf(z)y = z-- -- ( 0 - b ) z - y - b f ( z )xD-Fig. 2 Geometric structure of the vector field.

I ) e li I1 r ,D+ : x > l
Do : 1x1 < 1D-:Figure 3 shows the projections of the chaotic attractor ontoz - y plane and y - z plane. At these parameter values, asym-x<-l.

metric chaotic attractors coexist located symmetrically with respect to the origin. In this case, trajectories starting from D+
do not enter D-. Therefore, for the sake of the simplicity, weconsider the case that the nonlinear inductor has asymmetricq5 - z characteristic as shown in Fig. 4(a). The symmetric caseis considered later.

(6)

Equilibrium in Do is
are represented as follows.

1 ~ ; ~ ~ i i a t i (o 4i i) is linear in the each region.

t Iiv origin 0 and equilibria in D’

P* : ( r ,y , z ) = ( f ( 1 + E a ) , 0,O)

(7)

Iligc~ncquationsare represented as follows.
in,’- ( U - b+ a b ) d + (1 + aub)m- ab = 0

Tin Do (8)

IT0

I0I
I

111 the following discussions, we consider the case that Eq.
( 7 ) (Eq. (8)) has one real and a pair of complex conjugate
cig(.ri\aIiies A,,, g o f ju,, (A,, g, f ju,).
1,f.t E: ant1 E; be the eigenspxes in D* corresponding-1 50-I

1.5-1.5

1 50-

(4(b)
Fig. 3 An example of the chaotic attractor( a = 0.5, /3 = 0.02, a = 0.3 and b = 1).

t he real eigenvalue X, and the complex-conjugate eigenvali i c x nl, k ,7q,, respectively. Similarly, Let E,” and E,” be thetoc.igcrispaces in Do corresponding t o A, and o o f j w 0 ,respect,ively.
‘l’lic rigrnspaces are given by the following equation.

(a) Projection onto z - y plane.

(b) Projection onto y - z plane.

bI:-?: ; { . r k ( 1 + ~ a ) } + ( 1 - o ~ - u ~ ) y + ( 2 o , - a + b ) z = O ,zy-E,”:--lob
A, - abAo(A, - ab),++E; : abz (gi +U,” - 1)y ( U - b - 20,)z = 0.
(10)l~iirtheriiiorelet L t be an int,ersection of E: and x = f l . ThenI,: is represented as follows.
/,;:.1.=fI,( 1 - g,? - w,”)Y(4(b)Fig. 4 The 4 - i characteristics of the nonlinear inductor.

+ (20, - + b ) =~ f a b (11)
UFigure 2 shows the geometric structure of the vector field at(.r3 b)=(O.5, 0.1. 0.3, 1). At these parameter values, eigenv i i I i i ( ~ sa r e caluculated as follows.
((1% E .

A, = -10.97,

A, = 0.37,

C,

IDEAL MODEL
In order to investigate the phenomena generated in our model
deeply, we use a degeneration technique. That is, the case whereLz -+ 0 as shown in Fig. 4(b) is considered. This idealization

fj ~ =, 0.13 f j0.95,

oo fj u o = -0.28 f jl.13.

(12)
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iepiesents that the saturation of the nonlinear inductor is ideal.
In this case, the solution of Eq. (2) seems to be well explained
hy the following Ideal Model.
[Ideal Model]
1 when 4 = @,(in D + )
dv
C-=dt

2Tio

dio
L o x = rio - U .
2. when g!~ < @, (in D")

d4 = -Rio-dt
c-dv =dt

dz0Lodt

R+ ~4Consider the solution having the initial condition o n L,,,. t h a t
is the solut.ion starting from ( z l y,Z ) = (1, go. 0) (yo < 0). Let
A be the value of y at an initial point on L,,, whose flow shoul~lbe tangent to the transitional condition z = 0 from U' t o D J .

zo= (r-R)ro-v+

Note t.liat il in 'D
d$/dt = 0 in Eq. (2).

Fig. 5 Projection of the vector field onto y - z plane.

1R-4.
L1

(J When yo > A : The solution hits L,, again witliout reachin?
the boundary 2 = a .
@ \$'hen yo < A : The solution hits the hountlary L,,, an(lenters Do. Then, it. hits the plane T = 1 and enters L)+ again.
And it ret.urns back to L , again.

can be obtained by using the relation

These equations in D+ and Do are connected by the following
transitional conditions.

D+ - Do: when z, decreases and reaches - @ / L l .
D" D+:when 4 increases and reaches @.

-Therefore, one-dimensional PoincarC map can lie derivol a sfollows.

L'ia the rescaling (3), the normalized equation is represented

F : Lma? follo\vs.

1. when z = 1, (in 0')y=i =(1.5)

2. when T < 1, (in D o )and the transitional conditions are follows.

D+D o: z = a ,Do+ D'

yo - F(Yo)(20)

where yo is the y-coordinate of the initial condition on L,,,a i ~ lF ( y o ) is the y-coordinate of the point on L,,, to which the solw
tion leaving L , returns back.
Usually, we have t o solve implicit functions numerically i norder to obtain the time when solutions hit boundaries. Hence.
we can not represent PoincarC maps explicitly. However. i n t h i >case we can represent the Poincark map explicitly by taking t h camount of time the solution stays in Do a s the parameter.
Figure 6 shows an example of the Poincard map. A t tltese
parameter values it satisfies the Li-Yorlie's period three con(lit ion[5].

z-y + UZ.

- L,,,

n: z = 1.

x(17)

WErT

Though we omit the detailed proof due to space restrictions,
we can verify the justifiability of the Ideal Model by proving thefollowing.
limX, = -cc

a i 00
These equations are proved by expanding the eigenvalues into
the power series of E and show that the vector field of Eq. (4)converges to that of the Ideal Model.

-1.1--+YFig. 6 An example of the PoincarC map F( a = 0.5, a = 0.25 and 6 = 1).
Figure 7 shows the one-parameter bifurcation diagram of thePoincark map F . Figure S shows the experimental and the associated simulated results.
Last,ly, the symmetric case is considered. Figure 9 s h o ~ r san
example of the chaotic attractor in the sy~mnetriccase. 111 thiscase, the one-dimensional Poincard map can be derived sinularly. However, it has infinitely many extrema and its rigorous
analysis seems to be difficult.

POINCAR.E MAP
Fignre 5 shows the projection of the vector field onto y - zplane. Define

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I

0J

0.1

0.2

0.15

0.25

-aFig 7 One-parameter bifurcation diagrams of F

L~-1.5.

1.. -J11

1.5-1.5--tx(3)(b)-Fig. 9 An example of the chaotic attractor in thesynimetric case ( a = 0.5, a = 0.35 and b = 1).
( a ) Projection onto z - y plane.

( I ) ) Projection onto y - z plane.

CONCLUSIONS
tlrk article we have analyzed an chaos-generating au-111

I O I I ~ I I I ~ O circuit
II~which contains a nonlinear inductor. By con-bi(li>riiigtlic cas? that the nonlinear inductor is assumed to haveiil(,iil 5 i i t i i r i i t i o n i - i characteristic, one-dimensional PoincarCiiiiili i'iiii I K(Icri\.ed~
strictly and it is proved to be chaos in the-{'IIW 1 1 1 l,i-1'orlip wit 11 coinput,er assistance. The justifiability ofI III>i ~ l ~ ~ ; i l i ~ i ofi f i tlrro i i nonliiiear inductor is verified theoretically, 1 1 1 ~ 1 r x i w i i t i i c i i t ally.

R E F ER.ENC ES[ I ] li. !\liiyiiiiiii, Ti. Araki and RI. Morisue, "Non-Periodic Oscilliitioii
at.Josephson
AutonomousSystem,"'l'i'cllllici\l Report on Nonlinear Problem of IEICE,vol.
S l,l'-S!)-(i, pp. 35-:39, Sep. 1989.

I'[(4S. lll;il~il,'1'. Saito and S. hlori, "Chaotic Phenomena in a
( ' i r c i i i t \ v i l l i n Negative Resistance and an Ideal Switch ofl)io(li~.~."l'rans. IEICE, vol. E 7 0 , pp. 744-754, Aug. 1987.

[:I] 1'. N i b l i i o , N. Iiialm3 S. hlori and T . Saito, "Rigorous Anal\.>(.> 0 1 \ Y i i i ( I o w in a Symmetric Circuit,," Proc. ISCAS'89,Ill). 2 1 ~ ~ 1 - 2 l 5 419SI).
.

N. Inaha and S. Mori, "Chaotic Phenomena
I~c~iir-Dii~i~risional
Circuit with Two Ideal Diodes,"l)i~oc,I:( '(''r1)'8!1, pp. l.j7-1613 1989.

[4] j'. N i b l i i o .
III[ 5 ] '1'.

(b)-'Fig. 8 Experimental and the associated simulated results( a = 0.5 and b = 1).
( a ) Experimental results.

ii(b) Simulated results.
(1) 1-period(a). = 0.1500,(b)a = 0.2323.

(2) 2-period(a). = 0.2000,(b)a = 0.2760.

(3) 4-period(a). = 0.2150,(b)a. = 0.2855.

(4) Chaos(a). = 0.2400,(b)a = 0.3094.

(5) 3-periodic window (a). = 0.2475,(b)a = 0.3266.

1'. I,i i i l i t l J . i\. Yorlie, "Period Three Implies Chaos,"
11. Monthly, vol. 82, pp. 98.5-992, 1975.

.,\IIII~I'. hlat

(6) Chaos in the sense of Li-Yorke(a). = 0.2600,(b)a = 0.3398.

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