CHAOTIC PHENOMENA IN NONLINEAR CIRCUITSWITH TIME-VARYING RESISTORS
Yoshifumi Nishio and Shinsaku Mori
Dept. of Electrical Engineering, Keio University, Yokohama, JAPAN

There are four circuits satisfying the above conditions andthey are shown in Fig. 1. In the figures TVRs represent TimeVarying Resistors. The resistance R,(t) and the conductanceG,(t) of TVRs vary with time as shown in Fig. 2. The function representing the variation of the TVRs is square wave
with angular frequency wt, and duty ratio p , .
At first, we approximate the U - i characteristics of thediodes by the following twesegment piecewise linear functions(see Fig. 3).

Abstract: In this article, four simple nonlinear circuitswith time-varying resistors are analyzed. These circuits are akind of parametric excitation circuits and chaotic attractors
are observed in three circuits. In order to analyze these circuits a degeneration technique is used, that is, diodes in thecircuits are assumed to operate as ideal switches. Thereby
the Poincard maps are derived as one-dimensional maps andchaotic phenomena are well explained. Moreover, justifiabilityof the analyzing method is confirmed by circuit experiments.

INTRODUCTION
Recently, many chaos-generating systems have been p r e
posed and analyzed in various fields. For parametric excitation circuits, the Duffing-Mathieu equation have been confirmed to generate chaotic attractors by computer calculations

By changing the variables,
U, = v x , ,[l]. However, there are few discussions about the chaos inparametric excitation systems.
In this article, simple nonlinear circuits with time-varying
resistors are analyzed. All these circuits consist of two memory elements, a diode and a time-varying resistor. These circuits are a kind of parametric excitation circuits and chaotic
attractors are observed in three circuits. In order to analyze
these circuits in detail, we use a degeneration technique, thatis, we consider the case that the diodes in the circuits are assumed to operate as ideal switches. In this case, the Poincar4
maps can be derived as one-dimensional maps and chaotic
phenomena are well explained. This analyzing method havebeen proposed by Inaba et al.[2] and have been confirmed tobe extremely effective to analyze some circuits including adiode. Moreover, the justifiability of the analyzing method isconfirmed by circuit experiments.

i, =gvy,,dr’E,

the normalized circuit equations are given as follows.

I. Circuit 1.

{

XI

=

Y1 -Dl(4Yl=-21-Ylfl(r)11. Circuit 2.

{’= -zzfz(.)
Yz = 5 222

C I R C U I T MODEL- Yz - OZ(22)(4)

111. Circuit 3.
Consider the circuits satisfying the following conditions
1. The circuit consists of two memory elements, a diode, andx3=-23f3(7)Y3=

2 3 - D3(Y3)- y3

a Time-Varying Resistor (TVR).

IV. Circuit 4.
2. The circuit is governed by a twedimensional nonlinear
differential equation.

$4=Y4

si4

=

-24

- Y4f4(T) - D 4 ( Y 4 )( 6)

3 . Both of the diode and the TVR are connected in parallel

where f , ( r ) corresponds to the function switching the TVRs
and is shown in Fig. 4. The function O,(.)corresponds to thewith a capacitor or in series with a inductor.
91-645128/92$03.000 1992 E E E

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(a) Circuit 1.

(b) Circuit 2.

L3

I:.

id22T
(a) i = l , 4.

Gi(t)

.
I

2T

(b) i=2, 3.

Fig. 2 Characteristics of the T V F k(c) Circuit 3CL4A.

IR4W

I.

'(d) Circuit 4.

Fig. 1 Circuit model (TVR is Time-Varying Resistor).
(a) i = l , 2.

characteristics of the diodes and is represented as follows.

v d i ( ii)h>Define the following subspaces.

hr,~{{( ( 2 1 ,Y,)IG < 11(2 = 1 , 2 ){(G,Yt)lYt > E * }(2= 3,4),Y J I G 2 11

(2= 1,2){(rt,{(xt, Yt)lYt

i;

5 E t 1 (2 = 3 , 4 ) .

(8)(b) i=3, 4.

Becanse Eqs. (3)-(6) are piecewise linear and f , ( ~ takes)
only two constant values, the general solutions can be given.

Fig. 3 Characteristics of the diodes.
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The experimental and computer calculated results areshown in Figs. 5(a) and (b), respectively. In circuit experiments, TVR is realized by using an analog switch. We foundchaotic attractors from the circuits 1-3. In the circuit 4, wecannot found chaotic attractors and for any parameter values
the solution converges t o the origin or diverges. This reason
is explained as follows. Because the diode in the circuit 4 isconnected in series with the TVR, the stability of the origin
is decided by the sum of resistance of the diode when it is offstate and resistance of the TVR. The resistance of the diodewhen it is off-state is relatively large. Hence, the negative
resistance of the TVR is needed t o be larger value in orderto generate the oscillation, that is, R d 4 < r 4 - . However, if
the oscillation amplitude becomes larger and the diode turns
on, the solution diverges immediately because the resistanceof the diode when it is on-state is relatively small.

Fig. 4 f , ( ~associated)with R , ( t ) and G , ( t ) .

IDEALIZATION OF DIODES0-In the following, we analyze the circuits 1-3 by using adegeneration technique, that is, we consider the case that thediodes in the circuits are assumed to operate as ideal switches.
This degeneration technique have been proposed by Inaba etal. and have been verified to be effective to analyze chaos insome circuits including a diode.
Idealized diodes operate as follows.

XI(a) ( U ] = 0.04, b l = 0.58, w1 = 1.4, p l = 0.5)(b) (a1 = 0.11, b l = 0.60, W I = 1.4, pi = 0.5, €1 = 0.02)'24

1 . The diode takes either ON-state or OFF-state.

2. The diode at ON-state operates as a constant voltage
source with voltage V , and the diode at OFF-state operates as open.
-15 I-10
O -+x2(a) ( a 2 = 0.05, b2 = 1.30, w2 = 1.5, p 2 = 0.5)(b) (QZ = 0.10, b2 = 1.30, w2 = 1.5, p 2 = 0.5,

3. The diode turns off when the current through it becomeszero and turns on when the voltage across it reaches V,.

7

This idealization corresponds to the limit (Gd,, Rd,)+(oo, CO)
in Fig. 3.
In the case that the above idealization technique is used, thecircuit equations in the regions M , are degenerated as follows.
I. Circuit 1.
21 = 1(9)y'1= -1 - Y l f I ( 7 )= 0.01)

11. Circuit 2.

-{

22= 1

y2= 1

111. Circuit 3.
x3(a) ( a 3 = 0.01, b3 = 0.48, w3 = 1.25, p3 = 0.5)(b) ( a 3 = 0.01, b 3 = 0.35, w3 = 1.25, p 3 = 0.5, € 3 = 0.02)

The transitional conditions t o the regions M , from N , aregiven as follows.
I. Circuit 1.

Fig. 5 Chaotic attractors obtained from (1) Circuit 1,(2) Circuit 2, (3) Circuit 3.
(a) Experimental results. (b) Computer calculated results.

Y1 = 0(12)+MT)=0(13)11. Circuit 2.
Y2

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27r

111. Circuit 3.
23=1h4

2

These conditions are derived from the relation id,(v,) = 0(z=I, 2) or vd,(i,) = V , (i=3).
In the case that this idealization method is used, the circuit
equations are degenerated piecewisely and one-dimensional
PoincarC maps can be derived strictly. We explain the derivation of the PoincarC map corresponding each circuit model.

v9"

T
7r

1. In the circuit 1, when the solution enters the region NIfrom MI, namely when the diode turns off, ( q ,yl)=(l,0) is satisfied. Hence, only the phase of the square wave
switching the TVR decides the following motion of thesolution. Then the Poincark map can be derived as onedimensional map which transforms the n-th phase #ln
as shown in Fig. 6.
into the (n 1)-th phase/-+0lr-27rdlnFig. 7 Poincard map (circuit 1).
(al = 0.11, bl = 0.6, w1 = 1.4, p1 = 0.5)

Y1

sow$1"$++I)

Fig. 6 Derivation of the Poincark map.
2. In the circuit 2, the solution in the region MZ must go
through the point ( 2 ,y)=( 1, -aa). Hence, when the solution is on the point (1, -az), only the phase of the squarewave decides the following motion of the solution. Therefore, the PoincarC map can be derived as one-dimensional
map.

-w1

Fig. 8 1-parameter bifurcation diagram (circuit 1).
(a1 = 0.11, bl = 0.6, p1 = 0.5)

3. In the circuit 3, when the solution enters the region NJ
from M3, namely when the diode turns on, (23, y3)=(l,0) is satisfied. Hence, only the phase of the square wave
decides the following motion of the solution. Therefore,the Poincard map can be derived as one-dimensional map.

out circuit experiments and confirmed the justifiability of theanalyzing results.

An example of the PoincarC maps obtained from the circuit
1 is shown in Fig. 7. Though we omit the figures of thosefrom the circuits 2 and 3, their PoincarC maps have many
discontinuity and the orbits seem to be more complicated.
One-parameter bifurcation diagram is shown in Fig. 8. We
can see that the chaotic states and the periodic states appear
in a complicated manner. Moreover, for w z 3 the Poincark
map is homeomorphic and quasi-periodic attractors are also
generated .

REFERENCES[l] R. Tomiyasu, T. Kitagawa and M. Itoh, "Chaotic Solutions in the Duffing-Mathieu's Equation," Trans. IEICF
Japan, vol. J71-A, No. 6, pp. 1337-1338, June 1988.
[2] N . Inaba, T. Saito and S. Mori, "Chaotic Phenomena ina Circuit with a Negative Resistance and an Ideal Switchof Diodes," Trans. IEICE, vol. E 7 0 , pp. 744-754, Aug.
1987.

CONCLUSIONSIn this article, we have analyzed four simple circuits withtime-varying resistors. By using a degeneration technique
we derived the one-dimensional PoincarC maps and confirmedthat three circuits model generate chaos. Moreover we carried
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