Chaos in Self-Oscillatory Circuits
Containing Time Delay
Takashi Maruyama, Naohiko Inabat, Yoshifumi Nishio and Shinsaltu Mori
Department of Electrical Engineering Keio University
3-14-1 Hiyoshi, Kohoku-ku, Yokohama, 223 JapantDepartment of Information Science IJtsunomiya University
2753 Ishii-machi, Utsunomiya, 321 Japan
ABSTRACTThis paper gives rigorous analysis of chaos in two typesof self-oscillatory circuits containing time delay. The governing equation of each circuit is described by 2nd-order diffrrrntial equation containing time delay. By using mapping
mrthod, we can derive the 1-dimensional Poincark map explicitly from each circuit, and the Poincart. map is proved tohave a positive Liapunov number with computer assistances.
Moreover the proposed circuits are realized easily so that thetheoretical results can be verified experimentally.

jI

1

1C i B E R I T 0 F,

(1) Circuit-1(2) Circuit-2
Fig.1

Circuit models

1. INTRODUCTION

Many self-oscillatory systems containing time delay havehecn studied. Recently, chaotic phenomena are observed insuch systems[l]~[3].For example, Ueda et al.[2] and Koudaet a1.[3] have investigated the Minorsky's system in detail.
The Minorsky's system is an example of self-oscillatory systems, and the governing equation (the Minorsky's equation)
is described by the nonlinear difference-differential equation.
It exhibits chaos when that system has the large time delay[2] or the higher-order nonlinear characteristics[3], andthesc results are verified by the computer simulations. However the theoretical evidence of chaos has not yet been clarified, since it is very difficult to solve the nonlinear differencediferential equation strictly.
On the other hand, 1-dimensional mappings are mathematically studied in detail. For example, the sufficient condition where the mapping has a positive Liapunov number
is given by Lasota, Li and Yorke[4][5].
In this paper, we propose two types of self-oscillatory
circuits containing time delay which exhibit chaos, and analyze them strictly. The governing equation of each circuit isdescribed by 2nd-order differential equation containing timedelay. By using mapping method, the Poincark map can bederived explicitly as a 1-dimensional map from each circuit.
Furthermore we obtain the chaos generating condition where
the mapping has a positive Liapunov number, and show thechaos generating region in parameter space by computer calculations. Lastly the theoretical results are verified by circuit
experiments.

Fig.2 shows the switching operation and the pulse generating operation.

v,,(a) if v <the switch is connected to -g (circuit-1)
the pulse is not added (circuit-2)(b) if o 2 &,the switch is connected to G (circuit-1)
the pulse(V) is added (circuit-2)
In the case where time delay(Td) is contained in this operation] chaotic phenomena are generated in each circuit.

,Av/')<lthreshold voltage
it is chosen properlyiI,!

.',,

2. CIRCUIT MODELS
Fig.2
Fig.1 shows the circuit models. In this figure, -g is alinear negative conductance, and each constant is selected sothat o can vibrate. Each circuit consists of a -gLC oscillator
and a loop which controls an amplitude of circuit.

Switching operation (Circuit-1)
and pulse generating operation (Circuit-2)

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The normalized equation of the circuit-1 is represented
as follows:2+ f(z(T -7d))z+z = 0(1)and f(u) is switching operation and is represented bywhere

On the other hand, the normalized equation of the circuit2 is represented as follows:
2 - 262 f z = f(z(7 - T d ) )

(2)

-X+

and f(u) is pulse generating operation and is represented by
Fig.4

a ifu>10 otherwise

Vector field on the z - i: plain

where- g) = - 2 6 ,&G

V . G = LY.

As an initial condition, the following case is considered:Fig.3 shows an example of the chaotic attractor generated
in each circuit.

(7.,z, 2 ) = (O,zo, 0).

(3)

w e define L and L t h as

L = { z , Z I z < 0 , z = 0)
L t h = (2, z I = 1).
Furthermore let M be the value of z at an initial point on
L whose flow should be tangent to the line L t h . M is represented explicitly as{M = - exp -$(rW1- 0 ) } where O = arctan -.

61(4)Now we discuss the solution whose initial condition isrepresented by Eq.(3). The following movements of the sclutions are divided into two cases (I) and (11).

(I) Case M < 20 < 0 ( 0 in Fig.4)
The solution crosses L again at r = 2*/wl without
reaching the threshold L t h .
Fig.3(11) Case zo < M ( Q in Fig.4)
The solution starting from L crosses L t h at (r,z,z) =(71,z1,1). The solution continues to rotate divergently
around 0 until 7 = q T d . The solution crosses L t hagain at (r,z,z ) = ( T ~ T ~x2,, 1). Then, the solution
rotates convergently around 0 from (T,z, z ) = ( T ~Tdj 23, 2 3 ) to (7172
Td, 2 4 , 24).
For the sake of simplicity, we consider the case where(z4,Z4) satisfies the following condition:

Chaotic attractor++++ +3. DERIVATION

OF POINCARE MAP AND ITSANALYSIS

The Poincark map of each circuit can be derived explicitly as a 1-dimensional map. In this section, we explain thederivation of the Poincark map and its analysis by using thecircuit-1.
Fig.4 shows the vector field on the z - Z plain.

2 4> O , Z 4 < 1 orz4

< 0, i 4 < 0.

When this condition is satisfied, the solution starts torotate divergently and crosses L again at (7,z,Z) =(71+ + +72

73

T d , z5,o).

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Therefore, the Poincark map F can be defined as a 1dimensional map:

F : L -+ L ,As concerns the Poincar6 map F, we obtain below threeequations:

XO + F(xo)zo-+M+Olim

where 2 0 is the z-coordinate at the initial value on L andF ( z o )is the z-coordinate at the point on L which the solution leaving L returns back. F can be represented explicitly.
However the concrete representation of F is omitted due tospace restrictions. Fig.5 shows an example of F .

dzo

= exp

(s)
>1

[ M < xo 5 F2(M)]

(8)

(9)

dF
[w17;1 # 81.
(10)dxo
Eq.(8) shows that the value of F is continuous at M . Eqs.(9)and (10) show that the differential coefficient of F is discontinuous at M .
In the following discussions, we assume that the following
condition is satisfied:lim

W+M+O--1.3

F ( z 0 ) = zo+M+O+
lim F(xo)//-- -COThis assumption is verified by computer calculations.
Then, ifcldzo

Fig.5

<-1
io=F(M)is satisfied, Eq.(7) is satisfied. Therefore we regard Eq.(12)as the chaos generating condition. Fig.6 shows the chaos
generating region in two-parameter space.
Fig.7 shows oneparameter bifurcation diagram.
Similarly, it can be proved that the circuit-2 exhibits
chaos in the sense of theorem-1.
Lastly some of the typical experimental results and thecorresponding simulated results are shown in Fig.8. In theseexperiments, 61and Td are fixed, and 62 is used as a parameter. Experimental results match the theoretical ones very
well both qualitatively and quantitatively.

PoincarC map(61 = 0.015, 62 = 0.9, W I T d = T )In discussing about F , the definition is given.
[Definition 11An interval J in L is said to be invariant if F ( J ) c J.
And the invariant interval J is said to be stable if there
is some interval J' 3 J and some positive integer nsuch that F " ( X ) c J for all X E J', where F" denotes#n-times composite of F .
In the following discussions, we consider the case where
the two extrema M and S satisfy the following condition(see Fig.5):
F ( M ) > S.
(5)
DefineJ = [F(M), F 2 ( M ) ] .

(6)

If Eq.(5) is satisfied, the interval J is invariant and stable,
and further F has a unique extremum F ( M ) in J .

0. 3 4
7r[Theorem 1][4][5]
We assume that F : J + J . Ifs00. D l 5

Fig.6

Chaos generating region(WITd = T )61

is satisfied. It is known that F has the unique absolutely continuous invariant measure and also F is#ergodic.

-0. 0 3

Then if Eq.(7) is satisfied, the mapping has a positive
Liapunov number. Therefore, in this paper we regard thatchaos is generated when Eq.(7) is satisfied.
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4. CONCLUSIONS

We hzve proposed two types of self-oscillatory circuits
containing time delay which exhibit chaos and have analyzed
them strictly. By using mapping method, we could derive the1-dimensional Poincard map explicitly from each circuit, andthe Poincark map was proved to have a positive Liapunov
number with computer assistances. Moreover the theoreticalresults were verified by circuit experiments.

REFERENCES[l] T . Maruyama, N. Inaba, Y. Nishio and S. Mori : "Chaos
in an Auto Gain Controlled Oscillator Containing TimeDelay", Trans. IEICE, vol. J 72-A, pp.18141820, Nov.
1989

Fig.7

[ 2 ] Y. Inoue and Y. Ueda : "Self-Modulation Phenomena

I-parameter bifurcation diagram(61 = 0.015, w l r d = T )in a Self-Oscillatory System Containing Dead Time",
Trans. IEICE, vol. J 64-A, pp.791-792, Sep. 1981.
[3]A. Kouda and S. Mori : "Oscillations in a Self-Oscillatory
System Containing Time Delay", Trans. IEICE, vol.
J 72-A, pp.309-318, Feb. 1989.
[4] A. Lasota and James A.Yorke : "On the existence of in-variant measures for piecewise monotonic transformations", Trans. Amer. Math. Soc., ~01.186,pp.481-488,
Dec. 1973.
[ 5 ] Tien-Yien Li and James A.Yorke : "Ergodic transforma-tions from an interval into itself', Trans. Amer. Math.
Soc., ~01.235,pp.183-192, Jan. 1978.

I-/

#?

~Fig 6-2

Periodic S O I U L I ~ I I
w i c h period 2( A ) 6:. = 0.140(B)6: = 0.1'24
i-I -Fig 8-3 Periodic solution
w:b period 3( A ) 6 , = 0.190(B)62 = O.li3

Fq.b---. Chaos( A ) 6 , = 0 570( B ) 6: = 0.55C

Fig.8 Experimental results(B)and corresponding simulated results(A)(6,= 0.015, u1Td = r )1058

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