ICHAOTIC PHENOMENA IN AN LCR OSCILLATORWITH A HYSTERESIS INDUCTOR
Yoshifumi Nishio and Shinsaku Mori
Dept. of Electrical Engineering, Keio University
3-141 Hiyoshi, Kohoku-ku, Yokohama, 223 Japan
Tel. +81-45563-1341 Ext. 3319

ABSTRACT:In this article, an extremely simple LCR oscillator with a hysteresis inductor is analyzed. It is of great interest that this two-dimensional autonomous circuit generates
chaos. This circuit is described by second order differential
equation piecewisely. Therefore, the Poincark map can be derived as one-dimensional map strictly. For this PoincarC map
the parameter region for which chaotic attractors are generated is obtained under simple assumption where the definitionof chaos is that Poincark map possesses the unique absolutelycontinuous invariant measure.

can be derived as one-dimensional map strictly. For thisPoincark map we obtain the parameter region for whichchaotic attractors are generated under simple assumption
where the definition of chaos is that Poincark map possesses the unique absolutely continuous invariant measure[41.

CIRCUIT MODEL
The circuit model is shown in Fig. 1.

INTRODUCTION
Recently, various chaotic phenomena have been analyzed theoretically. Wider knowledge of this interesting
nonlinear phenomena will be needed for many fields. Forexample, when engineering systems need very accurate
control, we have t o consider the influence of the slightnonlinearity of the characteristics of the elements, transfer function and so on. As one effective approach toword
these problems it is important to analyze simple and native chaos generating systems and to make clear the condition of the generation of chaos.
In this aticle, an LCR oscillatot including a nonlinear inductor with hysteresis saturation characteristics isanalyzed. This circuit,model consists of only three elements and it is extremely simple twedimensional autonomous circuit. Generally, for autonomous systems tohave chaotic behaviour, the dimension of the phase spacemust be three or more due to the non-crossing trajectory
property. However, this circuit model generates chaos dueto the existence of the hysteresis loop. Newcomb et al. andSaito have proposed twedimensional chaos-generating autonomous circuits with hysteresis resistor [1]~[3]. However, such circuits are not considered to be in substance
two-dimensional because the hysteresis loop of resistors
generates reactive power. Moreover, in their models current through the hysteresis resistor or voltage across it
jumps when the solution on a branch moves to the otherbranch. On the other hand, in our circuit the dimension of
the system does not increase because the hysteresis loop ofinductors corresponds to resistance. Moreover, any variables in this circuit does not jump and hence this circuit isconsidered to be very native circuit. It is great interestingthat this simple and native circuit generates chaos.
This circuit. model is described by second order differential equation piecewisely. Therefore, the PoincarC map-gFig. 1 Circuit model.
This circuit consists of only three elements, that is, a linear
negative resistor, a capacitor, a nonlinear inductor withhysteresis saturation characteristics. The circuit equationis represented as follows.

{ Ic;:v-z(4)(1)

where U is the voltage across the capacitor, 4 is the fluxof the hysteresis inductor and i(4)is the current throughthe hysteresis inductor. The 4 - i characterist,ics of thehysteresis inductor is shown in Fig. 2. This nonlinear inductor exhibits constant hysteresis saturation characteristics as Fig. 2; no minor loops exist. We call each linear
branch as P+, P-, O+ and 0-.
By changing the valuablues;

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.

T0-p-1-3

01p3+xFig. 3 An example of the chaotic attractors( a = 4, p = 1.3, a = 0.12).
Fig. 24 - i characteristics of the histeresis inductor.
Uthe circuit equation is normalized as follows.

where H(x) represents hysteresis function and is given as
follows.

I

1(1 - a)(1+ P )2( 1 - a)(l- P )on the branch Of : H(x) = x f
2

on the branch P* : H ( z ) = az f(4)The transitional conditions between each branch are given
as follows.
P++0-: x = 1
O-+P+: x = lO-'+P-:
z=-p(5)p-+o+: x=-1o++p-: x z - 1o + + p + :x = p .
Eq. (3) is linear on each branch and the general solution
on each branch can be given. An example of the chaotic
attractors obtained by calculating the general solution ofEq. (3) is shown in Fig. 3.
Though Eq. (3) &second order equation on the eachbranch, the phase space is not simple twwdimensional
plane due to the existence of the hysteresis loop. Therefore, it is difficult to understand the motion of the solution.
Then, consider the state space to which H(z)-axis is a ppended as shown in Fig. 4. Note that H ( z ) corresponds
to the current i through the hysterisis inductor and it is
not a independent variable but a dependent variable.
Fig. 5 shows an example of the chaotic attractor in thestate space in Fig. 4. The motion of the solution is explained as follows. Consider the solution starting from theinitial point on the branch P-. Because both equirihria on
the branches P- and O+ exist near the boundary x = -1,Fig. 4 State space with H(x)-axis.
the solution oscillates between two branches P- and 0'.
Here the amplitude of the solution becomes larger by energy supplied from the linear negative resistor. When thesolution reaches the boundary x = p, it enters the branch
P+. For some proper parameter values, at this moment
the amplitude of the solution decreases and it starts tooscillate between P- and 0-. When the solution rota.tes
the hysterisis loop, energy is consumed as hysteresis loss.

POINCARE MAP
Define the following two half line as Fig. 6.

{

L+ : x = - - 1 , y > o .
L- : x = 1 , y < o .

(6)where L+ is the transitional condition from P- to 0' andL- is that from P+ to 0-. Consider the solution having

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map.

F : L+ U L- + L+ U L-,

yo + F(y0).

(7)

An example of the Poincark map is shown in Fig. 7.

a..\TI

1-Fig. 5 An example of the chaotic attractors(a= 4, p = 1.3, a = 0.12).

-3

03

YoFig. 7 Poincark map F(a= 4, p = 1.3, a = 0.12).

We define ymoz and ymin as Fig. 8. The value of y,(or-ZJ~,~)can be obtained as the y-coordinate of the point
at which the solution starting from (2, y) = (p,0) reaches
L+ (or L - ) via O+ and P- (or P'). Here we define thefollowing intervals on L+ and L-.
J + = [Ymin, ymoz] c L+,
J - = [-ymo=, -ymin] c L-(8)VYo)
Fig. 6 Derivation of the Poincark map.
the initial condition on L+, that is, (I, y) = (-1, yo) (yo >0). At first, define Y A as the y-coordinate of the initialpoint whose flow should be tangent to the transitional
condition from O+ to P+; 2 = p. The value of yA is given
implicitly.
The motion of the solution is distinguished to the following cases (see Fig. 6).

a2Ymin/0Yo(I) when yo < Y A : The solution reaches z = -1 without
reaching the trsnsitional condition 2 = p and enters P-.
The solution in P- reaches L+ again.
(11) when yo 2 Y A : The solution hits I = p and enters
P+. The solution in P+ reaches L-.
Fig. 8 Invariant intervals.

Namely, the solution starting from I/+ never fail to reachL+ or L- at some time. Since L- and L+ locate symmetriIn the case of F(yma,) < -ymal, the solution diverges toinfinity. For these parameter values the energy supplied
from linear negat.ive resistor is considered to be larger thanthat as hysteresis loss.

cally with respect t o the origin, the solution starting fromL- also reaches L+ or L-. Moreover, a point on L+ or Lcan be represented by its y-coordinate. Therefore, we candefine the Poincark map as the following one-dimensional

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\In the case of F(ymoz) > -ymol, interval J + U J - isinvariant with respect t o the mapping F and the solution
starting from J + U J - make an attractor on J + U J - . The
gradient of F on J+ U J - determins the stability of theat t ractor.
The differential coefficient of F is shown in Fig. 9.

Fig. 9 Differential coefficient of F
(a= 4, p = 1.3, a = 0.12).
Then, we assume t$at the following condition is satisfied.

IDivergence[ Condition 1 1

For all parameter values the following equation is satisfied for almost all yo.
(9)

In this case the following theorem is satisfied.
[ Theorem 1 1If the PoincarC map F satisfies the following conditions,
F possesses the unique absolutely continuous invariant
measure and chaos is generated.

(b)
Fig. 10 Chaos generating region(a) ,B = 1.3, (b) ,B = 1.2.

.

The chaos generating region is shown in Fig. 10. In
this figure the boundary between Divergence region andthe region under Divergence region represents the parameter values satisfying F(ym,,) = -ymo+. The boundary
between Chaos region a$d the region located on the left
side of Chaos region represents the parameter values satisfying that t,he left side of Eq. (10) equals to one. As far
as we carry out computer simulations, Eq. (11) is always
satisfied in Chaos region. When a or p is large, that is,
the hysteresis loss is large, chaos is generated for relatively
large parameter region.

generates chaos. Since this circuit is described by second
order differential equation piecewisely, the PoincarC map
can be derived as one-dimensional map strictly. For thisPoincard map the parameter region for which chaotic attractors are generated under simple assumption. Our future research is to carry out circuit experiments to confirm
the theoretical results.

REFERENCES[l] R. W. Newcomb and N. El-Leithy, "A Binary Chaos Generator," Proc. ISCAS'84, pp. 856-859, 1984.
[2] T. Saito, "On a Hysteresis Chaos Generator," Trans. I EICE Japan, Vol. J67-A, No. 12, pp. 1162-1167, 1984.
[3] T. Saito, "On Chaos Generating Circuits Described bya Piecewise-Linear Constrained Equation," Trans IEICE

Japan, Vol. J71-A, No. 4, pp. 965-972, 1988.

CONCLUSIONSIn this article, an extremely simple LCR oscillator witha hysteresis inductor have been analyzed. Though thiscircuit model is twc-dimensional autonomous syst,em, it[4] A. Lasota and J. Yorke, "On the Existence of the Invari-ant Measure for Piecewise Monotonic Transformations,"
Trans. Amer. Math. Soc., Vol. 186, pp. 481-488, 1973.

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