IEEE TRANSACTIONSON CIRCUITSANDSYSTEMS,VOL.

37, N0.4,

APRIL

1990

473

Rigorous An&yses of W indows
in a Symmetric Circuit
YOSHIFUMINISHIO,NAOHIKOINABA,SHINSAKU

Abstrocr -Two types of windows are found in a symmetric circuit, whichcannot be seen in the logistic map and are inherent in symmetric structureof this circuit. One type of window exhibits complex bifurcation phenomena such as the symmetry breaking and the symmetry recovering in its own
region, while the other type of window appears when one chaotic attractor
bifurcates to two distinct periodic attractors. The purpose of this paper isto derive a one-dimensional Poincare map from the circuit by the degeneration technique and to prove rigorously that two types of windows appear
alternately and infinitely many times inside some windows by applying acertain scaling mechanism on the Poincare map.

I.

INTRODUCTIONYMMETRIC
structures are often observed in thechaos-generating circuits [l]-[8]. For example the V-I
characteristic of nonlinear resistance in the circuit discussed in [2]-[4] is symmetric with respect to origin. The
circuit equations of these systems are invariant to thetransformation (x, y, z) + (- x, - y, - z). The “symmetry” is one of the features of the chaos-generating circuits.
Therefore, it is of great importance to investigate thebifurcation phenomena based on the symmetry of circuits,
such as the symmetry breaking, the symmetry-recovering[3], and so on.
In the meantime, some symmetric one-dimensional maps
have been studied [9]-[ll].
For example, Testa et al. [ll]investigated the cubic map and discovered two types of thewindows. May [9] has introduced the term “window” forthe interval in parameter space in which the periodic
solution remains stable [12]. It is called “window” since
the area of its existence on a one-parameter bifurcation
diagram is placed in the area of chaos like windows. The
two types of the windows discovered by Testa et al. [ll]cannot be seen in the logistic map, namely, one exhibits
complex bifurcation phenomena such as the symmetry
breaking and the symmetry recovering in its own region,
while the other is the windows appearing when one chaotic
attractor bifurcates to two distinct periodic attractors [ll].
These windows have interesting bifurcation structures andseem to be deeply related to bifurcation phenomena in

S

MORI, AND TOSHIMICHI

symmetric real physical systems. However, there are few
discussions to fill- in the gap between the windows inmathematical models and those of real physical systems.
In this paper we analyze two types of the windows in asymmetric circuit described by a differential equation in arigorous way. These windows are similar to those appearing in the cubic ‘map. For understanding our study easily,
at first we introduce some experimental and numericalresults.
Let us consider the circuit in Fig. 1 whose structure issymmetric with respect to the origin. This circuit consistsof three memory elements, one linear negative resistance,and one nonlinear resistance consisting of two diodes. The
negative linear resistance is realized by using the linear
region of the negative impedance converter made of an
operational amplifier. The measured I-V characteristic of
the nonlinear resistance is given in Fig. 2(a). W e approximate the I-V characteristic by the following 3-segment
piecewise-linear function as shown in Fig. 2(b):Then the circuit dynamics is described by the following
piecewise-linear third-order ordinary differential equation:di,L,z=u+ri,di,L2x=u-ud(i,)(2)
By changing the variables such that
i, =

g

c
,I

vx;

i,=-

JL,c

-=a.
L,

vy

=2

1

d--

r
Manuscript received December 5, 1988; revised September 3, 1989.
This paper was recommended by Associate Editor T. Matsumoto.
Y. Nishio and S. Mori are with the Department of Electrical Engineering, Keio University, Yokohama 223, Japan,
N. Inaba is with the Department of Information Science, Utsunomiya
University, Utsunomiya, Tochigi 321 Japan.
T Saito is with the Department of Electrical Engineering, Housei
University, Koganei-shi, Tokyo 184, Japan,
IEEE Log Number 8933900.

SAITO

=1

-=
’ L,

p

.
)

.

u=vz;

t=&c

’
!m
-=
rdL,

.
y9

-=;
.

(3)

(2) is normalized as

0098-4094/90/0400-0473$01.00

i=ax+z3=z-f(Y)i=-x-py
01990 IEEE

(4

474

IEEE TRANSACTIONS

Circuit model.

v,/(b)
=F
V

rd

0

ii

-v

(a)

AND

SYSTEMS,VOL.

37, NO. 4, APRIL 1990

Type 2 which associates with Fig. 4(i). In Fig. 5, for theparts (ai) and (b,), we give an initial condition (x, y, z) =(- l,O,O) for (4) and plot the x-coordinate of the crosssection when the flow intersects the plane of z = -0.2.
Next we increase the parameter (Y and choose the point at
which the calculation at the previous parameter value ends
as a new initial condition. For the parts (a2) and (b,), wegive an initial condition (x, y, z) = (l,O,O) and performthe similar calculation. The difference between these figures means that two distinct attractors coexist. In order tocharacterize these windows, some symbols are introduced:+=Hw
Fig. 1.

ON CIRCUITS(W

Fig. 2. The 1-V characteristic of the nonlinear resistance in Fig. 1.
(a) Measured I-V characteristic. (b) Approximated I-V characteristic.

where

Pl:
P2:
Cl:
C2:

state where one periodic attractor exists;
state where two distinct
periodic
attractors
exist;
state where one chaotic attractor exists;
state where two distinct chaotic attractors exist.
(6)

By using above symbols, these windows are represented asfollows (see Fig. 5).
The window of Type 1:
Cl + Pl + P2p=gc2
Window

Fig. 3 presents experimental results and Fig. 4 presents
the associated computer simulation of (4). All parametervalues are denoted in each figure where parameter r or (Y isvaried. The resistance of the diode when it is off is denotedby rd, and it is determined by measuring the I-V characteristic and applying the method of least squares [7]. Since(4) is piecewise-linear, the general solution in each region
can be obtained. The times when the solution crosses theboundaries are obtained by solving the implicit equations
numerically. There exists small amount of errors between
the experimental results and numerical ones. It seems to be
due to the stray capacitance of the diodes. These results
agree well at least qualitatively. Equation (4) is justifiablein discussing the circuit in Fig. 1.
At the parameter value in figures (b)-(e) and (f) of Figs.
3 and 4, there exist two distinct attractors whose flows
have different initial conditions. Because of the symmetryof the system, these two attractors locate symmetricallywith respect to the origin. The circuit exhibits a variety ofinteresting phenomena such ‘as the chaotic phenomenon,
the symmetry-breaking
transition phenomenon [13], thesymmetry-recoveringcrises [13] and some windows [12].
Especially, the difference between the shape of the windows in Fig. 4(g) and (i) draws our attention. The periodic
window in Fig. 4(g) is symmetric, while the periodic window in Fig. 4(i) is asymmetric. We call the window in Fig.
4(g) the window of Type 1 and also the window in Fig. 4(i)the window of Type 2. These two windows seem to haveconsiderably different natures. In order to investigate detailed structures of these two types of the windows, we
make the one-parameter bifurcation diagrams from calculating (4). Fig. 5(a) presents the window of Type 1 whichassociates with Fig. 4(g). Fig. 5(b) presents the window ofGenerating+ Cl + Cl
Regionand the window of Type 2:
Cl+p2Period-DoubliygC2
Window

Generating--) cl.
RegionThese windows are characterized as interesting phenomena
where the n.umber (one or two) and the nature (periodic or
chaotic) of the attractor change in a complicated manner,which are considered to be inherent in the symmetricstructure of the system.
The purpose of this paper is to analyze these windows
rigorously by using a degeneration technique shown in [7],[14]-[17]. This degeneration means that the diode in thecircuit is assumed to operate as an ideal limiter. In thiscase, we can derive a one-dimensional Poincare map described implicitly by using the general solution of thepiecewise-linear differential equation and it well explains
the above-mentioned phenomena. Especially, it is proved
rigorously, that two types of the windows appear alternately and infinitely many times inside some windows
under some reasonable assumptions by utilizing a certain
scaling mechanism on the Poincare map. That is, it isclarified that the number and the nature of the attractor
change in a complicated manner. In addition, we confirmed that the generation of windows is concerned with acertain universal constant [18], [19] from the numericalcalculation.
We also confirmed that the two types of the windows
appear in some simple symmetric systems, for example,
double scroll equation [2], Sparrow’s equation [20], and soon. It seems that the two types of windows are one of thetypical behaviors which characterize symmetric chaos-generating systems.

NISHIO

et id.:WINDOWS

IN A SYMMETRIC

475

CIRCUITI(4

Fig. 3. Experimental results projected onto (il, u)-plane. L, = 603.2 mH, L, = 206.5 m H and C = 0.0069 PF (p = 2.92).
(a) r = 3.11 kQ
a = 0.287. (b) r = 4.38 kG!
(Y= 0.423. (c) r = 4.82 k0
a = 0.470. (d) r = 4.90 k0 . a = 0.478.
(e) r = 507 kQ
a = 0.496. (f) r = 5.24 kO
a = 0.515. (g) r = 5.50 k0
a = 0.542. (h) r = 5.76 kB
a = 0.570.
(i) r = 5.86 k0
a = 0.581. (j) r = 5.95 k0
a = 0.590.

II.

IDEAL MODEL AND GEOMETRIC STRUCTURE O F
THE VECTOR FIELDS

Though the mapping method is usually used for the
analysis of chaos, it is very difficult to prove the generationof chaos in this case because the system to be analyzed is a
two-dimensional discrete dynamical system and the mathematical theories for two- or higher dimensional discrete
dynamical systems are not sufficiently developed. Therefore, in order to investigate the phenomena observed fromour circuit, we must use the degeneration technique shownin [7], [14]-[17], which means that the nonlinear resistancein the circuit operates as an ideal limiter as shown in Fig.
6(a). An equivalent circuit of the nonlinear resistance isshown in Fig. 6(b). W e have already verified mathematically that if the operation of the diode is sufficiently sharp,

the behavior of the solution of a circuit family including a
diode can be explained by idealizing the diode [14], [16],[17]. The idealized equation is given by the following.
Ideal Model@

When i, >. 0 ( ud(i2) = V) is satisfied:di,L,x=ri,+u

duCz = - i, - i,.

476

IEEE TRANSACTIONSON CIRCUITSANDSYSTEMS,VOL.

37, NO. 4, APRIL

1990

(4
(f)

(b)

1

(d)

(4
(j)

T

0

Fig. 4. Numerical results obtained by calculating (4). L, = 600 mH, L, = 200 mH, C = 0.0069, RF, and rd = 1.46 MO(/3 =3.0 and y = 470). (a) OL= 0.150. (b) a=0.275.
(c) a= 0.300. (d) a= 0.308. (e) a=0.350.
(f) cy=O.400.
(g) a = 0.412. (h) OL= 0.460. (i) a = 0.471. (i) a = 0.490.

@

When i, = 0 (Iud( i2) 1c V) is satisfied:connected by the followingdi,L,~=ri,+u

When i,<O( ud(i2) = - V) is satisfied:di.
L,$=ri,+u
di,
L,-$=u+V
C~=-i,-i,.

These equationsin the regions

conditions:@ + @j: when the current i, (through the nonlinear
resistance) decreases and becomes 0;@) + @J: when the voltage u (across the nonlinear
resistance) increases and reaches a constantv;
0 + 0: when the voltage u (across the nonlinear
resistance) decreases and reaches a Constant- v:@ + a): when the current i, (through the nonlinear
resistance) increases and becomes 0.
(9)CYz = - i,.
@transitional

(8)
0,

0,

and

0

By changing the variables in (3), the Ideal Model
are , normalized as the following.

(8) isNISHIO et al.: WINDOWS IN A SYMMETRICCIRCUiT(4

Fig. 6.

b,

Idealization of the nonlinear resistance in Fig. 1. (a) Idealized
I- V characteristic. (b) Equivalent circuit.

Fig. 5. One-parameter bifurcation diagrams obtained by calculating (4).
(Subscripts 1 and 2 denote different initial conditions.) (a) The windowof Type 1. (b) The window of Type 2.

(4

(f)

(h)

(d)

fsi L
(4
(3

T
-2

d

0

0
+X

2

Fig. 7. Simulated results obtained by using the ideal model (10). L, = 600 mH, L, = 200 mH, C = 0.0069 PF (/3 = 3.0).
(a) a=O.150. (b) o= 0.275. (c) a=0.300. (d) cu=O.308. (e) a= 0.350. (f) a = 0.400. (g) (Y= 0.412. (h) a = 0.460. (i)(Y= 0.471. (i) a = 0.490.

47xIEEE TRANSACTIONS

0

When y>O
[“I

@

r a

0

qrx+1/aAND

SYSTEMS,

31, NO. 4, APRIL

VOL.

1990

Since the circuit equation is piecewise-linear, the generalsolutions in each region are explicitly given as follows.

1

where

When y<O
0

1;-1
The transitional

ON CIRCUITS

X-l/a

Ii

Y + l/dz+lconditions

fd 4

FIW=
(10)
1.

are also represented byfi,( 7)fi,(d Ifiyb) =?‘p’, e ‘p7 sin wpr, e0p7cos wpr

dfl,( 4>f&)fi,W = 7
Q-0:

z=-1,

@-)@I

y=o.

01)

Fig. 7 l&esents computer simulation results of (10).
These results agree with those of (4) both qualitatively andquantitatively.
This agreement shows that this method
using the Ideal Model is justifiable to analyze the circuit inFig. 1.
In the following discussions we analyze the circuit inFig. 1 by using the Ideal Model of (10).
This equation contains two parameters, (Y and /3, and issymmetric, that is, (10) is invariant with respect to thetransformation(x, y, z) + (- x, - y, - z). Define:D+={(x,y,z):[;I{(x,y,z):

y=O}

. D-=

{(x,y,z):

y<O}.

[ I

fi,( 7) = [e”JsinW,7,dfzr( 7)f2,b)

X3-&+(/3+1)h-a/3=0.

[

(15)

z,+1

(16)

1

where x0, yo, z. are initial conditions.
Let E’( P * ) and E*( P * ) be the eigenspaces in D *corresponding to.the real eigenvalue A, and the complexconjugate
eigenvalues
respectively.
The
up + ja,,eigenspaces are explicitly given by the following equation:E’(p*):x T xp
7=X-=

ZTl

Y r ypP(y

hp(Xp-4

x3X,-(A;-aXp+l)(yrYp)(13)

are roots of the

Yo+yp

z+l

E*(P*):On the other hand, A, and up * jw,
following eigenequation:

I

x0+xp

=Fl(~)XF~'(0)XP

In the following discussions, we consider the case that 0
has a pair of complex-conjugate eigenvalues a, -t jw, andboth P+ and P.- have one real and a pair of complex-conjugate eigenvalues A, and up + jo,. Then a, and o,, areexpressed as

2,"e”JcosW,r]

x-txp
i 1

0

P*=(+X,,kY,,kl)=
0 = (O,O,O) c D”.

i=---jy.

P

The subspaces D+, D”, and D- represent the regions 0,
0, and @ in (lo), respectively. Equation (10) is piecewise-linear and it has a unique equilibrium in each region.
Let P+ be the equilibrium in D+ and also let 0 and Pbe the equilibria in Do and D-, respectively. The equilibria are explicitly given by=F~(~)xF~'(O)xfi,( 7)
Fiw = f*,(T)y ,f Y02)‘dfd d7.

= -Pf*,(d-

where

y>O}

Do=

1

+(a-A,)(zR)
We define the following

=o.

(17)

subspaces.

B’={(x,y,z):z=fl,y=O}c’*=((x,y,z):D”nE*(P*)}.

(18)

The subspace B’ means the transitional
condition@‘- 0
while ‘Bmeans the transitional
condition
Q-0.
Fig. 8 shows the structure of the vector field. Since (10)is symmetric, we show only the regions D+ and D”.
We describe the flow of the solution for the initialcondition on Do near 0. The solution rotates divergentlyNISHIO

et d.:

WlNDOWS

IN A SYMMETRIC

479

CIRCUIT

where rA is given by the following

implicit.-z,=[o,~]xF,(o)xF;~(~~)x[

equation:“,I,o<w,T~<T.
(21)

W e consider the solution whose initial condition is represented by (22), namely the solution having the initialcondition on L’:(7, x, Y2-q = (0, x0,0,-

Structure of the vector field and typical trajectories.

I
Fig. 9.

Projection of the vector field onto x--z plane.

around 0 constrained onto D“ until it hits the line B+ (orB- ). W e consider the case that the solution hits B+. Then
one of the diodes turns on and the solution gets into D+(the region a). Note that the flow exists always between
two eigenspaces E*(P’)
and E*(P-),
because C’ arelocated sufficiently far from the flow. In D+ the flow has
two components in the sense that its projection onto
E*( P+ ) rotates divergently around P+ and its projection
onto E’(P+) moves away from P+. At some time the flow
hits the plane Do. Then the diode turns off and thesolution gets into D”. The solution starts to rotate around
0 constrained onto D” again. If the solution in D” hits
B-, it gets into D-. Owing to the symmetry of (lo), thebehavior of the solution in D- is similar to that in D+.
III.

DERIVATIONO F THE POINCAR~ MAP

Fig. 9 shows the projection of the vector field onto x - z
plane. W e define the following subspaces on D”:
L+={(x,y,z):

x>o;

y=o,

z=-zm}

L-={(x,y,z):

x<o,

y=o,

z=Zm}.

(22)

If the parameter values are chosen properly, the solution
which leaves the line L+ hits L+ again at some time.
1) 0 < x0 G A: The solution rotates divergently around
0 constrained onto Do and it hits L- at the time 7 = r.
without reaching the transitional condition z = - 1.
2) A < x0: The solution hits the line B- at (7, x, y, z)= ( rl, xi, 0, - 1) and enters D-. The solution which enteredD- hits the plane Do at (7, x, y, z) = (rl + r2, x2,0, z2).
W e consider the case that the point (x, y, z) = (x2,0, z2)
exists in the hatched region illustrated in Fig. 9. In the
figure, the boundary curves R, and R, are represented asfollowing equations using parameters u1 and u2, respectively :7.
Fig. 8.

ZJ.

R,: [;I=F2(-u,)XF~‘(O)x[ m1

R,: [;I=F2(-u,)xF;‘(O)x[ 1

0 < w,u2 < T, -1~

z”

,

-“1

,

z < Z,,,.

Other boundaries are L+, L-, B-, and z-axis (z > Z,). If(x2,0, z2) exists in the above-mentioned region, the solution which returns back to Do starts to rotate divergently
around 0 constrained onto D” and hits L- at (r, x, y, z)= (rl + 72 + 73, x3,o z )Therefore, a disc;etemone-dimensional map which transforms a point on L+ into a point on L- can be defined:
f: L+ -+ L-,(24)

Xo+X3=f(Xo)where x0 is the x-coordinate of the initial point on L+
and x3 = f (x0) is the x-coordinate of the point on Lwhich the solution leaving L+ hits L-. Since (10) is
symmetric, we can definef^: L- + L+, x0 + f^( x0)(19)Hereafter, the points on L+ and L- are denoted bythese x-coordinates.
Let A be the x-coordinate of the initial point on L+
whose flow should be tangent to the line B- at x = 0. W e
can give A as follows:~=[l,0]XF,(0)XF~1(7,)X[“,] (20)

(23)

(25)

where the map f” transforms a point on L- into a point on
L+. The map f has the same form as f.
The concrete representations of f are as follows.
1) O<x,<A:f (x0) =x3-

(26)

The value of xs is represented as follows:
x,=[I,o]xF~(T~)xF~~(o)x

[ 1
-xl

m

(27)

480

IEEE TRANSACTIONS

where r. is given by the following~,=[~,~]xF~;,(~~)xF~~(~)x

implicit

ON CIRCUITSANDSYSTEMS,VOL.

37, NO. 4, APRIL

1990

equation:<7r.
[ m1 0<ooro
(28)
-xi

,

2) A -C x0: ,The solution leaving L+ hits Bat(7, x, y, z) = ( rl, x1, 0, - 1). The value of x1 is represented
as follows:~,=[~,~]xF~(T~)xF~~(~)x

1 m1 (29)[ m1) 0<0,r1<7T.
(30)
-x;

Fig. 10.

where r1 is given by the following-~=[o,~]xF~(~~)xF~~(o)~implicit

equation:-xl

Next, the solution which enters D- hits D” at (7, x, y, z)= (ri + r2, x2,0, z2). The values of x2 and z2 are represented as follows:

An example of the Poincak map f, and the invariant
J+ and J-.

J- = [fa(x,,In the following
implicit

equation:

r

1

x1+

Y,=[~,~,~]xF~(T~)xF~~(~)x

xp
Yp

= L+,

>, -x,,]CL-.

fa(J+)

= J-

(37)

A(J-)

c J+.

(38)

Define:(32)

F&of,:

L++L+,xo~Fa(xo)Next, the solution hits L- at (T, x, y,.z) = (pi + 72 +r3, x3,0, Z,). The value of x3 is represented as follows:&ff,of;:

L--+L-,

where r3 is given by the following
Z,,,=[OJ]XF~(~~)XF;~(O)X

implicit

fia( J- ) c J-.

,

(34)

At last, f(xo)is obtained as
f(x0)

=x3.

(35)

Hereafter, we fix Z,,, = 0.2 and p = 3 and choose CYas abifurcation parameter. Let a,,bethe maximum value of(Y such that a one-dimensional map f can be defined, thatis, the point (x, y, z) = (x2,0, z2) exists in the hatchedregion in Fig. 9.
IV.

GLOBALBIFURCATIONS

Fig. 10 shows an example of f obtained by calculating(26)-(35). The implicit equations are solved numerically.
First the global bifurcations are discussed. We parameterize f and [ by the parameter CYand write f, and fi
instead of f and f: respectively. f, or fi has one extremum in L+ or &-. Let Q be the point on L+ at which(39)

F,(J+)cJ+

equation:-cIl.
[ 1 0<w*r3;:

x,+&(x,).

From (37) and (38)(33)

x~=[~,O]XF~(~~)XF;~(O)X[::](36)

discussions, we consider the case,1
O<opr2<27r.

9 x,,]

is satisfied, then0iintervalsf, has an extremum and then - Q is the point on L- at
which fi has an extremum. Let - x,, be f,(Q).
Hereafter we use a superscript “+” for an interval which
is a part of L+ and similarly we use “-” for an interval
which is a part of L-.
Define:J+ = [ L( - x,,)where r2 is given by the following

--+X

L+

(40)

That is, J+ (or J-) is an invariant interval of F, (or &).
We give the definition on a periodic point of F,.
Definitionof Periodic Point

A point k is said to be m-periodic point if k = Fam(k)and k # F,‘(k) for l<l<m. k is said to be stable if
1DFam( k) ) < 1, where DF,“( k) denotes the differential
coefficient of F,” at k.
Fig. 11 shows the one-parameter bifurcation diagramson L+. We take the initial condition x0 = Q in J+ in Fig.
11(a) and x0 = - Q in J- in Fig. 11(b). We can find that
there are differences between Figs. 11(a) and (b), whichmeans that two distinct attractors coexist in this map. At
these parameter values, two asymmetric attractors are located symmetrically with respect to the origin in the circuit.
We begin with the consideration of the phenomena in
the range (YG (Y,~ (see Fig. 11). Let Ax, be the point in J+
such that fa(x,) = - x,, then f,( - x,) = x,, so thatF,(xf) = x,. Therefore, X, is a fixed point of F,. For

NISHIO

et Ul.: WINDOWSIN A SYMMETRIC

481

CIRCUITL-

0v-x:,x,-x,

X2
-X

L+

Fig. 13.

“0.15

Fig. 11.

0.3

&bl

4.1

0.45 ffmax-ofOne-parameter bifurcation diagrams on L+. (a) Initial condition is xc = Q. (b) Initial condition is xc = - Q.

Proof that F, satisfies the Li-Yorke’s

period 3 condition.

For (~,i < (Y-C CY,,, J,’ and Jb+ are not the invariant
intervals of F,. Namely, the two distinct chaotic attractors
collide with each other and are united as one chaotic
attractor. Such phenomena are called the symmetryrecovering crisis by Kitano et al. [13].
V.

Two TYPES OF WINDOWSAs shown in Section I, for (~,i -C LY< (Y,,, two interesting windows appear, which are the main objects of thispaper. In this section we clarify the mechanisms and thestructures of them.
First we give the accurate definition of a window.
Definition of W indows
It is said that a m-periodic window appears in someinterval .[cY=,(~~1,if there exist CY,, (Y,, and 0~~such that:L+

Fig. 12.

f, for abl < a < q.

(Y< (~~i,x, is a stable fixed point, and the solution asymptotically approaches x,. When (Y exceeds o~~i,x, becomes
unstable. At this parameter value, a symmetry-breaking
transition occurs.
For (Eli < (Y-C (Eli, F:(Q) < x, is satisfied and thenfa(Ja+>CJL
i fa(Jb’) cJ,-

(41)

L(Ja-) =Jb+i L(Jb-) cJ,+
where

J,’= [L< - xmax),X,]3Jb+= [x,3hlJ = J+
Jo- = [Lbm&-x,],

J; = [+q,--m,]

CJ

.-

(42)

(see Fig. 12). Therefore, for (Eli < CY< CY,~:
Ex.u(Ja+kJa+
K(Jb+)

= Jb+

(43)

and two distinct attractors exist in J+: one passes throughJ,’ and Jb- alternately, and the other passes through Jb+
and Ji alternately. Since F, is like the logistic map [12] inthe interval J,’ or Jb+, F, is considered to exhibit perioddoubling bifurcation route to chaos in each interval in thisparameter range.

1) At (Y= (Y,, a tangent bifurcation occurs. Fu has two
m-periodic points near Q for (Y, c (Y< (Ye.At least for(Y, < (Y< CY, one is a stable m-periodic point and theother is an unstable m-periodic point. Let the former
be x, and also the latter be x,.
2) At (Y= (Y,,,,x, is equal to Q (superstable).
3) At CY= CY~,F,“(Q) is equal to x,. That is, the chaotic
attractor in this window collides with the unstablem-periodic point x, and the chaotic regions widen toform a single band. This phenomena is called theinterior crisis [21].
4) For (Y, < (Y< (Ye, there exists the interval J such that
F,(J) c J; Q E J and x, 4 J, that is, an attractor ofF, exists in this interval.
5) For (Y, < (Y< (Ye,F, satisfies the Li-Yorke’s period 3
condition [22].
Proposition 1
F, satisfies the Li-Yorke’s period 3 condition for (~,i <a -e a,,,~ where (~,i is a parameter value at which the firstsymmetric 3-periodic attractor appears.
Proof: W e define X,,,x n+1=

f(X,)...X,cJ+
f(X,)...X,cJ-’

as follows.
n=O,fl,+2;...

(44)
For aa1 < a < amax, f, or fi is shown as Fig. 13 andthere exists the point X0 on L+ such that the points Xi,

482(.1) :~~I

IEEE TRANSACTIONSON CIRCUITSANDSYSTEMS,

Fig. 15.

intervals Jc and J;

VOL.

31, NO. 4, APRIL

1990@2) ~~-:

0.11

cl.415

-

b.in
0

0.475
-Cl

0.47

(a)

Invariantgenerated near f, 0 i 0 f,(Q).

(b)

Fig. 14. One-parameter bifurcation diagrams obtained by using theideal model. Magnificationsof the region Wl and W2 in Fig. 11.
Subscripts 1 and 2 denote different initial conditions x0 = Q and
x0 = - Q, respectively. (a) The window of Type 1. (b) The window ofType 2.

X,, and X, satisfy-x,<x,<-x,<x*.

(45)Consider the points X-r, X-,, X-s, and X-,. For (~,r <a < a,,, these points can be chosen for which the following equations are satisfied:--xLFig. 16.

The neighborhoodof F,(Q)(see text)

x,<-X-l<Xz-XX,<X-,<Xf
x,<-x-,<-Xl
Xf < x-4 -c - x-1(46)

where xf is a fixed point of F,. Hencex,<-x~,<-x,<x~,<x/<x~,<-x~,~x,(47)’
is satisfied. Since X-, = F,( X-,),= F,3(&)
F,‘( X-,)> X-z, > F,( X-,)

X,, = F:( X-,),

and X,’ F,‘( X-4)

(48)

is satisfied.
Therefore, F, satisfies the Li-Yorke’s
period 3 condiQ.E.D.
tion for (~,r < (Y< a,,.
There seems to exist infinitely many Type 1 and Type 2
windows for (~,r < (Y< (Y,,. Examples of such windows
are shown in Fig. 14. These figures are magnifications of
the regions marked with Wl and W2 in Fig. 11 andassociate with Fig. 5. At first, we explain the mechanism of
the window of Type 1. Thereafter, we will explain thewindow of Type 2.
A.

forms J- into J+. f, 0 fz 0 f, in the neighborhood of Q isshown in Fig. 15. The following situation is observed bycomputer calculation.
Situation I
1) At (Y= (~,r, f, 0 i 0 f, is tangent to the Line-A+ in
the neighborhood of Q. F, has two periodic points near Q
for (~,r < (Y< (Y,,. At least for (~,r < (Y< o,,,r one is a stable
periodic point and the other is an unstable periodic point.
Let the former be x,r and the latter be x,~~.
2) At (Y= (~,r, x, = Q. That is, f, 0 f, 0 f,(Q) = - Q issatisfied.
3) At (Y= (~~r, F,‘(Q) = xul.
That is, the following conditions are satisfied.
1) At a := (~,r, the composite map F,’ is tangent to theLine-B+ (A in Fig. 16) and a tangent bifurcation occurs.
2) At (Y= (~,r, F,‘(Q) = Q is satisfied (II in Fig. 16).
That is, Q becomes a superstable fixed point. If (Y increases beyond (~,,,r,three extrema appear near Fa( Q)( C in
Fig. 16).
3) At (Y= adI, F:(Q) = x,r (F in Fig. 16) and theinterior crisis occurs.
4) Define the following intervals as shown in Fig. 15:

The Window of Type 1

We discuss the mechanism of the window of Type 1 asshown in Fig. 14(a), which is a magnificationof the3-periodic window in Fig. 3(g).
For the sake of the simplicity, we introduce some symbols. Let Line-AZ and Line-A- be the lines which satisfy
f(x) = - x and f(x) = - x, respectively. Let Line-B+ andLine-B- be the lines which satisfy F,(x) = & and r’,(x) =x, respectively.
Consider the composite maps f, 0 A 0 f, and A 0 f, 0 d.
The former transforms J+ into J- and the latter trans-

J; = [&,xul] c J+, J; = [ - &,-xul] c J-

where
fa+f,(%l)=-x,1.

fa+fa(J:)

= JC

(49)

For (~,r < (Y< adI

f’bf,“!i<

J;)

= J:.

PO

Then,
F:(J:>

c J3+,@;( J;)

c J;.

(51)

NISHIO

ef al.:WINDOWS

IN A SYMMETRIC

483

CIRCUIT

L-x
L‘

Fig. 17.

Therefore,0

a 3-periodic window

L+

Note that the composite map f, 0 i 0 f, in J: has thesimilar form to the original map f, in J+. Therefore,
f, 0 fi 0 f, in Js+ exhibits various phenomena observed in
the original map f,, such as, a symmetry-breaking transition and a symmetry-recovering crisis.
The W indow of Type 2

Next, we discuss the mechanism of the window of Type
2 as shown in Fig. 14(b), which is a magnification of the
2-periodic window in Fig. 3(i).
Consider the composite maps F,“.= A 0 f, 0 d 0 f, and
82 = f,o do f,o L. The former transforms J+ into J+
and the latter transforms J- into J-. F,’ in the neighborhood of Q is shown in Fig. 17. The following situation isobserved by computer calculation.

*x

Thereafter, the point in J- is transformed into a point in
J+ by A (the points in Jf are marked with 0 in Fig. 18).
Fig. 18 clearly shows that the orbit starting from Q makes
the periodic attractor which returns to Q through four
periodic points in J- and J+ alternately. Note that thepositions of the periodic points in J+ are different fromthose in J-. Namely, in this case two distinct attractors
coexist. Since Fa2: J4+ -, J4+ is similar to the logistic map,
it exhibits a period-doubling bifurcation route to chaos.

Situation 2
1) At (Y= (Y,~, F,’ is tangent to the Line-B+ and atangent bifurcation occurs. F,’ has two 2-periodic points
near Q for (Y,~ < LY< (Y&. At least for (Y, < (Y-z CX,one is astable 2-periodic point and the other is an unstable 2-periodic point. Let the former be xs2 and the latter be x,,~.
2) At (Y= LY,,,~,x,i = Q (superstable).
3) At (Y= cxd2, F,“(Q) = xU2, at which an interior crisis
occurs.
4) For (Y02< (Y< (Yd2, the following invariant intervals
can be defined in J + as shown in Fig. 17:

VI.

SCALING MECHANISMS OF WINDOWSIn the previous section, it is shown that the similar
structure to the global bifurcation diagram of F, is observed inside the window of Type 1. Actually, inside ‘thiswindow, the window of Type 1 and Type 2 appear infinitely many times. It means that in this circuit the.
number (one or two) and the nature (periodic or chaotic)
are changed in an extremely complicated manner.
In this section in order to confirm this fact, we prove byusing a simple scaling mechanism that there exists a sequence of the windows inside the window of Type 1 inwhich two types of windows appear alternately and infinitely many times.
As mentioned in the previous section, the composite
map f, 0 fi 0 f, makes the invariant interval J: near Q fora al < (Y< ad,. Define:5; = [c$,2, xU2] c J+, 5; = [ - gU2,- xU2] c J-

T,f,+f,:*Ia
Tol-faof,of,:

where
Therefore,-

Fig. 18. f, and i at a = a,*. “0” and “ 0” represent the points in
J- and J+. respectively. The solid line and the broken line represent
the mappings f, and fi, respectively.

appears for cz,i < (Y<

adl*

B.

Q

Invariant interval Jc generated near F:(Q).

(52)
43
a”,>
= x,2.
a 2-periodic window appears for aa < (Y<ad2-

For the purpose of simple explanation of the mechanismof this window, the maps f, and j” at (Y= (Y,,,~are shown
being overlapped in Fig. 18. W e consider the orbit startingfrom Q in J+; The point Q is transformed into a point in
J- by f, (the points in J- are marked with l in Fig. 18).

J;-5;
.

J; + J:.

(53)

Moreover, we rewrite J: = [iUl, x,i] as I+ = [f, m] andalso J; = [ - iU1, - x,i] as I- = [ - I, - m]. In this section,
we consider T, and ?a only for (~,i < a < (Ydl.
Theorem 1
If T, satisfies the following (Condition), there exist thewindows which appear in the following order:

Type 1:
3x?
Type 2:

‘;xg

;x\5

>x;/

Zx?

...
‘;x$

n=3,4,5;..
where the numbers represent periods of the windows.

7

.*.

/

...

3Xn
7\

h

3x5

n ’

r”’
...

(54)

484

IEEE TRANSACTIONSON CIRCUITSANDSYSTEMS,VOL.

37, NO. 4, APRIL

1990(a) c.

T,.

(a)
Fig. 19.

T, changes from Tam, to Tad,.

Condition
1) The Schwarzian derivative [23] of the map T, isnegative.
2) The map T, has a unique extremum Q in I+.
3) As (Y increases continuously, the map T, changes
continuously from Ta, to T,,, as shown in Fig. 19 where
TaGI = - x is satisfied for only x = m and TadlcXj = - 1
is satisfied for only x = Q.
It is known that if the Condition
1 is satisfied,
T’ o.fa o T, o . . . has at most one inflection point betweenn
its adjacent extrema.. The above mentioned
confirmed numerically.

(Condition)is

Proof:
Fig. 20(a) and (b) show T,: I+ --j I- and ?a 0 T,: I+ +I+ at (Y= 1yr2 (the symmetry recovering crisis). Define theinterval 1: as shown in Fig. 20(a):Fig. 20.

T,, ?a 0 T,

andT, 0 ?a 0 T, -at(Y= Q.

(c) T, 0 T, 0 r,.

(b)

to

Z2+ = [12, m21
From (55)-(57)whereTa(Z2) =Ta(mz)The following=-I,(Z2<m2).

the following

equations are satisfied:(55)

equations are satisfied.
ItoT,(u=+=m2(56)(~-X(Q)

lacadl=m(57)

and for (Y,~<: (Y< adl:

Let us consider the form of T, 0 fm 0 T, in 12 for (Y,~< (Y<ad1 by the chain ruleD{Ta~fa~Ta(x)}=DTa{fa~Ta(x)}~D{~a~Ta(x)}.

TaofaoTa(e,)=TaofaoTa(e2).e,<x<Q‘oror

Q <x < e2e2<x-cm2.

(61)
The composite map T, 0 ?a 0 T, has three extrema in 1: asshown in Fig. 20(c).

(64)

Then, as LYincreases, there exists a process such that theneighborhood of T, b fa 0 T,(Q) ascends and intersects theLine-A + at some parameter value. Moreover, if Condition
1 (negative Schwarzian derivative) is satisfied, T, 0 T, 0 T,
touches the Line-A + in the neighborhood of Q at a uniquepoint (tangent bifurcation).
Let the tangent bifurcation
parameter value be (Y= LY,~. Define the interval 1: foraa < a -c ad1 as shown in Fig. 21(a):13’= [13,m31(60)
There exist the two points e, and e,(Z, < e, < Q < e2 < m2)
in 1; such that pa 0 Ta(ei) = Q(i = 1,2) and>T,oFaoT,(Q).

where
Taofb,~Ta(l,)=T,0faoTa(m3)Then, there exists a parametersuch thatp9~k(Q>
Hence, Situation=-13(f3<m3).

(65)value (Y&((Y,3 < (Yda< (Ydi)

lacod3=-m3.

(66)

1 occurs in T, 0 fa 0 T, in the neighbor-

NlSHlO

ef 01.: WINDOWSIN A SYMMETRIC

485

CIRCUITI

1, Q m
\I:?

m

4
-x

(b)

hood of Q for (~,s -=c(Y-C czd3.Namely, the yindow of Type
1 appears for (~,s < cx< ad3. Since (T, 0 TJ3(13) = I,, theperiod of this window is 3 X 3 with respect to the original
map F,.
Next, we consider the form of the composite map
fao Tao fao T, in 1; for ad3 c (YC (~~r. At (Y= ad3, theinterior crisis occurs and the above-mentioned window
disappears (see Fig. 21(a)). For ad3 < cw< tidl, similarly, thefollowing equation is satisfied:

(‘.)

id)

Fig. 23. Some examples of the windows which appear in the orderrepresented by (54). Initial condition is x,, = Q in the above figures andit is x,, = - Q in the below figures. (a) 3 x 3-periodic window of Type 1.
(b) 3 x2-periodic window of Type 2. (c) 3 x5-periodic window of Type
1. (d) 3 X3-periodic window of Type 2.

where

D{~a~Ta~~a~Ta(x)}<o.. .1, -c x e e3 orori >o..* e,-cx<Q

Q -CX c e4e4<x<m3

if~~Ta~~a~Ta(14)<faoTa+Ta(~).

=?a~TaOfa~Ta(m4)(67)

whert e3, and e,(f, < e3 < Q < e4 < m3) in I3+ satisfy
I”a 0 T, 0 Ta(ei) = - Q(i = 3,4). The composite
map
T, 0 T, 0 T, 0 T, has three extrema in I: as shown in Fig.
21(b).
From (66), (63), and Fig. 20(a), the following equations
are satisfied:=fa~TaOfa~Ta(er)

-+Ck
(‘4

(8)

(70)

Then, as (Y increa!es, ther: exists a process such that theneighborhood of T, 0 T, 0 T, 0 T,(Q) ascends and intersects
the Line-B+ at some parameter value. Let the tangent
bifurcation parameter value be (Y= CY,~.Define the interval
Ii for (Y,~ < (Yc (~,,r as shown in Fig. 22:=l,(l,<m,).
(71)

Then, there exist a parameter value CY~,+((Y,~< ad., < 0~~~)such that
J~~Q~N~(Q>Jnzad,=m4.

(72)Hence, Situation 2 occurs in I’, 0 T, 0 -ih 0 T, in the neighborhood of Q for (Y,~ c (Y< ad4. Namely, the window ofType 2 appears for (Y,~ < (Yc ad4. Since (T, 0 fa)2(14) = I,,
the period of this window is 3 ~2 with. respect to theoriginal map F,.
Note that the condition of fa oATu0 i!a 0 T, in 1: (seeFig. 22) is the same as that of T, 0 T, in 1: (see Fig.
20(b)). Therefore, the above-mentioned process can beapplied to fa 0 T, 0 ? 0 T, in I: and we can repeat thesimilar discussion ir&itely
many times. As a result of this,
Q.E.D.
we obtain Theorem 1.
Fig. 23 shows some examples of the windows obtainedby computer calculation, which appear in the order represented by (54).
Next, we show that the generation of these windows isconcerned with a certain universal constant. This universal

IEEETRANSACTIONSON CIRCUITSAND SYSTEMS,
VOL. 37, NO. 4, APRIL 1990

4X6

TABLE ICALCULATION O F 6:

results of this paper may contribute to the study of bifurcation phenomena in symmetric chaos-generating systems.
To show the universality of our results may be our future
problem.
ACKNOWLEDGMENT

The authors would’like to thank Assoc. Prof. Hiroshi
Kawakami, Tokushima University, Japan, Assoc. Prof.
Tetsuro Endo, National Defense Academy, Japan, and Dr.
Ryuji, Tokunaga, Waseda University, Japan, for theirvaluable discussions.
constant is obtained by calculating the following equation:
s(i)

=

Qi+l)
%(i+2)-%i- %(i+(73)
1)

where (Y,,,~represents the parameter value at which 3 x iperiodic attractor for the window of Type 1 and 3 X i/2periodic attractor for the window of Type 2 become superstable. The result obtained by calculating (73) with acomputer is listed in Table I. It seems that S(i) converges
to 3.0654. . . , when i + 00.
W e could not get the accumulation point of the windows
analytically. By numerical calculation (Y,,,~seems to conve.FFe to adl; the parameter value at which the interior
crisis occurs.
RemarkIt is known that the accumulation point of the sequenceof the tangent bifurcations for the logistic map is theparameter value at which the interior crisis occurs [19],[24]. (For a two-dimensional
map similar discussion has
been done [25].)
Because our Poincart map f or f^ is like the logistic map,a part of the theory on the successive tangent bifurcations
for the logistic map may be used for our system. However,symmetry of the system is considered in our discussion
and we show that the number of the attractors changes in
a complicated manner. Moreover, we discuss the sequenceof the tangent bifurcations based on the real physicalsystem.
VII.

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CONCLUSIONSIn this paper, two types of the windows generated in asymmetric circuit have been investigated. By using a degeneration technique, we make clear the mechanisms ofthese two types of windows and prove rigorously that two
types of the windows appear alternately and infinitely
many times inside some windows under some reasonableassumptions by utilizing a certain scaling structure on the
PoincarC map.
The one-dimensional Poincare map derived from ourcircuit is like a combination of two logistic maps and it
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“On higher dimensional constrained autonomous chaotic[I71
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v41 R. L. Devaney, Au Iutroductiorl to Chaotic D~vrmmical S.vstems.
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NlsHloef ‘11.: WINDOWSIN A SYMMETRIC

CIRCUIT

Yosbifumi Nisbio received the B.E. degree inelectrical engineering from Keio - University,
Yokohama Japan, in 1988. He is currently working towards the Master’s degree in the Department of Electrical Engineering, Keio University.
His research interests are in nonlinear circuits.

487
Shinsaku Mori received the B.E., M.E., and Ph.D.
decrees in electrical enaineering from Keio Univeysity, Yokohama, Japan, in-1957, 1959, and1965, respectively.
Since 1957, he has been engaged in research at
Keio University, mainly on nonlinear circuits
theory and communication engineering.

3
Naohiko Inaba received the B.E., M.E., and Ph.D.
degrees in electrical engineering from Keio University, Yokohama Japan, in 1984, 1986 and1989, respectively.
In 1989 he joined the Department of Information Science at Utsunomiya University, where heis currently a research associate. His researchinterests are in nonlinear circuits.
Toshimicbi Saito, for a photograph and biography, please see page 409 of
the March 1990 issue of this TRANSACTIONS.