WINDOWS IN A NON-AUTONOMOUS CIRCUIT WITH SYMMETRY
Miyoshi T.: Sekilcawa M.!Sat0 T.) Inaba N.! and Nishio Y.7*tS§ Faculty of Engineering, Utsunomiya University, Utsunomiya, 321-8585 Japan

YFaculty of Engineering, Tokushima University, Tokushima, 770-8506 Japan
ABSTRACT
Symmetric structure exists in many chaos-generating
circuits. In those circuits, interesting bifurcation
phenomena based on the symmetry of circuits, suchas symmetry recovering, symmetry breaking and soon, are observed. In the present study, two types ofinteresting windows are observed in a circuit withsymmetry which we propose. This circuit has a pairof diodes. We consider the case in which the pair ofdiodes operates as a pair of ideal switches. In thiscase, the Poincark map can be derived rigorously asa one-dimensional map. We prove that a countably
infinite number of two types of windows are generated alternately by applying the theorem written in
our paper[l] t o the one-dimensional map.
1. INTRODUCTION

1-0.

Symmetric structure exists in many chaos-generating
circuits. Chua’s circuit is an example. The governing equation is invariant against the transformation(2,y, z ) + (-x, -y, - z ) because the characteristicof the nonlinear element is symmetric with respectt o the origin[2]. In circuits having such symmetry,interesting bifurcation phenomena such as symmetry recovering and symmetry breaking occur.
We proposed a three-dimensional autonomous
circuit whose governing equation is invariant against
the transformation (2,y, z ) + (-x, -y, - z ) and analyzed generated windows[l]. A window is a periodic phase surrounded by chaotic phases. By analyzing the Poincarh map, it was clarified that certain
two types of windows are generated by the target
circuit. Furthermore, it was clarified theoreticallythat a countably infinite number of the two types ofwindows are generated alternately.
On the other hand, symmetric structure exists also in non-autonomous systems such as Duffing equation and a forced Rayleigh equation. They
are invariant against the transformation ( t ,x,y) +(t+27r/w, -2, -y), where w denotes the angular frequency of the external force.
In the present study, we analyze windows which
are generated by a forced Rayleigh circuit having apair of diodes. By numerical calculation, it is clarified that the circuit generates two types of windows.

IFigure 2: v-id characteristic of ideal diode pair.
One type is a window in which symmetry breaking
bifurcation occurs. The other is a window where theshape of the orbit stays unsymmetric with respect t othe origin. Those are the same as the autonomous
circuit mentioned above.
The circuit is analyzed assuming the pair of diodest o be a pair of ideal switches. In the domain whereone of the diodes is on, the governing equation isdegenerated t o a one-dimensional equation becauseof the assumption. Therefore the Poincark map isderived rigorously as a one-dimensional map. By
applying the theorem written in our paper[l], it isproved that a countably infinite number of the two
types of windows are generated alternately.
2. TARGET CIRCUITThe target circuit is shown in Fig. 1,where - g N , g E ,v, i, i d , E , and wf denote linear negative conductance, positive conductance, the voltage across thecapacitor C , the current through the inductor L , thecurrent through the pair of diodes, the amplitude ofthe external force, and the angular frequency of theexternal force, respectively. We approximate thatthe v-i characteristic of the pair of diodes is shownin Fig. 2. Let us change variables as follows.

*E-mail: miyoshi@every.is.utsunomiya-u.ac.jp+E-mail: mooskQevery.is.utsunomiya-u.ac.jp$E-mail: teching@every.is.utsunomiya-u.ac.jp§E-mail: inaba@is.utsunomiya-u.ac.jp

0-7803-7057-0/01/$10.00 02001 IEEE.

Figure 1: Target circuit.

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We obtain the following governing equations. As
Fig. 2 shows, the branch where one diode is on isdenoted by +on and the other -on.

{+ o n : x = 1,off: x - 2 6 ~ + x = E s i n w r ,-on: x = - 1 .

(2)While the diode pair is +on (-on), x equals 1 (-1)
because the voltage across the capacitor is constrainedby the threshold voltage of the diode. Note that thedimension of the equation is degenerated to one on
the branches +on and -on. T h e solution of Eq. (2)is connected by the following condition.

ioff-++m: X = l ,+on+off: x=2S+Esinwr,off -+ -on : x = -1,-on -+ off : x = -2S+ E s i n w r .

C+ and C - describe the connection position determined by Eq. (3). C+ corresponds t o the +on -+off connection and is on the plane D+.C - corresponds t o the -on -+ off connection and is on theplane D-.
Fig. 3 shows the structure of the vector field oforbits. In the figure, two examples of orbits aredrawn. We treat the case in which 0 < WE< 1. Let
us consider the solutions whose initial points are on
the line C+. The solutions whose initial points areon the line C+ are classified into the following twocases.

3. POINCARE MAP

While the diode pair is +on (-on), the dimensionof the constrained equation is one. Therefore thePoincark map is derived as a one-dimensional map.
We use some symbols to define the Poincark map.
D+, D , and D- denote three subsets of r-x-x space where the diode pair is +on, off, and -on,respectively, as follows.

i(x = - 1)

Figure 3: Orbits.

Let us call Eqs. (2) and (3) the constrained equations.

D+ = ( r , z , k ) I k = l},D= ( r , x , k ) 11x1 < l ) ,D- = ( r , x , k ) I j. = 1).

D-D+(x = 1)
(3)

The solution leaves Po a t r = T O , goes throughthe domain D , where the diode pair is o f f ,and hits the plane D+ on PI a t r = 71, when
the diode pair is +on. The solution moves
constrained by the plane D+ and hits again
the curved line C+ on P2 a t r = 7-2.

(4)

We treat the case in which 0 < 6 < 1. In that case,eigenvalues of the constrained equations in the domain D are conjugate complex numbers S .tjy (y =d m ) . Because Eq. (2) is piecewise linear, thesolution is derived analytically in each domain. It isdescribed as follows.

The solution leaves POa t r = 70,goes throughthe domain D , where the diode pair is off,and hits the plane D- on PI a t r = 71, when
the diode pair is Ton. The solution moves constrained by the plane D - and hits the curved
line C- on P 2 at r = 7 2 .
Because the system is invariant against the transformation (7, x, k ) -+ ( r 2.rr/w, -2, - k ) , the solutions
whose initial points are on the line C- are classified similarly. Therefore the Poincark map T can be
defined as the following one-dimensional map.

+

where P = (ro,xo, 20) denote the initial condition
and X = ( 6 ,w, E). Let each solution in each domain where the diode pair is +on, off, and -on
be denoted by4+on)T,(doff & o f f ) T , and&o?)T,respectively. Each solution is derived analytically. Its formula is omitted owing tolimited space.
Let us define curved lines Ci and C- as follows.

T : (C'T ( 0 )=U C-) -+ (C+ U C - ) , 6 H T ( 6 ) ,wro/2.rr mod I, for P
O E C+,(wro/2.rr mod 1) 1, for Po E C - ,wr2/2.rr mod 1, for P 2 E C + ,(wr212.rr mod 1) 1, for P 2 E C-.

++(7 )1344

(4(b)

Figure 6: Enlarged picture of Poincark map.

Iof the auxiliary lines and the Poincark map. Now
we introduce some symbols. Let P and Q denotethe values of 0 that take the extrema1 values in thesections shown in Fig. 6 (a) and (b) respectively.
They are used as the initial values t o obtain Fig. 5(a) and (b) respectively. Let p denote the value of Ba t an intersection of Line-A and the Poincark map
as shown in Fig. 6 (a). a(# p) denotes the valuesof 8 which satisfies T ( a ) = T(/3).The section J is
defined as follows.

,1eFigure 4: Poincark map.

= [ff,PI.

J(9)Shown in Fig. 6 (b), 0,y, and J’ are defined similarly. Considering Eq. (8), the following equations
hold.

y=Figure 5 : Bifurcation diagram.

c =The derivation of the Poincark map T’isomitted. InFig. 4,an example of the Poincark map is shown.

1+/3+2

mod 11+ 1,

mod1 +1,(10)Furthermore, it is evident from Eq. (8) that the sections J and J’ of the Poincark map are the same inshape. In the neighborhood of the parameter value
a t which Fig. 6 is given,

4. TWO TYPES OF WINDOWSIn the following discussions, a parameter E is varied and (w,S) is fixed t o (0.3, 0.29). Fig. 5 showsa one-parameter bifurcation diagram obtained fromthe Poincark map T . Let Bn = T”(80). In Fig. 5 ,8500 ...8l000 are plotted. Note that different initialvalues are used between Fig. 5 (a) and (b). The
initial values are explained below.
Now detailed explanation for bifurcation phenomena observed in Fig. 5 is given by analyzing
the Poincark map in the following. Because Eq. (2)is invariant against the transformation (7,x,2 ) +(T 2n/w, -2, -i),the following holds.

I T ( P )I 0.

(11)

T ( J ) c J‘.

(12)

T ( J ’ ) c J.

(13)

Therefore
Similarly++ 1 mod 2) mod 1
T(O + f mod 1) mod 1,
for 0 5 0 < 1,mod1)+l}
mod1,
for 1 5 0 < 2.
T((0+={T(O(8)
Let us draw auxiliary lines Line-A and Line-B as
shown in Fig. 4. Line-A is T ( 8 )= 8 0.5 and LineB is T ( 8 ) = 8 - 0.5. Fig. 6 is a enlarged picture
which shows the neighborhood of the intersections+1345

Thus T 2 ( J ) c J and T2(J’)c J’. It follows thatwindows appear on J in the neighborhood of this parameter value. Fig. 7 shows the map T when E =E,1 = 2.987 and when E = E,,, = 3.029. T touches Line-A and Line-B when E = Eal. T ( P ) = yand T ( Q ) = a when E = E,,,.
T 2 ( J )c J andT2(J’)C J’ holds while E,, < E < E,,,.
A tangent bifurcation occurs when E = Eal.
The shape of the composite map T 2 is shown by Ain Fig. 8 (a). There exists a stable fixed point as illustrated with B in Fig. 8 (a),while E,, < E < Ebl.
At that time, the shape of an attractor is symmetric7mTm-Line-A’‘4b.9ie 0.980 97

0.97

T‘(811‘101II

rm TtmLine-B

0.91
1.41

-e

1.48

0.92

0.92

0 97

0.92
0 9 2 T(Q)(b)

P-00.97

(b)

0.98

T%II

0.91

1.41-e1.48

0.92

r0.92092

0 9i

0 92

0.97

(b’)

Figure 7: Enlarged picture of Poincark map. (a)(b)
E = E,1 = 2.987. (a’)(b7)E = E,,= 3.029.

Figure 8: Composite map T 2 . (a) A: E = E,1. B:
Eal 5 E < Ebl. C: E = E b l . (b) Ebl 5 E < ET1.
(c) E = E,1. (d) E = E,,,.

with respect to the origin in the phase space of thegoverning equation.
A symmetry breaking bifurcation occurs when
E = Ebl. T h e fixed point becomes unstable asshown by z* in Fig. 8 (b). Illustrated with x, andxb in Fig. 8 (b), two stable fixed points appear. At
that time, the shapes of attractors are unsymmetric with respect t o the origin. Varying the value ofa parameter, chaos appears by the route of period
doubling bifurcation.
A symmetry recovering crisis[l] occurs when E =E,1. The extremal values of the map T 2 go outside the sections I and I’ as shown in Fig. 8 (c).
T h e unsymmetric chaotic attractor bifurcates into
the chaotic attractor whose shape is symmetric withrespect to the origin.
An interior crisis[l] occurs when E = E,,,.
Theextremal values of the map T 2 go outside the section J as shown in Fig. 8 (d). There disappears thechaotic attractor which existed while E,1 < E <Fig. 9 (a) and (b) are enlarged pictures which show
W1 in Fig. 5 . Fig. 9(c) and (d) are enlarged pictures
which show W2 in Fig. 5 . Note that different initialvalues are used between Fig. 9 (a) and (b). And also
between (c) and (d).

Emax.

It has been proved[l] that a countably infinite
number of certain two types of windows are generated alternately, if the shape of the map in the section J is like a parabola, if there exists the case inwhich the map touches Line-A and Line-B as shownin Fig. 7 (a) and (b), and if there exists the case inwhich the extremal values of the map hit the end
point of the section as shown in Fig. 7 (a’) and (b’).
Thus, in the non-autonomous circuit proposed in thepresent study, it is clarified that a countably infinite
number of certain two types of windows are generated alternately. In fact, the two types of windows areobserved a s shown in Fig. 9. One type is a window
in which symmetry breaking bifurcation occurs, andwhich is called Type 1. The other is a window where
the shape of the attractor stays unsymmetric withrespect t o the origin, and which is called Type2.

(b)( 4Figure 9: Windows. (a)(b) Type 1. (c)(d) Type 2.
5 . REFERENCES[l]Y. Nishio, N. Inaba, S. Mori, and T. Saito,“Rigorous analyses of windows in a symmetriccircuit,” IEEE Trans. Circuits Syst., vol. 37,
no. 4, pp. 473-487, April, 1990.

[2] L. 0. Chua, M. Komuro, and T. Matsumoto,“The double scroll family,” IEEE Trans. Circuits Syst., vol. CAS-33, pp. 1072-1118, Nov.,
1986.

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