INTERNATIONAL JOURNAL OF CIRCUIT THEORY AND APPLICATIONS, VOL. 15,189-192 (1987)

LETTERS TO THE EDITOR

SELF-SIMILAR BIFURCATION STRUCTURES FROM CHUA’S CIRCUIT

LIN YANG AND YOULIN LIAO

Department of Electrical Engineering and Computer Sciences and Electronics Research Laboratory, University of California, Berkeley, C A 94720, U.S.A.

Self-simifar structure is one of the features associated with chaotic phenomena. A self-similar bifurcation structure from computer simulations of the 1 - D double scroll Poincare‘ map”* (called the 1-D map in the following) is observed on the a-P plane (the two-dimensional bifurcation diagram on the plane defined by the two bifurcation parameters a and P ) .

In Figure 18(a) of Reference 1 (the a-P diagram), there is a sequence of patterns having a self-similar structure, that is each pattern in the sequence is a scaled-version of the orange-yellow pattern in Figure 18(f) of Reference 1. In those 1-D map Figures, different colours are used to represent different states of bifurcation. The repetitive orange-yellow patterns which represent period-2 (orange) and period-4 (yellow) attractorst are resketched in Figure 1 to emphasize this property of repetition of

I -D map

100

7 5

P 5 0

25

0

0 10 20 30 40

a

Figure 1. Illustration of the self-similar structures in the a-B bifurcation diagram

?Here by a period-n attractor we mean an attractor with a periodic orbit that goes through the PoincarC plane n times.

0098-9886/87/020189-04$05.00 0 1987 by John Wiley & Sons, Ltd.

Received 31 October 1986

190 LETTERS TO THE EDITOR

patterns in the double scroll region. Besides the patterns shown in Figure 18(a), more patterns similar to that in Figure 18(f) are found from further computer simulation. We have found that the self-similar patterns appear at areas around (31.5, 73.5), (16, 29-75), (11.9, 19-95), (10.13, 16-1), (9.135, 14.1), (8.58,12.96), (8.243, 12.268), (8.027, 11.872) . . . , on the a-P plane. The patterns are distributed along a slightly curved line tending toward the origin that has decreasing slope as a and P decrease. The slope (APrI)/(Aa") = ( P n ~ l - ~ n ) / ( a n ~ l - a n ) , where n increases as a and P decrease, approaches 2.0 as n goes to m.

If we use Aa,, or A@, as a measure of the distance between two adjacent patterns, then Aa,,/Aan-l or Ab,,/Afln-l gives us a measure of the convergence rate of the sequence of patterns. We found that both ratios approach about 0.6 as (a$) approaches the origin.

It should be pointed out that in Reference 1, for reasons of realizability under the parameter range in which we are interested, modifications were imposed on the original definition of the 1-D map presented in Section 7 of Reference 1 by (7.1) to (7.9). The modified 1-D map was used to produce Figure 18 in Reference 1.

The two assumptions in Section 7 of Reference 1 associated with (7.1) guarantee that the I -D map 7c* is a well-defined continuous function, but, unfortunately, they are not always valid for the parameters (m,,, ml, a, 1) of interest. For instance, for the parameters we used, the spiral CXL intersects the line L 2 and the point D is not located on the left of A in some cases. The computer simulations of the a-b diagram, as shown in Figure 18 in Reference 1 were obtained by extending the definition of the 1-D map as follows:

7c* : B T 1 , -+ BIAl, (1) -

where BIAl, is shown in Figure 2. As suggested in Reference 1, for computational simplicity, x* can be transformed into

7c* : X(O)NI, + X(O)N1,

I

D l-unit

A 1-

Figure 2. Geometrical interpretation of the definition of the I - D Poincark map n*

LElTERS TO THE EDITOR 191

(see Figure 2). K* is now a well-defined piecewise-continuous @fiction and preserves the qualitative behaviour of the double scroll family.

The difference between the original and the modified 1-D map is that the domain has been changed from m, (see (7.10) of Reference 1) to B’N,. The reason for this is that Y(u) must be on the segment X(O)N,,. Hence, in Figure 2,p‘ = n ; ( p ) and (X(p), Y(p)) is the graph of K*. The modified I[* is shown in Figure 3(a) from which one can see that the map has many discontinuous points. These discontinuities are due to the fact that the points on segment a& of CIA;, hit the segment ml, before they are mapped onto the segment X ( 0 ) N I , by K*. With this in mind, we define an auxiliary map K’, which is a direct extension of (7.7) in Reference 1, as follows:

-

- K’ : ON1, -+ ON1, (3)

For this map, Y(u) is found by going backward (clockwise in Figure 1) along the trajectory for points on segment & of CIA;, and going forward for the other points on aim until the trajectory intersects X(O)N1,.

Analytically, for K’, we have

I y(u> I exPCal(n - arg y(u>>l, I y ( 4 I exp[-a1(n + arg y(u))l,

Y’(u) = for y E 1

for y E 11 (4) i where I and I1 are areas indicated in Figure 2. Equation (4) is validated by the same condition assumed for (7.7) of Reference 1 (i.e. I y t I is relatively large compared to the other eigenvalues). Now K’ is a well-defined function. Figure 3(b) shows K’ with the same parameter values used in Figure 3(a).

Having defined K’, the modified I - D map is defined analytically as

for Y’(u) 2 X ( 0 ) i exp[2naln*] y‘(u) Y’(u), for Y’(u) < X ( 0 ) Y*(U) =

where n* is the integer such that:

exp[2nol(n* - l)]Y’(u) < X ( 0 ) 5 exp[2naln*]Y’(u)

1 4 map

I / / / 1

0 .000 I I I I

0 .000 0 . 5 0 0 1.000 1.500 2 . 0 0 0 2.500

X

Figure 3(a). I -D map n* at m,, = -1/7, ml = 217, rr = 10.0 and fi = 15.0

192

2.500

2.000

1.500

Y 1.000

0.500

0.000

LE’ITERS TO THE EDITOR

1-D map

0.000 0.500 1 .000 1.500 2.000 2 . 5 0 0 X

Figure 3(b). Same as Figure 3(a) but with the extended definition of I-D map

REFERENCES

1. L. 0. Chua, M. Komuro and T. Matsumoto, ‘The double scroll family’, IEEE Trans., CAS-31, 1072-1118 (1986). 2. Mireille E. Broucke, ‘One parameter bifurcation diagram for Chua’s circuit’, ZEEE Trans., Circuits and Systems, to appear.