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a=0.7 b=1
15
10

5
0
-5
-10

-15
-20 -15 -10 -5 0 5 10 15
a=0.7 b=0.9998
15

10
5
0
-5
-10

-15
-20 -15 -10 -5 0 5 10 15 20
FIGURE 2.3
Plot of the Mira curve for a = 0.7. The starting point coordinates are (0, 12.1). Top panel: b
= 1, bottom panel: b = 0.9998.

2.8.2 Hénon’s Model

The coordinates of the Hénon’s orbits are generated iteratively through the
following system of nonlinear difference equations:

2
xk( + 1 ) = ax k( + 1 ) − b y k( ( ) (− xk( )) )
(2.45)
2
yk( + 1 ) = bx k( + 1 ) + a yk( ( ) (− x k( )) )
where a ≤ 1 and b = 1 − a 2 .
Executing the following script M-file illustrates the possibility of generating
two distinct orbits if the starting points of the iteration are slightly different
(here, a = 0.24), and the starting points are slightly different from each other.
The two cases initial point coordinates are given, respectively, by (0.5696,
0.1622) and (0.5650, 0.1650). See Figure 2.4.

a=0.24;
b=0.9708;

© 2001 by CRC Press LLC