Clas Blomberg - Physics of life-Elsevier Science (2007)

308 Part VII. Non-linearity
shall identify a number of such events with an approach and later divergence along the
same directions. By looking at many points of that kind one gets a picture of linear features
close to a periodic orbit, which information then is used for stabilisation.
Stabilisation of unstable orbits
There are always possibilities by modifying parameters to stabilise the unstable periodic
orbits. That is what we want to take up here.
The procedure can conveniently be demonstrated for what is known as the Henon map:
x n1 ax n by n y n1 x n (28.13)
For values of a above a certain threshold (1.06), this shows a chaotic behaviour. In
studies of this map, b is normally kept fixed, usually put equal to 0.3, while a is varied.
There are always two fixpoints of this equation:
x
1 b
y
2
0 0
( 1
b)
4
2
a
(28.14)
As in other cases, one makes a linearisation around the fixpoints
x n x 0 z n y n y 0 h n (28.15)
Close to the fixpoint, one gets linear expressions
j n1 2 x 0 j n b h n h n1 j n (28.16)
The conventional stability analysis works with the matrix:
⎛2x0
b⎞
⎝⎜
1 0⎠⎟
with eigenvalues:
l
2
12 , x 0 x 0 b
(28.17)