Find all the fixed points of the Henon map and show that the

Find all the fixed points of the Henon map and show that they exist only if a > a0, where a0 is to be determined.

Solution

Formulae for the H´enon map
Using the second of equations (4.4) to eliminate y from the first, gives a
quadratic expression for x:ax2 + (1 b)x 1=0.
Remembering that y = bx, there are two solutions (x±, y±), namely
x± = (1 b) ± p(1 b)2 + 4a

2a, y± = bx±.
Provided that a > 0, they satisfy x+ > 0, x < 0
Let’s consider the parameter b to be fixed and investigate the dependence
of the fixed points on a. Obviously they exist only for (1b)2+4a 0,
or
a a0 = 1
4
(1 b)
Two-dimensional systems
n fixed points period m<n new points period n

1 2 – 3 2

2 4 2 2 1

3 2 2 – –

4 8 4 4 1

5 2 2 – –

6 16 4 12 2   

7 30 2 28 4
8 64 8 56 7
Table: Table of periodic orbits, H´enon map, a = 1.4, b = 0.3.
If b = 0.3, then a0 = 0.1225. The birth of a pair of fixed points at a
critical parameter value is typical of a tangent bifurcation. As with onedimensional
maps, the question of stability must be settled using calculus.
It is clear that one of the fixed points must move discontinuously as
a passes through zero.11 This is because the curvature of the parabola
changes sign; indeed, for a = 0 the parabola degenerates into a straight
line.