# Henon strange attractors

You see below the familiar parameter and dynamical planes of the Henon map.
Unlike the Henon map page the right window shows
only the results of plotting
*it* = 3 10^{6} successive points obtained by iterating of the
Henon map with *(a, b)* corresponding to the white cross position
(click mouse to choose new parameters or zoom the pictures).

The black regions (to the left)
with the smooth "Henon-swallow" structures on the dynamical plane (to the right)
correspond to chaotic dynamics. In this black chaotic sea there are many
small regions ("shrimps" or "swallows") with periodic motion (dotted
structures in the right window).
The matrix

| 2x b |
J = | |
ij | 1 0 |

has eigenvalues *λ*_{1,2} = x +- (x^{2} +
b)^{1/2}. Therefore for *a = 1.4* and *b = 0.3* the
fixed point *x*_{2} = y_{2} = -0.883896 is unstable
with *λ*_{1} = 0.1559 and *λ*_{2} =
-1.9237 . The 1st figure to the left below shows the results of
successive iterations of the Henon map for *a = -1.4*
and *b = 0.3* starting from the fixed point *x*_{2}
(marked by the *x* label).
Similarly obtained plots starting with other initial values are almost
identical (exept for an initial transient), suggesting that the figures is
an attractor. The next figures are successive blow-ups of the squared regions
in the preceding figure. Scale invariant, Cantor-set-like
structura transverse to the linear structure is evident. Thus the attractor
is strange with dimension between one and two.
The last four images are scaled by the factor of *0.15* .
(click mouse to zoom the pictures)
[1] *M.Henon* "A two-dimensional mapping with a strange attractor"
Comm.Math.Phys. **50**, 69 (1976).

Contents
Previous: The Henon map
Next: Structure of the parameter space of
the Henon map

*updated* 27 June 2004