Henon strange attractors
You see below the familiar parameter and dynamical planes of the Henon map.
Unlike the previous Henon map page the right window
shows only the results of plotting
it = 3 106 successive points obtained by iterating of the
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).
has eigenvalues λ1,2 = x ± (x2 +
b)1/2. Therefore for a = 1.4 and b = 0.3 the
fixed point x2 = y2 = -0.883896 is unstable
with λ1 = 0.1559 and λ2 =
-1.9237 . The 1st figure to the left below shows the results of
successive iterations of the map started at the fixed point (marked by the
x label). Plots started at another 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)
∂ x'/∂ x || ∂ x'/∂ y
∂ y'/∂ x || ∂ y'/∂ y
 M.Henon "A two-dimensional mapping with a strange attractor"
Comm.Math.Phys. 50, 69 (1976).
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updated 27 June 2004