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 10^{6} 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 "Henonswallow" 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
J_{ij} =
(

∂ x'/∂ x  ∂ x'/∂ y

∂ y'/∂ x  ∂ y'/∂ y


) =
(


)

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 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 blowups of the squared regions
in the preceding figure. Scale invariant, Cantorsetlike
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 twodimensional mapping with a strange attractor"
Comm.Math.Phys. 50, 69 (1976).
Contents
Previous: Structure of the parameter space of the
Henon map
Next: Nonlinear resonance
updated 27 June 2004