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).

The matrix
    Jij = (
∂ x'/∂ x   ∂ x'/∂ y
∂ y'/∂ x ∂ y'/∂ y
)   =   (
2x   b
1 0
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)

[1] M.Henon "A two-dimensional mapping with a strange attractor" Comm.Math.Phys. 50, 69 (1976).

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updated 27 June 2004