The two-parameter diffeomorphism of Henon is

*x' = -x ^{2} + by + a
y' = x*.

The parameter space of the Henon map is organized in a very regular way. Complicated pleopodic shrimplike structures are shown below (computation algorithm and coloring scheme is explained in "swallows" and "shrimps"). Every shrimp consists of a main body of periodicity

A striking regularity is that essentially all isoperiodic shrimps appear
aligned along a very particular direction *α* defined by the (red)
line *b = - 0.583a + 1.025* (see Fig.1 at the bottom).
A portion of the line you see above.
This pictures is a magnified view of a domain around the *8'*-shrimp
in the second applet. The dense "foliation" of roughly parallel lines seen to
the left and to the right of the *8'*-shrimp in the second applet
originated from shrimps located almost exclusively along *α*.
There are also a secondary *β* direction "perpendicular" to
all legs going down and a further direction, defined by thouse legs
going to the right.

From the picture one recognizes that most of the dynamics known for unimodal
maps (exemplified here for the quadratic map along *b = 0*) corresponds
to "traversing the foliation of legs" coming down from shrimps located in the
domain above. The apparently random way with which periodic windows appear
embadded in the chaotic domain along *b = 0* is in fact a consequence
of the ordering of shrimps along *a*.

*A remark* Due to slow convergence near the period doubling
bifurcations there are black strips between zones of different periodicities.
As since shrimps are rather slow the last applet makes only *300*
iterations in every pixel. If you don't like the strips use
"Shrimps Hunter" with *it = 2000* .

[1] *J.A.C.Gallas*
Structure of the
parameter space of the Henon map Phys.Rev.Lett., **70**, 2714(1993).

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