Structure of the parameter space of the Henon map
Jason A.C. Gallas

The two-parameter diffeomorphism of Henon is
    x' = -x2 + 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 k plus an infinite succession of adjacent domains having periods k 2n, i.e., following a period-doubling route to chaos from the main body k. Most of shrimps consist of four main legs made of infinite number of continuous and smouth segments resulting from intersections with legs from other shrimps.

"Shrimps Hunter" controls Click mouse in window to find period p of the point. Click mouse + <Alt>(<Ctrl>) to Zoom In(Out) 2 times. Hold <Shift> to modify Zoom In/Out x4

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.

Via Caotica
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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updated 16 June 2004