Simple Chaos - The Hénon Map
Hénon's images are among the best known in twentieth century mathematics...
The Hénon map - at least one version of it - with parameters a and b is the map
H: (x, y) -> (y, 1-ay2+bx)
from the plane to itself. The map can be constructed in three stages
H: (x, y) -> (y, y) -> (y, 1-ay2) -> (y, 1-ay2+bx)
which I'll picture in a moment. This map is a slight variation of one first occurring in a classic paper by M. Hénon as a discrete approximation (Poincaré return map) of a certain complicated three-dimensional flow. It is one of the basic objects of study in the theory of dynamical systems. The map itself is relatively simple, but upon iteration produces extraordinarily complex phenomena.
It is a product of the age of computers and of computer graphics. Although discovered as long ago as 1976, very few of the phenomena observed computationally have been rigorously explained, and even now new computational approaches to understanding this map are being found. I am not entirely familiar with the state of mathematical rigor in the investigations, and will not attempt to distinguish empirical observation from rigorously verified fact. Some might therefore call this an article about physics!
The map H is in fact one of the simplest possible non-linear 2D transformations. Each application of H to a point is not hard to visualize in itself. The problem, as it often is in dynamical systems, is to understand the long-term behavior of points when H is repeatedly applied. The problem is all the more difficult since varying the parameters has a great effect on long-term behavior. Here are some of the many questions one would like answered:
These questions are among the most intriguing in mathematics. The literature on this topic is vast and I have nothing very new to say about it, but after seeing David Ruelle's article "What is a Strange Attractor?" in the August 2006 Notices of the American Mathematical Society, I thought there was room for a discussion with more pictures (even though there are many good pictures already in the literature).
I'll start by looking at the case that Hénon did, that with a=1.4 and b=0.3. Here is a picture of the three stages of the construction of H(x,y) from (x,y):
Note that the result of the first two stages is a point on the parabola y=1-ax2, and that if b is small the final result is not far away from it. The image of a rectangle can be constructed similarly:
For these values of a and b, under iteration of H a given initial point will either pass off to infinity or approach a rather odd curve known as a strange attractor. The curve can be visually approximated by plotting a large number of points (2,000 for the image on the left below) in the trajectory of the origin (0,0).
On the right is a close-up look at part of the first image with a larger number of points plotted. It suggests that there is a very complicated fine structure to the attracting curve, and indeed it was exactly this that Hénon's original paper suggested by even more close-up examinations. Hénon's images are among the best known in twentieth century mathematics.
Not all points will be attracted to the Hénon attractor. I do not know any good way to tell whether a point is actually attracted to it, but we can tell whether a point escapes to far away in a reasonable amount of time. We can do this for a square array of points and color them if they do not escape to far away in a reasonably short time, as in this image:
In other words, the points in the red region should approximate the basin of attraction of the Hénon attractor. The structure of this basin seems in itself interesting - its shape changes drastically as a and b do. At any rate, knowing the basin suggests another way to approximate the attractor itself - we first construct a quadrilateral that covers the curve and lies inside the basin of attraction ...
... and then iterate the map on this region.
The attractor - the limit of the images - looks likely to be a very thin curve with several layers. A cross section through several layers looks like a Cantor set, and each layer looks somewhat like a parabola nested inside other layers. The attractor is stable under the map H.
Hénon and Smale
As with many examples of a dynamical system, the principal goal of research so far has been to describe the restriction of H to the attractor in terms of symbolic dynamics, that is to say to describe the points of the attractor in terms of strings of symbols, here 0 and 1, and to describe H in terms of shifts of these strings. In practice, one wants to generate the strings in terms of a directed graph of transitions between nodes that can be described without too much trouble. The simplest example in this sense is Smale's horseshoe, where the strings are all sequences of 0 and 1. The Hénon maps in general seem to be much, much more complicated. The most sophisticated attempt to deal with them is the pruning front conjecture of Cvitanovic. My original goal in starting this article was to explain the pruning front, but that proved to be too hard for the moment. But I want to give some idea, necessarily vague because brief, of how symbolic dynamics enter the problem.
The first question is that of seeing how strings of 0 and 1 arise. To do this for the parameters that I have been working with is too difficult to explain here, so I'll look instead at a simpler case. Suppose now that a=6 and b=0.9. For these parameters the Hénon map no longer has an attractor, but it has something analogous - its non-wandering set. Every Hénon map with b not equal to 0 is invertible, and the inverse can be easily found by solving
X = y
for x and y to get
x = (Y-1+aX2)/b
The non-wandering set is the set of all points P that don't get pushed off to infinity under iterates of either H or its inverse. In other words, the points Hn(P) are precisely those that remain within a bounded region of the plane for all integers n, both positive and negative. For our original parameters the wandering set coincides with the attractor, but for other parameters it may not be an attractor. For certain values of the parameters a and b the non-wandering set may be described by an interesting limit process.
Every Hénon map H has two fixed points, obtained by solving
x = y
which leads to a quadratic equation for y:
y = 1-ay2+by
with roots y = [(b-1) +/- sqrt((b-1)2 + 4a)]/2a. In our case, we get the fixed points (0.4,0.4) and (-0.41666,-0.41666). The map in the vicinity of these fixed points is essentially the linear map determined by its Jacobian matrix
Its determinant is -b and its roots are therefore real and of opposite sign. More precisely the eigenvalues at the first fixed point are -4.9806 and 0.18069, those at the second one -0.17394 and 5.1739. Both fixed points are hyperbolic, which means attraction along the eigenlines for the small eigenvalue and repulsion along the other.
Under the map, a small circle around each fixed point is pushed into an ellipse by H, and under iteration is stretched out into the unstable manifold of the point. Conversely, the circle is pushed out into the stable manifold of the point by the inverse map. I repeat: the stable manifold of a fixed point is the curve of points that are pulled towards it by H and the unstable manifold is the curve of points that are pushed away from it by the inverse of the map. In the neighbourhood of the point the first is asymptotic to the line of eigenvectors with absolute value less than 1, and the second with absolute value greater than 1.
The pictures below show what happens to some small circles upon iteration a few times of H. The red shapes are the pre-images of the circles.
But here is the point. If a point lies outside the colored region shown below, then it must pass off to infinity eventually. Therefore, the non-wandering set must be contained in that region, which I'll call M.
But it must also be contained in H(M), hence the intersection M ^ H(M), which breaks up into two regions I'll call M0. and M1. ...
... as well as H-1(M) ^ M, which breaks up into two regions M.0 and M.1 ...
as well as H-1(M) ^ H(M) which breaks up into 4 regions M0.0, M0.1, M1.0, and M1.1.
Continuing, every point in the non-wandering set may be labeled by a doubly-infinite string of 0 and 1, separated in the middle by a decimal point. The string to the right of the decimal point specifies the past of the point and that to the left its future. Applying H to one of these points amounts to a shift of the string. This Hénon map is a very natural realization of Smale's horseshoe.
Back to Hénon
We know now how to describe the non-wandering set of at least one Hénon map in terms of symbolic dynamics. Why can't we do this for the original map?
The construction for a=6 and b = 0.9 depended on the configuration of the unstable and stable manifolds for the two fixed points of the map - they had to enclose a more or less rectangular region. Here's what happens for a=1.4 and b=0.3:
We don't get that nice box. In general, suppose we fix a and increase b from 0 on up. For b=0 the Hénon map smashes everything down onto the parabola y=1-ax2, and the map on that parabola is essentially the 1D map associated to it. The non-wandering set is described by one-sided strings of 0 and 1. For small non-zero values of b it is known that there is an attractor. As b is raised, a certain critical value will be passed after which a horseshoe appears and the non-wandering set is described by all two-sided binary strings. For a=1.4 this critical value is just above 1.3. In between 0 and the critical value just about nothing is known rigorously. It has been conjectured that the non-wandering set is describe by a proper set of binary decimal strings, one which is obtained from the set of all strings by Cvitanovic's pruning process, and there is much evidence for this conjecture.
The literature is vast, and I have listed here only the few items I have used extensively.
NOTE: Those who can access JSTOR can find some of the papers mentioned above there. For those with access, the American Mathematical Society's MathSciNet can be used to get additional bibliographic information and reviews of some these materials. Some of the items above can be accessed via the ACM Portal, which also provides bibliographic services.
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