Simple Chaos - The Hénon Map
Hénon's images are among
the best known in twentieth century mathematics...
Bill Casselman
University of British Columbia, Vancouver, Canada
cass
at math.ubc.ca
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:
-
What is the long term behavior of a given point
as H is repeatedly applied to it?
- How
does this behavior vary as the point is varied?
- How do the answers to the first questions
vary as the parameters a and b vary?
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.
- How can one describe the attractor more precisely?
- How can one
describe how H acts on the attractor itself
(which is where, you might say, all the action is)?
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
Y = 1-ay2+bx
for x and y to get
x = (Y-1+aX2)/b
y = X
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
y = 1-ay2+bx
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
0 1
b -2ay
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.
References
The literature is vast, and I have listed here only
the few items I have used extensively.
-
Bill Casselman,
`Picturing the horseshoe map',
Notices of the American Mathematical Society,
May 2005.
-
Predrag Cvitanovic, Gemunu Gunaratne,
and Itamar Procaccia,
`Topological and metric properties of Hénon-type
strange attractors',
Physical Review A 38, 1503 - 1520.
This contains the most detailed account I have seen
of the pruning front conjecture.
-
Predrag Cvitanovic and many others,
Classical and quantum chaos, available
at ChaosBook.org.
A huge but very readable introduction.
-
M. Hénon,
`A two-dimensional mapping with a strange attractor',
Communications in Mathematical Physics
50 (1976), 69-77. Original introduction,
with calculations and pictures for a=1.4, b=0.3.
-
John Milnor and William Thurston,
`On iterated maps of the interval', page 465
in Lecture Notes in Mathematics 1342,
1988. This is the classic account of the 1D case.
-
David Ruelle,
`What is a strange attractor?',
to appear in the Notices of the American Mathematical Society,
August 2006. This is a short summary of the state of rigorous results for
some classes of strange attractors.
Bill Casselman
University of British Columbia, Vancouver, Canada
cass
at math.ubc.ca
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. |
