Applied Math T-shirt 1999

[ti5] This years applied math T-shirt shows schematic pictures of the stable manifold of a hyperbolic fixed point of an area preserving map. The stable manifold of a fixed point is the set of points that are eventually mapped into that fixed point. The invariant manifold is the boundary of the colored region. The fixed point, which is not part of the picture, would be located to the right of the highest point of the picture. The half circle that extends to the right from this point is added to the picture for artistic reasons (in particular to obtain a closed curve that could be "filled"), it is not part of the manifold, which cannot intersect itself. The specific data telling us which loops to draw was obtained from a classical example of a chaotic map introduced by Hénon in 1969, (Quart. Appl. Math. 27, p291). Today we call that map after its inventor. As a system parameter is changed the number of "loops" in the manifold decreases, the parameter for the T-shirt pictures is close to the situation where the maximal number of loops exist.

[k60lev1234typeCol] The invariant manifold is a line of infinite length, therefore only part of it is shown. Distance along the invariant manifold is measured in terms of the number of iteration steps that are needed to get to a certain segment of the manifold close to the fixed point. This number is called the type. The different types are given different colors, where blue is type 1 and red is type 4. In this picture also part of the unstable manifold is shown: the vertical line. Every intersection of the stable and unstable manifolds is a homoclinic point. Homoclinic points are forward and backward asymptotic to the fixed point. The main picture shows the manifolds up to type 5, but for a smaller parameter, i.e. there are fewer loops.

[ti6] The sequence of pictures shown on the T-shirt shows loops up to type 3, 4, 5, and 6. Increasing the type increase the amount of details shown, in principle the amount of detail is unbounded, but it can obviously not be resolved on a T-shirt. The picture up to type 7 is already at the limit of a typical screen resolution. On the right you can see the loops up to type 6. On the T-shirt we also showed the actual formula for the iteration of the map in a somewhat cryptic form. It reads

d2 x = k - x2 - 2x ,
which is the second difference form of the map. The d2 represents the second difference operator, such that in longer form
xn+1 - 2xn + xn-1 = k - xn2 - 2xn .
If x0 and x1 are given all the other xn with n>1 can be calculated from this rule. k is the above mentioned parameter.

These pictures grew out of a research project about Homoclinic Bifurcations for the Hénon Map (pdf3) which is the work of David Sterling Holger Dullin, and James Meiss. The goal of this research is to gain a better understanding of how hyperbolic invariant sets are formed in chaotic systems. Even though the Hénon map is one of simplest and oldest examples of a chaotic systems there are still things to learn about it. Our knowledge about simple examples like this one will eventually help to understand nonlinear dynamical systems that are all around us.
Thanks to all the people who helped to figure out the final design, and a special thanks to Jo Iwanski for actually putting the pictures onto the T-shirts.
The T-shirt is for sale for $12 in the Applied Math office, ECOT 225.


Last modified: Wed Oct 21 18:45:53 MDT 1998