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- From: dtang@virtual5.harvard.edu (Diane Tang)
- Newsgroups: sci.fractals
- Subject: Billiard Balls and Lyapunov exponents
- Message-ID: <1993Jan8.161122.28062@das.harvard.edu>
- Date: 8 Jan 93 16:11:22 GMT
- Article-I.D.: das.1993Jan8.161122.28062
- Sender: usenet@das.harvard.edu (Network News)
- Distribution: na
- Organization: Aiken Computation Lab, Harvard University
- Lines: 36
-
- Hi -- I realized that I wasn't very clear in my first posting, so I'm
- going to try again.
-
- I'm working on a project involving billiard balls in a convex space,
- i.e., an ellipse, stadium, circle, etc. Currently, my program just
- calculates the trajectory (given an initial velocity and position).
- When the ball collides with a wall, the angle of incidence = angle of
- reflection. There's no gravity, not acceleration in fact, and no
- spin. I do realize that this problem is pretty much deterministic.
-
- However, even though it may be deterministic, I think that the
- lyapunov exponents for the ellipse and stadium are positive (and 0 for
- the circle), and I want to calculate them.
-
- My current implementation is to take two points with slightly
- different starting points, iterate them through the system, and
- calculate the "x" and "y" exponents. i.e.,
- l_x = 1/n sum( log_2 d_(i+1)_x - d_i_x )
- --------------------
- d_i_x
- where d_i_x = separation between the 2 pts at the ith iteration.
-
- The problem with this implementation is that the direction of the
- largest change is probably not in the x or y direction.
-
- Alternatively, I could just calculate the largest exponent (just do
- the total change), but I would really like to be able to get two
- exponents.
-
-
- Any ideas, suggestions, references, etc.?
-
- Thanks a lot in advance!
-
- -- Diane
- dtang@husc.harvard.edu
-