A reference for this approach is the Guckenheimer & Holmes book,
"Nonlinear Oscillations ...", section 6.8. I agree, it is "heuristic".
The point -3/4 can be dealt with similarly. The function f(z) = z^2-3/4
has fixed point -1/2 with eigenvalue -1. So we use f(f(z)) to get
eigenvalue +1. Transform coordinates z = -1/2 + y so that we are
interested in what happens for y near 0. Consider the
transformations z^2 - 3/4 + i*r, for small, positive, r. After some
algebra, we get the differential equation:
dy 4 3 2 2
---- = y - 2 y + 2 I r y - 2 I r y - r
dt
(Here I = sqrt(-1).) The time the trajectory spends near 0 (say between
-1 and 1) is the integral
1
/
| 1
| ----------------------------------- dy
| 4 3 2 2
/ y - 2 y + 2 I r y - 2 I r y - r
-1
Its asymptotics are now shown in the articles called
"Asymptotics of an integral" in sci.math.symbolic. We conclude that
if n is the number of steps required for divergence, then
n is asymptotic to Pi/(2*r). Actually, this is the number of steps
for the composition f(f(z)), so the number of steps for f(z) itself
is double this, Pi/r. So: Mr. Boll's asymptotic behavior is vindicated.
This is a heuristic argument, so there could be some work done to make it
rigorous...
Now, is -5/4 the next spot to study? Or how about -2 ?
--
Gerald A. Edgar Internet: ed...@mps.ohio-state.edu
Department of Mathematics Bitnet: EDGAR@OHSTPY
The Ohio State University telephone: 614-292-0395 (Office)
Columbus, OH 43210 -292-4975 (Math. Dept.) -292-1479 (Dept. Fax)