Re: [Dynamical Systems] Convergence to a fixed point
- From: israel@xxxxxxxxxxx (Robert Israel)
- Date: 2 Dec 2005 19:30:10 GMT
In article <dmp8t1$i3n$1@xxxxxxxxxxxxxxxxxxxxx>,
LordBeotian <pokispy76@[CANCELLA QUESTO]yahoo.it> wrote:
>
>"LordBeotian" <pokispy76@[CANCELLA QUESTO]yahoo.it> wrote
>
>>I have a differetiable map T with an unstable fixed point x_0.
>
>I mean "hyperbolic fixed point".
>
>> How can I estimate how many iterations does it take for a point in the
>> stable manifold at a distance O(1) from x_0 to reach a distance
>> O(epsilon)?
>
>I mean the stable manifold *of x_0*.
Let S be the set of eigenvalues of the Jacobian matrix of T at x_0
that have absolute value <= 1, and r the maximum of the absolute
values of these. If r < 1, then for typical starting
points y_0 in the stable manifold the sequence of iterates y_n
should have ||y_{n+1} - x_0||/||y_n - x_0|| -> r as n -> infinity.
For any r' with 1 > r' > r, the number of iterations to go from
distance 1 to epsilon should be less than ln(epsilon)/ln(r') for
sufficiently small epsilon.
Robert Israel israel@xxxxxxxxxxx
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
.
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