Re: how define a partition for symbolic dynamics in the logistic map ?

On 23 Nov 2003 07:43:03 -0800, Claus <[EMAIL PROTECTED]> wrote:
> Hi,
> 
> I am comparing vertical slices in the logistic map and want to compare
> the symbolic dynamics (or kneading interval) of 1 slice to
> another..that being the case, for each iterate, how do I define my
> partition IN LAYMAN'S TERMS ?  Do I select my parameter as the
> partition, the number 2 (1/2 of the Y axis), or something else ?

If you have a one-dimensional map, like the logistic map, 

        x(n+1) = F( x(n) )

then you can define a partition by breaking it at the critical points,
the local maxima and minima in the domain (or a discontinuity
in the function or slope). 

This will form a generating partition.   So if F(x) = a x(1-x)
then you would put your partition break point at x=1/2.  
(For the logistic map this happens to be independent of the parameter
a, but this isn't true in general). 

There are other generating partitions (perhaps an infinity
of them) but they can be more complicated and harder to find.

A generating partition is special in the sense that, in layman's
terms, the sequence of symbols is "equivalent" to the sequence of
actual orbits in the continuous valued map.  Other arbitrary
symbolizations do not have this property.

Finding the generating partition for higher than 1-d systems is
substantially harder.

There are no firm constructive mathematical results at all, and only
in very recent years are there numerical algorithms to come up with
approximations/guesses. 

If you know the spectrum of unstable periodic orbits then there is
one method I know.  I recently published an even more approximating
method for the case of having just time series of observed data.

When I was in grad school, generating partitions were sort of like the
loch ness monster, magical things that may be 'out there' somewhere,
but nfew knew what they look like or how to find them reliably.

> Claus Hammer