Re: logistic map - periodic locations beyond critical point

[EMAIL PROTECTED] (Robert Israel) wrote in message news:<[EMAIL PROTECTED]>...
> In article <[EMAIL PROTECTED]>,
> Claus <[EMAIL PROTECTED]> wrote:
> 
> >Thanks - so you are saying that between 3.7-4 its ALL periodic windows
> >??  Thats news to me.. Can you name a parameter value that is periodic
> >and has LOTS of iterates/points ??  I will try 3.900000001
> 
> Are you asking about periodic orbits, or about stable periodic orbits?
> There should be periodic orbits all over the chaotic region.
> For example, for r=3.9 the logistic map f: x -> r x (1-x) has two 
> unstable 3-cycles (approximately (.1326525274, .4487177896, .9647439150)
> and (.1809860054, .5780971634, .9512126972).  If you want stable 
> periodic orbits, one way is to look for values of r where 1/2 will
> be in a periodic orbit.  For example, at r=3.9, (f@@15)(1/2) < 1/2
> while at r=3.9001, (f@@15)(1/2) > 1/2 (where I'm using f@@n for f 
> iterated n times).  By the Intermediate Value Theorem
> there is some r between 3.9 and 3.9001 at which (f@@15)(1/2) = 1/2,
> and thus there is a stable periodic orbit of period 15 (or a divisor
> of 15).  It turns out to be approximately r=3.9000686655875783473 which
> has a stable periodic orbit of period 15.  
> 
> Robert Israel                                [EMAIL PROTECTED]
> Department of Mathematics        http://www.math.ubc.ca/~israel 
> University of British Columbia            
> Vancouver, BC, Canada V6T 1Z2

Dr. Israel,

I am looking for a periodic parameter value that has:
1- the most number of iterates in the map (out of curiosity, how many
iterates would this parameter generate?)
2- where I can reproduce the exact same iterates every time
3- is finite (not infinite, and not chaotic).  
That being the case is 3.99 for example this parameter value, or are
any of the parameters between say 3.9 -3.9999999999 periodic and
therefore would work ?