1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Generalizing recursion in mapping functions

  1. Oct 22, 2012 #1

    Pythagorean

    User Avatar
    Gold Member

    I have a mapping function:

    [tex]x_{n+1} = \mu (1-x_n) [/tex]

    I have some condition that occurs for:

    [tex] \mu (1-x_0) > 1 [/tex] (1)

    which is:

    [tex] x_0 < 1- \frac{1}{\mu} [/tex]

    but via the map function, there's an initial condition that leads to the above solution:

    **UNDER CONSTRUCTION, ERROR FOUND**
     
    Last edited: Oct 22, 2012
  2. jcsd
  3. Oct 23, 2012 #2

    Pythagorean

    User Avatar
    Gold Member

    Well... "solving" my error just confuses me more. Mapping functions with a shift in them are really unintuitive to me. So, from the top:

    Given:

    [tex] x_{n+1} = \mu (1-x_n) [/tex]

    Some special condition occurs at:

    [tex] x_{n+1} = \mu (1-x_n) [/tex]

    Which, in terms of initial condition, means that if:

    [tex] x_0 < 1- \frac{1}{\mu} [/tex]

    than the condition will be met. HOWEVER, there are also initial conditions that will map to the above space. How do I find them? For instance, I can reverse and apply the order of operations in the mapping functions (inverse scale, then shift opposite):

    [tex] x_0 < \frac{1- \frac{1}{\mu}}{\mu}+1 [/tex]

    but this gets unwieldy as I try to go back more and more iterations. Is there an inductive way to represent this "reverse mapping" series as a funciton of n (the number of iterations the mapping function requires). Or am I going about this all wrong?
     
  4. Oct 23, 2012 #3

    Stephen Tashi

    User Avatar
    Science Advisor

    It's unclear what you mean by mapping a condition to space. What space?
     
  5. Oct 23, 2012 #4

    Pythagorean

    User Avatar
    Gold Member

    the region defined by
    [tex] x_0 < 1- \frac{1}{\mu} [/tex]
     
  6. Oct 23, 2012 #5

    Stephen Tashi

    User Avatar
    Science Advisor

    I don't understand what you mean by "mapping" the condition to the space since it is the condition that defines the space.

    (If your are trying to ask a question about the attractors in the domain of iterated functions, it would be best to use the standard terminology for that subject - or give a link to a page that explains your question.)
     
  7. Oct 23, 2012 #6

    Pythagorean

    User Avatar
    Gold Member

    Well, I am, but I was trying to just focus on the micro-issue I'm having. I just want to find the basin of attraction for that region I defined, but im not sure if its really an attractor (it goes to infinity; this is the right side of the tent map for mu>2).

    Anyway, I will spend more time on it and if I don't get it, I will reformulate the question in more detail later.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Generalizing recursion in mapping functions
Loading...