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Math Help - Writing about a function. Help me understand function iteration notation

  1. #1
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    iaoalst
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    Writing about a function. Help me understand function iteration notation

    I have a kind of weird assignment. I have the following function:

    g(x) = αg(x/α)

    Where g apparently should be interpreted as the second iteration and not g squared. My assignment is to write as much as possible about this function and it's properties. The lecture was loosely connected to chaos, dynamical systems and the logistic function but I was told that it could be "solved" without knowing anything about dynamical systems or what was said during the lecture.

    I have some trouble understanding the notation. Is there anyway to rewrite in a normal IMHO more understandable form? Is g(x) = g(g(x)) ?

    Where should I begin? Can someone help me by at least pointing me in the right direction by letting me know how and if it can be rewritten?
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  2. #2
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    Re: Writing about a function. Help me understand function iteration notation

    Quote Originally Posted by xaosman View Post
    I have a kind of weird assignment. I have the following function:

    g(x) = αg(x/α)

    Where g apparently should be interpreted as the second iteration and not g squared. My assignment is to write as much as possible about this function and it's properties. The lecture was loosely connected to chaos, dynamical systems and the logistic function but I was told that it could be "solved" without knowing anything about dynamical systems or what was said during the lecture.

    I have some trouble understanding the notation. Is there anyway to rewrite in a normal IMHO more understandable form? Is g(x) = g(g(x)) ?

    Where should I begin? Can someone help me by at least pointing me in the right direction by letting me know how and if it can be rewritten?
    I think that your interpretation of the notation is correct.

    g(x)=ag(g(\frac xa))=ag^2(\frac xa) now replacing x with x / a

    g(\frac xa)=ag(g(\frac x{a^2})) substituting back gives g(x)=ag(ag(g(\frac x{a^2})))

    And by the same reasoning:

    g(x)=ag(g(\frac x{a}))=ag(ag(g(\frac x{a^2})))=ag(ag(ag(g(\frac x{a^3}))))=ag(ag(ag(ag(g(\frac x{a^4})))))...

    Oops. Out of time.
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