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How to determine Local Asymptotic Stability,Monotonic, Oscillatory, etc

  1. Jun 6, 2012 #1
    1. The problem statement, all variables and given/known data

    I'm trying to figure how to to determine if a function is Locally Asymptotic Stable or Instable.

    I also need to know if the approach of the function is monotone or oscillatory.

    2. Relevant equations

    Non-linear dierence eq. model : xn+1 = f (xn) (1)

    Linear dierence eq. model : vn+1 = f '(x)vn (2)


    If all solutions of (2) at x tend to 0 as n → ∞ then
    all solutions of (1) with x0 suciently close to x tend to x as n → 1
    [local asymptotic stability of x]
    approach to x is monotone if 0 < f '(x) < 1
    approach to x is oscillatory if 1 < f ' (x) < 0
    solution move away from x as n → 1, if |f '(x)|> 1 [instability of x]

    3. The attempt at a solution

    I am trying to figure out the following two conditions:

    f ' (K) >1
    i.e. 1 - r > 1, which then becomes r < 0

    f ' (K) < 0
    i.e. 1 - r < -1 , which then becomes r > 2

    Thanks
     
  2. jcsd
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