# Homework Help: How to determine Local Asymptotic Stability,Monotonic, Oscillatory, etc

1. Jun 6, 2012

### trojansc82

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