(adsbygoogle = window.adsbygoogle || []).push({}); 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 : x_{n+1}= f (x_{n}) (1)

Linear dierence eq. model : v_{n+1}= f '(x_{∞})v_{n}(2)

If all solutions of (2) at x_{∞}tend to 0 as n → ∞ then

all solutions of (1) with x_{0}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

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

Can you offer guidance or do you also need help?

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