IDC402 Non Linear Dynamics

2022-01-12

Stability of Orbits

If there is a $\delta$ error, will the orbit converge to given orbit?

Stability of fixed point

$F(x+\delta) = F(x)+F’(x)\delta + \mathcal{O}(\delta^2)$

Hence, $\delta F^n = (F’)^n$

Hence, if $||F’(x^*)||<1$, we have stability.

Stability of p Orbit

For stability of p Orbit, we find the stability of fixed points of $F^p$

$|\lambda_p|=\begin{cases} >1& \implies \text{unstable}
<1& \implies \text{stable}
=0& \implies \text{super stable} \end{cases}$

For the [2022-01-07#Tent Map|Tent Map], all orbits are unstable as $F’ = 2$ always.