Fixed Points of a system of diff.equations

Question

I have

$f'_1(t)=-af_1(t)f_2(t)+bf_3(t)$

$f'_2(t)=f_1(t)$

$2f'_3(t)=-f_1(t)$

How is it possible to evaluate fixed points of this system of equations and afterwards the stability of these points. I only know how to do it for simple equations like $x_{n+1}=f(x_n)$

Answer 1

First, calculate fixed points, then analyze the stability of the linearization.

Your fixed point is $f_1 = 0, f_3 = 0, f_2 = p$ if $b \ne 0$ and $p \in \mathbb{R}$.

Next we linearize,

$$\begin{pmatrix} -ap & 0 & b\\ 1 & 0 & 0\\ -1/2 & 0 & 0 \end{pmatrix}$$

Then calculate the eigenvalues for each $p$ to determine stability near each fixed point.