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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 $x_{n+1}=f(x_n)$

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What is the definition of a fixed point here –  Amr Nov 9 '12 at 1:23
    
That is the problem, I do not know it exactly. mathworld.wolfram.com/FixedPoint.html says that the derivative in this point has to be 0 –  Montaigne Nov 9 '12 at 1:28
    
Would there be a difference between a fixed point and a stationary point in this system of diff.equations ? Maybe it should mean stationary point, and not fixed point. –  Montaigne Nov 9 '12 at 1:30
    
Should the second and third equations be $f_2'(t) = f_1(t)$ and $2f'_3(t) = - f_1(t)$, or is it really derivatives on both sides? –  Christopher A. Wong Nov 9 '12 at 3:55
    
I think this is a mistake, the derivatives on the right sides can be cancelled. –  Montaigne Nov 9 '12 at 7:45
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1 Answer

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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.

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