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

  • 0
    What is the definition of a fixed point here2012-11-09
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    That is the problem, I do not know it exactly. http://mathworld.wolfram.com/FixedPoint.html says that the derivative in this point has to be 02012-11-09
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    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.2012-11-09
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    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?2012-11-09
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    I think this is a mistake, the derivatives on the right sides can be cancelled.2012-11-09
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    _Stationary_ and _fixed_ are used indistinctively in the literature. What you are looking are points that don't move in time, i.e. where the derivative is zero, hence you need to put the LHS = 0 and solve for $f_1$, $f_2$ and $f_3$.2012-11-09
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    I get for f_1=f_3=0 and f_2=c. How can i evaluate the stability of the points?2012-11-09

1 Answers 1

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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.