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Consider the 2 parameter family of linear systems

$\frac{DY(t)}{Dt} = \begin{pmatrix} a & 1 \\ b & 1 \end{pmatrix} Y(t) $

In the ab plane, identify all regions where this system posseses a saddle, a sink, a spiral sink, and so on.

I was able to get the eigenvalues as $\lambda = \frac{a+1}{2} \pm \frac{\sqrt{(a+1)^2 - 4(a-b)}}{2}$

but need help in finding the sink and source.

I got the spiral sink as: if $a \lt -1$

spiral source if $a \gt -1$

and center if $a = -1$

Can someone check this?

  • 2
    Note the title of your question. The trace of the matrix is $a+1$ and the determinant is $a-b$. The conditions will probably be in terms of these two quantities and not just in terms of $a$. Also, $p^2-4q$ should play a role as well where $p$ is the trace and $q$ the determinant (this appears in your formula somewhere).2011-08-02

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Summarizing the comments: the best way to begin is to look at determinant $a-b$ and trace $a+1$:

  • $a-b<0$: saddle
  • $a-b> 0$ and $a+1=0$: stable center
  • $a-b> 0$ and $a+1<0$: stable node or spiral, depending on $(a+1)^2-4(a-b)$ being positive or negative
  • $a-b> 0$ and $a+1>0$: unstable node or spiral, depending on $(a+1)^2-4(a-b)$ being positive or negative