Trace Determinant Plane Differential Eqns

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?

Solution 1:

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