Consider the two parameter family of linear systems.
$$y'_{1}(t)=ay_{1}(t)+by_{2}(t)$$ $$y'_{2}(t)=-by_{1}(t)+ay_{2}(t)$$
I know my determinant is $a^2+b^2$ and my trace is $2a$. To find my eigenvalues I think I look at $a\pm bi$.
In the $ab$ plane identify all the regions where this system possesses a saddle, a sink, a spiral sink.
How do I go about this?
Solution 1:
Consider that $$ (y_1+iy_2)'=(a-ib)(y_1+iy_2) $$ and $$ (y_1-iy_2)'=(a+ib)(y_1-iy_2). $$