Since it's quite a long time I've gone through mathematical physics problems, I'm quite rusted with those topics, so I welcome cheerfully all your answers:
For every $\alpha\in[0,1]$ we consider the following system $\left(\begin{array}{c}\dot{x}\\\dot{y}\end{array}\right)=\alpha\left(\begin{array}{cc}0 & -1 \\ 1 & 0\end{array}\right)\left(\begin{array}{c}x \\ y\end{array}\right)+(1-\alpha)\left(\left(\begin{array}{cc}-\frac{1}{10} & -1 \\ 1 & -\frac{1}{10}\end{array}\right)\left(\begin{array}{c}x \\ y\end{array}\right)-\left(\begin{array}{c}0 \\ x^2\end{array}\right)\right).$
a) For every $\alpha\in[0,1]$ determine whether the origin is stable, asymptotically stable or unstable.
b)Prove that for every $\alpha\in[0,1]$ the origin is not globally asymptotically stable.
Thanks in advance and regards.
-Guido-
Edit After consulting a textbook I've managed to solve completely part a), but still I cannot figure out the solution of part b). I'm afraid I'm completely stuck so I cannot show my work, because there is none. At any rate, as I said, my issue is part b) so I'm renewing my ask for help. Again Thank you and best wishes.
-Guido-