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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-

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What have you tried? –  Chris Taylor May 1 '12 at 11:53
    
I know how to linearize the system around the origin, an then how to solve point a). Point b) however is escaping my mind. In particular, even if it is quite embarassing to say, i don't understand what globally means in tht context. So, yes, point b) is my main issue to deal with. –  guido giuliani May 1 '12 at 12:07
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up vote 1 down vote accepted

If your textbook uses the same terminology as I am used to: "globally asymptotically stable fixed point" is one where for any initial value $(x_0,y_0)$, the corresponding evolution will converge to the fixed point as $t\to \infty$.

To show that a fixed point is globally asymptotically stable, one usually resorts to constructing a global Lyapunov functional for the dynamical system.

To show that a fixed point is not globally asymptotically stable, it suffices to find a trajectory which does not converge to the fixed point. In your case in particular, I'll give you the hint: how many fixed points does this dynamical system have?

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