0
$\begingroup$

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-

  • 0
    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 i$f$ 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.2012-05-01

1 Answers 1

1

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?