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I am a little bit unsure about the following problem:

Given the system:

$\dot{x} = y - x^3$

$\dot{y} = -x + 2x^3 -2y$

Using the Liapunov function $V(x,y) = x^2 + y^2$, show that the origin in the $(x,y)$-plane is asymptotically stable.

ATTEMPT AT SOLUTION:

OK, so the family of curves:

$V(x,y) = x^2 + y^2 = \alpha$, $0 < \alpha < \infty$

is a topographic system. We thus have:

$\dot{V}(x,y) = 2x\dot{x} + 2y\dot{y}$

$= 2x(y - x^3) + 2y(-x + 2x^3 -2y) = 2xy - 2x^4 - 2xy +4x^{3}y -4y^2 = -2x^4 + 4x^{3}y - 4y^2$

In order for us to achieve asymptotic stability, we must have that $\dot{V}(x,y) < 0$ for all points except the origin. However, I can not see this being the case here. Say we take $x = -5$, $y = -5$, then $\dot{V}(x,y) > 0$. So I must do something wrong here.

Any help/tips would be greatly appreciated!

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Never mind, I figured it out :). I was wrong in stating that this has to be valid in the entire (x,y)-plane. We only need to define an open neighborhood $N_{\mu}$ around the origin for which $\dot{V}(x,y) < 0$. In this case we achieve this by choosing $\mu = \sqrt{2}$. –  Kristian Mar 30 '12 at 12:19
    
You can answer your own question, since you solved it. –  Davide Giraudo Mar 30 '12 at 21:02

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