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Differential Equations

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Moderate

Quadratic Lyapunov Function: Stability for $(0,0)$

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Consider

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

By choosing a Lyapunov function of the quadratic form $V=ax^2+bxy+cy^2$ where $a>0, c>0$ suitably, we conclude that $(0,0)$ is

A

Asymptotically stable

B

Stable but not asymptotically stable

C

Unstable

D

None of the above