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Consider differential equation \begin{align}x'&=-t(x+y)\\ y'&=-y+x-y(y^2-6).\end{align} Can some one suggest a lyapunov function for it.

I have examined $V(x,y)=x^2+y^2$ , $V(x,y)=(1+t^2)(x^2+y^2)$ , $V(x,y)=(1+t^2)^2(x^2+y^2)$ but I could not solve the stablity with these functions.

Thanks

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  • $\begingroup$ Are you assuming $(0,0)$ is a stable solution? I ran a few sample solutions at fixed values of $t$, and it looks like the stable solutions are $(-2,2)$ and $(2,-2)$, not $(0,0)$. $\endgroup$
    – hasnohat
    Jan 13, 2015 at 8:16
  • $\begingroup$ Yes for $(0,0)$ $\endgroup$
    – Fin8ish
    Jan 13, 2015 at 8:37
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    $\begingroup$ As far as I remember, there's a kind of general recommendation. If equilibrium is hyperbolic and you know a Lyapunov function for its linearization, then the same Lyapunov function would work for non-linear system at least in a small neighbourhood of this equilibrim. $\endgroup$
    – Evgeny
    Jan 13, 2015 at 12:46

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