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I have this dynamical system in $\mathbb{R}^2$:

\begin{aligned} \dot{x}&=x \left( 1-x-\frac{1}{3} y\right)\\ \dot{y}&=y \left( \frac{3}{4}-y-\frac{1}{2}x \right) \end{aligned}

I found where $\frac{dx}{dt}$ and $\frac{dy}{dt}$ are $0$ and the nature of these points, after that I studied the global dynamical of the system now; I have to show that every points s.t. $\frac{dx}{dt}$ and $\frac{dy}{dt}$ are $0$ in that point is asymptotically stable in the future using an appropriate Lyapunov function, and then discuss the evolution (in the future) of a generic ($x_0$,$y_0$) with both $x_0$ and $y_0$ > $0$.

I really have a problem in finding the Lyapunov, anybody can help me?

Thanks

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  • $\begingroup$ " I have to show that every points s.t. dxdt and dydt are 0 in that point is asymptotically stable" Difficult to show this since only one fixed point amongst four is stable, the others being all instable. $\endgroup$ – Did Sep 6 '18 at 16:56
  • $\begingroup$ WP unhelpful?? $\endgroup$ – Cosmas Zachos Jan 31 at 14:53
  • $\begingroup$ And further. $\endgroup$ – Cosmas Zachos Jan 31 at 14:57

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