Lyapunov functions and phase portraits

A student asked me: "I have a question regarding Q3 on the 2008 M2AA1 paper. In the question you are asked to sketch the phase portrait of the system for various parameters. The question gives you two pictures of the lyapunov function for these parameters. I was wondering how one might use a lyapunov function to deduce the phase portrait as the question suggests. I have checked the answers for that question which don't give much detail, however the portraits look remarkable similar to the lyapunov functions. "

Let \(V(x)\) be a Lyapunov function, ie \(\frac{d}{dt}V(x)\leq 0\). Then \(V(x(t)\) must not increase when \(t\) increases. It is also insightful to note that \(\frac{d}{dt}V(x)=\nabla V(x) \cdot\dot{x}\leq 0\). Recall that \(\nabla V(x)\) is the normal to level set \(\{\tilde{x}~|~V(\tilde{x})=V(x)\}\) at the point \(x\). Thus \( \nabla V(x)\cdot \dot{x}\leq 0\) indeed means that the vector \(\dot{x}\) does not point in the direction of higher level sets of \(V\)and thus that if \(t\geq\tau\) then \(V(x(t))\leq V(x(\tau))\).