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 \(V(x(t))\leq V(x(\tau))\).

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 \(V(x(t))\leq V(x(\tau))\).

I am absoluetly worst in maths. I passed my maths exams with a lot of prayers and not by using my brain. So I am sorry but I really did not understand any of this.

ReplyDelete