Suggestion for a Lyapunov function

Question

Consider the differential system $$ x'=x+y $$ $$y'=x-y+xy$$

What would be a Lyapunov function for this system at $(0,0)$?

I have considered functions $V(x,y)=ax^{2n}+by^{2m}$ but none of them seems to be Lyapunov.

Answer 1

As it is indicated in comments Lyapunov function is usually used to infer the stability of the equilibrium. There is a useful theorem by Chetaev, which allows one to build a function (often called Chetaev's function in Russian-language literature) to prove instability. In your case you can take $$ V(x,y)=xy, $$ which is positive in the first quadrant, zero on its boundary, and the origin also belong to the boundary.

I have $$ \dot V=y^2+x^2+x^2y>0, $$ for $x,y>0$. Hence, by the aforementioned theorem, the origin is unstable.

Answer 2

We refer to Liapunov's technique to determine the stability only when the linearized system is not hyperbolic at the equilibrium point.

Now ,bearing this in mind, let's just check the linearized system, which can be derived by simply dropping the non-linear terms : $\left( \begin{array}{cc} 1 & 1 \\ 1 & -1 \\ \end{array} \right)$

This system has a positive eigenvalue ($\sqrt{2}$) and a negative one ($-\sqrt{2}$), so it's hyperbolic, and from the linearization we can state that the origin is a saddle (hence not stable).

Bottom line, search for a Liapunov function only when the equilibrium is not hyperbolic. If it is hyperbolic, such as your case, then you're a lucky boy! (or a lucky girl if you're a girl)

;)