Liapunov function (扁頭科學's question at Yahoo! Answers)

[Fernando Revilla]

Here is the question:

Construct a suitable Liapunov function of the form ax^2+cy^2,
where a and c are to be determined. Then show that the critical
point at the origin is of the indicated type.
1. dx/dt = -x^3+xy^2, dy/dt = -2x^2y-y^3; asymptotically stable

Here is a link to the question:

Differential equation (Liapunov function)? - Yahoo! Answers

I have posted a link there to this topic so the OP can find my response.

 

[Fernando Revilla]

Hello 扁頭科學,

If $f(x,y)=ax^2+cy^2$, then: $$L_v(f)(x,y)=\nabla f(x,y)\cdot (v_1,v_2)=(2ax,2cy)\cdot (-x^3+xy^2,-2x^2y-y^3)=\\-2ax^4+2ax^2y^2-4cx^2y^2-2cy^4$$ If $a=c=1$, we get $L_v(f)(x,y)=-2(x^4+x^2y^2+y^4)<0$ for all $(x,y)\ne (0,0)$. This means that $f$ is a Strict Lyapunov Function at $(0,0).$ As a consequence, $(0,0)$ is asymptotically stable.