For dx/dt = y^3 - 2*x^3
dy/dt = -3*x - y^3
Show that the origin is nonlinearly stable by nding the appropriate Lyapunov function.
Hey, I don't have it but let me try to rough it in ok:
So we have:
$\displaystyle x'=y^3-2x^3$
$\displaystyle y'=-3x-y^3$
In these, sometimes we can find a Lyapunov function of the form $\displaystyle ax^2+by^2$ for suitable choices of a and b such that $\displaystyle L(p_0)=0$ at the critical point $\displaystyle p_0$, $\displaystyle L(x)>0$ if $\displaystyle x\neq p_0$ in some region about $\displaystyle p_0$ and $\displaystyle F\cdot\nabla L \leq 0$ in that same region. If so, then the point $\displaystyle p_0$ is stable. Well the critical point is the origin so let $\displaystyle a>0$ and $\displaystyle b>0$ and then:
$\displaystyle F\cdot\nabla L =(y^3-2x^3,-3x-y^3)\cdot(2ax,2bx)=2axy^3-4ax^4-6bxy-2by^4$
$\displaystyle =2axy^3-6bxy-(4ax^4+2by^4)$
The last term is always negative. However, I can't show that for suitable choices of a and b, the entire expression is less than or equal zero in some deleted neighborhood of the origin.