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:
In these, sometimes we can find a Lyapunov function of the form for suitable choices of a and b such that at the critical point , if in some region about and in that same region. If so, then the point is stable. Well the critical point is the origin so let and and then:
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.