Proof

[squenshl]

Consider the system:
dx/dt = y, dy/dt = 2x - 4x3 - y.

I know that the Hamiltonian H(x,y) = y2/2 - x2 + x4 + y2/2 = y2 - x2 + x4. But how do I show that H is a Lyapunov function for this system. Please help.

[squenshl]

Is it:
d/dt H(x(t),y(t)) = d/dt(y2 - x2 + x4) = y dy/dt + dx/dt(-2x + 4x3) = y(2x - 4x3 - y) + y(-2x + 4x3) = 2xy - 4x3y - y2 - 2xy + 4x3y = -y2 < 0. Since dH/dt < 0, this is a Lyapunov function.

[trambolin]

Also show that H is always positive for nonzero x,y. Then you are done.

[squenshl]

Cool.
Cheers.