How can I prove H(x,y) is a Lyapunov function for this system?

[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.