Consider the Hamiltonian function $$H(x_1, x_2, y_1, y_2) = \frac{1}{2}(x_1^2 + x_2^2) + x_1^2 x_2 - \frac{1}{3}x_2^3 + \frac{1}{2} (y_1^2 + y_2^2)$$
a) The corresponding Hamiltonian system:
\begin{align} \dot x &= \frac{\partial H}{\partial \vec y} = (y_1, y_2)^T \\ \dot y &= -\frac{\partial H}{\partial \vec x} =(-x_1 -2x_1x_2, -x_2 - x_1^2+x_2^2)^T \end{align}
b) I found these critical points:
$(0, 0, 0, 0), (0, 1, 0, 0), (\frac{\sqrt 3}{2}, -\frac{1}{2}, 0, 0), (-\frac{\sqrt 3}{2}, -\frac{1}{2}, 0, 0).$
c) If we linearize this system around the critical point at the origin then we get: \begin{bmatrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ -1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ \end{bmatrix}
which has the following characteristic polynomial $$(\lambda^2 + 1)^2,$$ thus this is a non-hyperbolic critical point, so I used Lyapunov functions to investigate its stability. The Lyapunov functions I tried are: $$V = c_1x_1^2 + c_2x_2^2 + c_3y_1^2 + c_4 y_2^2$$ and $$V = c_1x_1^2 + c_2x_2^2 + c_3y_1^2 + c_4 y_2^2 + c_5 x_1^4 + c_6 x_2^4 + c_7 y_1^4 + c_8 y_2 ^4$$
I couldn't get these functions to work.
For the critical point $(0, 1, 0, 0)$ I got the following eigenvalues for its linearized system: $\lambda = \pm 1, \pm \sqrt 3 i.$ How do I go about classifying this critical point?
Thanks.