I want to know if it is possible to determine if the following Hamiltonian system has unbound solutions. Let us consider the Hamiltonian function
$$ H(x,y,p_x,p_y) = \frac{1}{2}(p_x^2+p_y^2) + \frac{x^2y^2}{2}. $$ Let us consider the Energy level $H = 1/2$. Then, the dynamics occurs in a closed and unbound set given by
$$ \{(x, y) \in \mathbb{R} : - 1 \leq xy \leq 1\}. $$
We say that a solution is generic if $x(t) $ or $y(t) $ are not the constant function zero. Does the Hamiltonian system have a generic unbounded solution?
I wonder if a good strategy is to show that the lyapunov exponents are positive for the the solution $(x,y,p_x,p_y) = (t,0, 1,0) $