Consider the ODE system $$ \dot{x}=y,\qquad \dot{y}=x-x^2. $$ Its Hamiltonian is given by $$ H(x,y)=\frac{y^2}{2}-\frac{x^2}{2}+\frac{x^3}{3}. $$ The solution that corresponds to $H=0$ and $x<0,y<0$ satisfies the equation $$ \dot{x}=y=x\sqrt{1+\frac{2\lvert x\rvert}{3}}. $$
It is said that this solution tends to infinity as $t$ grows.
How can I see that?