Consider $$ \frac{dx}{dt}=y,\quad \frac{dy}{dt}=x+x^2-y. $$ Describe the set of all bounded orbits. Hint, use $H(x,y)=\frac{y^2}{2}-\frac{x^2}{2}-\frac{x^3}{3}$.
The equilibria are $$ O_1=(0,0),~~~O_2=(-1,0). $$
The linearization matrix of the system at $O_1$ is $A_1=\begin{pmatrix}0 & 1\\1 & -1\end{pmatrix}$ and so $O_1$ is a saddle. The linearization matrix of the system at $O_2$ is $A_2=\begin{pmatrix}0 & 1\\-1 & -1\end{pmatrix}$, hence $O_2$ is a stable equilibrium.
So, I know that at least $O_1$ and $O_2$ are bounded orbits. How can I determine all others?
I think candidates only can be some orbits that start in $O_1$ and end in $O_2$. But which ones?
Here is a phase portrait.
