Chapter 5.3, Problem 1E

For the system

$\begin{array}{l}\frac{dx}{dt}=Y\\ \frac{dy}{dt}=x^3-x\end{array}$

(a)show that the system is Hamiltonian with Hamiltonian function

$H\left(x,y\right)=\frac{y^2}{2}+\frac{x^2}{2}-\frac{x^4}{4}$

(b) sketch the level sets of H, and

(c) sketch the phase portrait for the system. Include a description of all equilibrium points and any saddle connections.