Solution
= Solution
Take $H(x,y)=y^2/2+x^2/2-x^4/4$. Along a trajectory,
$$
\dot H=y(x^3-x)+(x-x^3)(y+ax-bx^3)=(x-x^3)(ax-bx^3).
$$
Integrating over a <periodic orbit> returns $H$ to its starting value, and hence
$$
\boxed{\oint_\Gamma(x-x^3)(ax-bx^3)\,dt=0.}
$$
This is the <energy obstruction for a cubic planar oscillator>.