= Solution
The equilibria are $\boxed{(0,0),\ (\pm\sqrt{2/3},0)}$. The Jacobian is $J=\bigl(\begin{smallmatrix}0&1\\2-9x^2&-1\end{smallmatrix}\bigr)$. At the origin it has <eigenvalues> $1,-2$ and tangent eigenvectors $(1,1)$ and $(1,-2)$, so the origin is a <saddle equilibrium>. At either nonzero point the <eigenvalues> are $\boxed{(-1\pm i\sqrt{15})/2}$, giving stable foci.
The <energy>-like function $H=y^2/2-x^2+3x^4/4$ has $\dot H=-y^2\leq0$. It is coercive at large distance, so trajectories remain bounded; there are no <periodic orbits> because their <energy> could not return to its starting value unless $y\equiv0$. The <stable manifold> of the <saddle equilibrium> separates the two attracting basins. Its unstable branches leave the origin and approach the right and left foci, respectively. Their negative <energy> near the origin prevents crossing back through $x=0$.
For either invariant manifold write $y=h(x)=sx+cx^3+O(x^5)$. Odd <symmetry> rules out even powers. Tangency to the vector field requires $h'(x)h(x)=2x-h(x)-3x^3$. Matching coefficients yields $s^2+s-2=0$ and $c(4s+1)=-3$. Hence
$$
\boxed{W^u:\ y=x-\frac35x^3+O(x^5),\qquad
W^s:\ y=-2x+\frac37x^3+O(x^5).}
$$
These are local graphs, not globally exact cubic curves.
\Image[/past-exam-of-the-topics-in-applied-mathematics-course-of-the-university-of-oxford/2007-damped-double-well.png]
{title=Damped double-well phase portrait: a saddle separates two stable foci, with cubic local invariant manifolds}
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