= Quartic double well with bilinear coupling
{title2=$g(x,y)=x^4+y^4-4a^2xy,\quad a>0$}
The <stationary points> of this <polynomial> are the origin and $(a,a)$, $(-a,-a)$. Its <Hessian matrix> at the origin has <eigenvalues> $\pm4a^2$, so the origin is a <saddle point>. At each of the other points the <eigenvalues> are $8a^2$ and $16a^2$, giving strict <local minima>. The identity
$$
g=(x^2-y^2)^2+2(xy-a^2)^2-2a^4
$$
shows that they are also the only <global minima>, with value $-2a^4$.
For $-2a^4<c<0$, the <level set> $g=c$ has two closed components, one surrounding each minimum. The zero <level set> has two lobes meeting at the origin with the coordinate axes as tangent directions. For $c>0$, there is one closed outer component. These statements also follow in <polar coordinates> from the quadratic equation in $r^2$, together with the absence of other <stationary points>.
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