Quartic potential with fourfold symmetry ($f=x^2+y^2-\frac{1}{2}(x^4+y^4)-b{x}^2{y}^2$)

For $b>0$, the origin is a strict local minimum. The four axis points at distance one have Hessian eigenvalues $-4$ and $2(1-b)$. The four diagonal points with $x^2=y^2=1/(1+b)$ have Hessian eigenvalues $-4$ and $-4(1-b)/(1+b)$. Thus for $b<1$ the diagonal points are the global maxima and the axis points are saddle points of a scalar function; for $b>1$ these roles reverse. In polar coordinates, $f=r^2-\frac{1}{2}q(\theta )r^4$, where $q=1+\frac{1}{2}(b-1){\sin }^2(2\theta )>0$. Maximizing over $r$ gives $1/(2q)$ and proves the global statements. At $b=1$, $f=\frac{1}{2}-\frac{1}{2}(r^2-1{)}^2$, so the entire unit circle is the maximum set.