= Sheared quartic double-well gradient flow
{title2=$V=(x^2-1)^2+(x+y)^2-1$}
The completed-square form gives global minima at $(1,-1)$ and $(-1,1)$, both with value $-1$, and a <saddle point> at $(0,0)$ with value zero. Negative levels split into two closed components; the zero contour joins their lobes at the saddle. The <gradient flow> $\dot x=-V_x,\dot y=-V_y$ decreases $V$ by the <gradient-flow dissipation identity>. A negative initial level traps each trajectory on its original side of $x=0$, because $V(0,y)\geq0$.
Back to article page
