Consider the function

$$V(x, y)=x^{4}-x^{2}+2 x y+y^{2}$$

Find the critical (stationary) points of $V(x, y)$. Determine the type of each critical point. Sketch the contours of $V(x, y)=$ constant.

Now consider the coupled differential equations

$$\frac{d x}{d t}=-\frac{\partial V}{\partial x}, \quad \frac{d y}{d t}=-\frac{\partial V}{\partial y}$$

Show that $V(x(t), y(t))$ is a non-increasing function of $t$. If $x=1$ and $y=-\frac{1}{2}$ at $t=0$, where does the solution tend to as $t \rightarrow \infty$ ?