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The critical-point equations are $2xy=0$ and $x^2+y^2=1$, so the points are $(\pm 1,0)$ and $(0,\pm 1)$. The Hessian isThus $(0,1)$ is a strict minimum, $(0,-1)$ a strict maximum, and $(\pm 1,0)$ are saddles. The local contour patterns are respectively ellipses, inverted ellipses, and crossing hyperbolas.

The gradient flow has $\dot{x}=-2xy$ and $\dot{y}=1-x^2-y^2$. HencePutting $u=y^2$ gives $x{u}^{\prime }-u=x^2-1$. Multiplication by the integrating factor $1/x$ yieldsand therefore the general trajectory is

$dU/dt=\nabla U\cdot (-\nabla U)=-|\nabla U{|}^2\le 0$. The descending gradient flow trajectory from $(2,1)$ stays in its bounded contour basin and tends to the only critical point there, $(1,0)$.

For a smooth function $f$ on a Riemannian manifold, its Riemannian gradient is the unique vector field satisfying $g({\nabla }_{g}f,V)=df(V)$ for every vector field $V$. It depends on the Riemannian metric, and its negative generates the downward gradient flow.

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 $V(0,y)\ge 0$.

For real $\mu ,\beta$ and $\lambda >0$, writing $A=x+iy$ gives a gradient flow with potential $V=-(\mu +\beta )x^2/2-(\mu -\beta )y^2/2+\lambda (x^2+y^2{)}^2/4$. The origin has exponential asymptotic stability for $\mu <-|\beta |$, is algebraically attracting at equality, and is unstable above it. Nonzero stable equilibria are the real pair $\pm \sqrt{(\mu +\beta )/\lambda }$ for $\beta >0$, or the imaginary pair $\pm i\sqrt{(\mu -\beta )/\lambda }$ for $\beta <0$, when their squared amplitudes are positive. Differentiating the two real equations gives eigenvalues $-2(\mu +\beta ),-2\beta$ on the real branch and $-2(\mu -\beta ),2\beta$ on the imaginary branch. For $\beta =0$, $\mu >0$ gives a radially attracting circle with neutral phase, so individual equilibrium points have Lyapunov stability but are not individually asymptotically attracting.

A nonsmooth gradient flow driven by the subdifferential of a convex function. Explicit discretization gives the forward subgradient step; implicit discretization gives the proximal operator. Monotonicity of a convex subdifferential underlies uniqueness and contraction properties.

If $f$ is bounded along a downward gradient flow, the total time spent where $\Vert {\nabla }_{g}f\Vert \ge \epsilon$ is at most $(\sup f-\inf f)/{\epsilon }^2$. The gradient-flow dissipation identity proves this by integrating the indicator of that region. It bounds all visits together, without requiring the visited region to be connected.

Total variation flow is the $L^2$ gradient flow of the total variation seminorm on a domain. Formally it is $u_t=\mathrm{div}(\nabla u/|\nabla u|)$, but the subgradient formulation handles zero gradients and jumps. Adding quadratic fidelity gives a strictly convex stationary total variation denoising problem.