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25 13 $begingroup$ I've been thinking about gradient flows in the context of Morse theory, where we take a differentiable-enough function $f$ on some space (for now let's say a compact Riemannian manifold $M$ ) and use the trajectories of the gradient flow $x'(t) = - operatorname{grad} f(x(t))$ to analyse the space. In particular the (un)stable manifolds $$W^pm(p) = { x in M | x to p textrm{ under gradient flow as } t to pm infty}$$ of critical points $p$ must fill up the whole space, which means that the gradient flow from each point must converge to a critical point of $f$ . Most references (I've been using Jost's Riemannian Geometry and Geometric Analysis ) simply claim that when $f$ is Morse (has non-degenerate Hessian at all critical points) the gradient flow always converges and then...