In differential topology, a mathematical discipline, and more specifically in Morse theory, a gradient-like vector field is a generalization of gradient vector field. The primary motivation is as a technical tool in the construction of Morse functions, to show that one can construct a function whose critical points are at distinct levels. One first constructs a Morse function, then uses gradient-like vector fields to move around the critical points, yielding a different Morse function.
Definition Given a Morse function f on a manifold M, a gradient-like vector field X for the function f is, informally:
away from critical points, X points "in the same direction as" the gradient of f, and near a critical point (in the neighborhood of a critical point), it equals the gradient of f, when f is written in standard form given in the Morse lemmas. Formally:
away from critical points, X ⋅ f > 0 , {\displaystyle X\cdot f>0,}
around every critical point there is a neighborhood on which f is given as in the Morse lemmas:
f ( x ) = f ( b ) − x 1 2 − ⋯ − x α 2 + x α + 1 2 + ⋯ + x n 2 {\displaystyle f(x)=f(b)-x_{1}^{2}-\cdots -x_{\alpha }^{2}+x_{\alpha +1}^{2}+\cdots +x_{n}^{2}}
and on which X equals the gradient of f.
Dynamical system The associated dynamical system of a gradient-like vector field, a gradient-like dynamical system, is a special case of a Morse–Smale system.
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