If you are just interested in the number of the fixed points of the flow, you can just put your question in terms of zeros of the vector field; in this case the right tool is the topological degree, which is stable under small perturbations of the field in the uniform norm; and in absolute value is (generically) a lower bound on the number of the zeros. Of course, if the vector field is variational (it's the gradient if a functional) much stronger invariants are available (all the Morse complex machinery).
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If you are just interested in the number of the fixed points of the flow, you can just put your question in terms of zeros of the vector field; in this case the right tool is the topological degree, which is stable under small perturbations of the field in the uniform norm; and in absolute value is a lower bound on the number of the zeros. Of course, if the vector field is variational (it's the gradient if a functional) much stronger invariants are available (all the Morse complex machinery). |
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