Topological Methods in Nonlinear Analysis

On trajectories of analytic gradient vector fields on analytic manifolds

Aleksandra Nowel and Zbigniew Szafraniec

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Let $f\colon M\to {\mathbb R}$ be an analytic proper function defined in a neighbourhood of a closed "regular" (for instance semi-analytic or sub-analytic) set $P\subset f^{-1}(y)$. We show that the set of non-trivial trajectories of the equation $\dot x =\nabla f(x)$ attracted by $P$ has the same Čech-Alexander cohomology groups as $\Omega\cap\{f< y\}$, where $\Omega$ is an appropriately choosen neighbourhood of $P$. There are also given necessary conditions for existence of a trajectory joining two closed "regular" subsets of $M$.

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Topol. Methods Nonlinear Anal., Volume 25, Number 1 (2005), 167-182.

First available in Project Euclid: 23 June 2016

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Nowel, Aleksandra; Szafraniec, Zbigniew. On trajectories of analytic gradient vector fields on analytic manifolds. Topol. Methods Nonlinear Anal. 25 (2005), no. 1, 167--182.

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