Sur les trajectoires du gradient d’une fonction analytique. (Trajectories of the gradient of an analytic function). (French) Zbl 0606.58045

Łojasiewicz, Stanisław

Semin. Geom., Univ. Studi Bologna 1982/1983, 115-117 (1984).

Theorem: Let f be an analytic function \(\geq 0\) in a neighborhood of 0 in \({\mathbb{R}}^ n\) such that \(f(0)=0\). Then there exists a neighborhood U of zero such that every trajectory \(y_ x\) of the system \(y'=-\text{grad} f(x)\) satisfying \(y_ x(0)=x\) is defined in [0,\(\infty)\), has finite length and tends to a point of \(Z=\{f(x)=0\}\) as \(t\to \infty\), with uniform convergence. We need the following inequality: If F is an analytic function in a neighborhood of 0 in \({\mathbb{R}}^ n\) such that \(F(0)=0\), then there exists \(0<\zeta <1\) such that \(| \text{grad} F(x)| \geq | F(x)|^{\zeta}\) in a neighborhood of zero. Remark: The theorem implies that every analytic set is (locally) a retract under deformation of its neighborhood.

MSC:

37-XX	Dynamical systems and ergodic theory
34C99	Qualitative theory for ordinary differential equations

Keywords:

analytic function; analytic set; retract