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

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