gradient system

A gradient system in ${ℝ}^n$ is an autonomous ordinary differential equation

$$\dot{x}=-\mathrm{grad}V(x)$$ (1)

defined by the gradient of $V$ where $V:{ℝ}^n\to ℝ$ and $V\in C^{\mathrm{\infty }}$. The following results can be deduced from the definition of a gradient system.
Properties:

• The eigenvalues of the linearization of (1) evaluated at equilibrium point are real.

• If $x_0$ is an isolated minimum of $V$ then $x_0$ is an asymptotically stable solution of (1)

• If $x(t)$ is a solution of (1) that is not an equilibrium point then $V(x(t))$ is a strictly decreasing function and is perpendicular to the level curves of $V$.

• There does not exists periodic solutions of (1).

Title	gradient system
Canonical name	GradientSystem
Date of creation	2013-03-22 15:14:25
Last modified on	2013-03-22 15:14:25
Owner	Daume (40)
Last modified by	Daume (40)
Numerical id	7
Author	Daume (40)
Entry type	Definition
Classification	msc 34A34