gradient system
A gradient system in ${\mathbb{R}}^{n}$ is an autonomous^{} ordinary differential equation^{}
$$\dot{x}=\mathrm{grad}V(x)$$  (1) 
defined by the gradient^{} of $V$ where $V:{\mathbb{R}}^{n}\to \mathbb{R}$ and $V\in {C}^{\mathrm{\infty}}$. The following results can be deduced from the definition of a gradient system.
Properties:

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The eigenvalues^{} of the linearization of (1) evaluated at equilibrium point are real.

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If ${x}_{0}$ is an isolated minimum of $V$ then ${x}_{0}$ is an asymptotically stable solution of (1)

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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 $V$.

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There does not exists periodic solutions of (1).
References
 HSD Hirsch, W. Morris, Smale, Stephen, Devaney, L. Robert: Differential Equations, Dynamical Systems^{} & An Introduction to Chaos. Elsevier Academic Press, New York, 2004.
Title  gradient system 

Canonical name  GradientSystem 
Date of creation  20130322 15:14:25 
Last modified on  20130322 15:14:25 
Owner  Daume (40) 
Last modified by  Daume (40) 
Numerical id  7 
Author  Daume (40) 
Entry type  Definition 
Classification  msc 34A34 