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gradient system

(Definition)

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

$\displaystyle \dot{x}=-\operatorname{grad}V(x)$

(1)

defined by the gradient of

where $V:\mathbb{R}^n\to \mathbb{R}$ and $V\in C^\infty$ . 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 is an isolated minimum of then is an asymptotically stable solution of (1)
If is a solution of (1) that is not an equilibrium point then is a strictly decreasing function and is perpendicular to the level curves of .
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.

"gradient system" is owned by Daume.

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34A34 (Ordinary differential equations :: General theory :: Nonlinear equations and systems, general)

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