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

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

defined by the gradient of $V$ 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 $x_{0}$ 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).

## References

Title gradient system GradientSystem 2013-03-22 15:14:25 2013-03-22 15:14:25 Daume (40) Daume (40) 7 Daume (40) Definition msc 34A34