A gradient system in $\mathbb{R}^n$ is an autonomous ordinary differential equation \begin{equation} \dot{x}=-\operatorname{grad}V(x)\label{eq} \end{equation}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 () evaluated at equilibrium point are real.
• If $x_0$ is an isolated minimum of $V$ then $x_0$ is an asymptotically stable solution of ()
• If $x(t)$ is a solution of () 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 ().

References

HSD

Hirsch, W. Morris, Smale, Stephen, Devaney, L. Robert: Differential Equations, Dynamical Systems & An Introduction to Chaos. Elsevier Academic Press, New York, 2004.