gradient system
A gradient system in is an autonomous ordinary differential equation
(1) |
defined by the gradient of where and . 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.