Gradient dynamical systems have no nonconstant recurrent solutions

Question

This is given as something we should intuitively understand, but I don't see how this is trivial. We were given that a solution is recurrent if $X(t_n) \to X(0)$ for some sequence from $t_n$ to infinity. Why is this true?

Answer 1

Consider the gradient function $\Phi = \Phi({\bf x})$,

$$ \frac{{\rm d}{\bf x}}{{\rm d}t} = -\nabla \Phi \tag{1} $$

Assume the system is recurrent

$$ \Phi({\bf x}(t_n)) = \Phi({\bf x}(0)) \tag{2} $$

But on the other hand

$$ 0 \stackrel{(2)}{=}\int {\rm d}\Phi = \int_0^{t_n} \nabla \Phi \cdot \frac{{\rm d}{\bf x}}{{\rm d}t} {\rm d}t \stackrel{(1)}{=} -\int_0^{t_n} \left| \frac{{\rm d}{\bf x}}{{\rm d }t}\right|^2{\rm d}t < 0 $$

So you get to a contradiction. A gradient dynamical system cannot have recurrent orbits!