A dynamical system defined by the Riemann zeta function Let  $\zeta$ be the  classical Riemann  zeta  function.
We  define  a  differential equation  on $\mathbb{R}^{2} \setminus \{1\}$  by $\dot Z= \zeta(Z)$. From a foliation point  of  view this  vector  field  can  be  counted  as  a  smooth  vector  field  on whole  $\mathbb{R}^{2}$  with  the  following  equivalent formulation(They  have the same trajectories).
$$\dot Z= \lVert z-1\rVert^2  \zeta(Z)$$
Then the field  has  a  saddle  point  at  $1$.
Are  there  some researches   about this  dynamical system?Are there  closed orbits for  this  equation?The  latter is  equivalent  to ask: "Are there  zeroes  of  the  Riemann  Zeta function whose  Taylor  expansion (after  translation to the  origin  and real  rescalling ) is   in the  form $"iz+...."$. Every  zero  of  a  holomorphic  map  with this linear part    is  necessarily a center,  a  singularity surrounded  by  a  band  of  closed orbits.
 A: This is not an answer, but rather a long comment.  Software like Mathematica can plot trajectories of this dynamical system, by (numerically) solving the system of ODEs.  Here are trajectories that start on a circle of radius $1/10$ centered around the lowest nontrivial zero of $\zeta(s)$  (sorry, I can't make myself call the variable $Z$.)

The picture near the zero is exactly what you expect when you think of approximating $\zeta(s)$ by the linear approximation: the ODE in $s=\sigma+it$ is very nearly $\sigma^\prime=\sigma$ and $t^\prime=t$, at least, up to the scaling and rotation of the picture by $\zeta^\prime$ evaluated at the zero.  Further from the zero, higher order terms come in to play.
Here are trajectories that start on a (larger) circle of radius $1/2$ centered on the pole at $s=1$.  This picture is also what you should expects when you think about the Laurent expansion of $\zeta(s)$ at $s=1$

Update (Things I should have remembered from complex analysis in my original posting):  The function $f(s)=(s-1)\zeta(s)$ is holomorphic in $\mathbb C\backslash \{1\}$.  Let $u(\sigma,t)$ and $v(\sigma,t)$ be the real and imaginary parts, as functions of $s=\sigma+i t$  (NB: $t$ is not time in this notation.)  From the point of view of complex analysis, it makes more sense to look instead at the Polya vector field $(u,-v)$ , which is conservative.  Indeed, with
$$
F(s)=\int_{s_0}^s f(w)\, dw=U(\sigma,t)+iV(\sigma,t)
$$
we have
$\triangledown U=(U_x,U_y)=(U_x, -V_x)=(u,-v)$
The trajectories for this vector field are the level curves of $V$.
I don't believe there are any closed orbits.
A: A colleague gives me this article: 
Does the Riemann zeta function satisfy a differential equation?
where the next formula appear
$$\zeta'(z)=\zeta(z)\sum\limits_{n=1}^\infty{\ln p_n\over1-p_n^z}.$$
So, your original problem symplifies to the system
$$\dot{y}=F(z)y^2$$
$$\dot{z}=y$$
(of course $F(z)=\sum\limits_{n=1}^\infty{\ln p_n\over1-p_n^z}$).
This system have been studied for $F$s in many spaces. But I can't tell what kind of function is it.
I hope it helps. About closed orbits, I really don't know (but I don't think they are).
