TITLE: A dynamical system defined by the Riemann zeta function
QUESTION [6 upvotes]: 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.

REPLY [2 votes]: 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).