Let $A$ be a Banach or a $C^*$ algebra. We consider the differential equation $$(*)\;\;\;\;Z'=Z^2-Z$$ on $A$.

Obviously the singularities of this systems are just the idempotents of the algebra.

It can be easily shown that the group of invertible elements is invariant under this flow. (**Edit:According to the answer of Robert Israel we get that the space of Left zero divisors is flow invariant too**.) Furthermore the group of invertible elements does not contain any periodic orbit(Except the trivial case of singularity $Z_0=1$ but strictly speaking a singular point can not be regarded as a periodic orbit).
Furthermore non of the following algebras can have a periodic orbit of the above systems:

1) The Matrix algebra

2)$C^*_{\text{red}} F_1=C^*_{\text{red}} \mathbb{Z}=C(S^1)$

Our questions:

1)Is there a Banach or $C^*$ algebra $A$ for which the system $(*)$ has a periodic orbit?

2)In the literature, are there some researches devoted to Kaplansky or Kadison Kaplansky conjecture via dynamical consideration of the equation $(*)$? As we see in this post, the three key elements of the Kaplansky conjecture is meaningfully involved with the dynamical interpretation of $Z'=Z^2 -Z$.These $3$ concepts are "Invertibles", "zero divisors" and "idempotent".

**Proof of the fact that the group of invertible elements of a $C^*$ algebra $A$ is invariant under flow of $(*)$:**

The group of invertible elements of $A$ is denoted by $G(A)$.Let $Z(t)$ be a solution of $(*)$ with $Z(0)=Z_0\in G(A)$. For some $t_0>0$, let $Z(t)\in G(A),\; \forall t\in [0,t_0)$ but $Z(t_0)$ is non invertible. Note that $W(t)=Z(t)^{-1}$ is a solution of $$(**)\;\; W'=W-I$$ Obviousely this vector field $(**)$ is a complet vector field, i.e. all soltions has maximal interval of definitions equal to $(-\infty, +\infty)$. In particular $W(t)$ is defined at $t_0$ hence $W(t)$ is bounded around $t_0$. This situation contradicts to the following lemma which is proved in Functional Analysis by W. Rudin.(10.17 lemma page 256).

**Lemma:** Let $Z_n$ be a sequence of invertible elements of a Banach algebra which converges to a non invertible element then the sequence $W_n=Z_n^{-1}$ is an unbounded sequence.

**Remark:** Please see the comment conversations to the following link as some suggestions for consideration of dynamical methods in the idempotent problem.