The forbidden set, solvability and stability of a circular system of complex Riccati type difference equations

In this paper, the circular system of Riccati type complex difference equations of the form

$$u_{n+1}^{(j)}=\frac{a_j{u}_n^{(j-1)}+b_j}{c_j{u}_n^{(j-1)}+d_j},n=0,1,2,\cdots ,j=1,2,\cdots ,k,$$

where $u_n^{(0)}:=u_n^{(k)}$ for all $n$, is investigated. First, the forbidden set of the equation is given. Then the solvability of the system is examined and the expression of the solutions, given in terms of their initial values. Next, the asymptotic behaviour of the solutions is studied. Finally, in case of negative Riccati real numbers

$$R_j:=\frac{a_j{d}_j-b_j{c}_j}{[a_j+d_j{]}^2},j\in \overline{1,k},$$

it is shown that there exists a unique positive fixed point which attracts all solutions starting from positive states.