Exploring Constraints, Solvability and Stability of a Circular System of Complex Riccati Type Difference Equations

Abstract

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 then the expression of the solutions are given in terms of their initial values. Next, the asymptotic behaviour of the solutions is studied. Finally, in case of negative Riccati real numbers
$$
R_j:=\frac{a_j d_j-b_j c_j}{\left[a_j+d_j\right]^2}, \quad j \in \overline{1, k},
$$
it is shown that there exists a unique positive fixed point which attracts all solutions starting from positive states.