I am working on the stability of the a discrete-time non-linear system described as follows.
$$x_1(k+1) = a_1 x_1(k) - \frac{x_1(k)}{x_1(k)+ x_2(k)}[a_2 x_3(k) + a_3 x_4(k) + a_4] + a_5 $$
$$x_2(k+1) = b_1 x_2(k) - \frac{x_2(k)}{x_1(k)+ x_2(k)}[a_2 x_3(k) + a_3 x_4(k) + a_4] + b_2 $$
$$x_3(k+1) = c_1 x_3(k) + \frac{x_3(k)}{x_3(k)+ x_4(k)}[c_2 x_1(k) + c_3 x_2(k) + c_4] + c_5 $$
$$x_4(k+1) = d_1 x_4(k) + \frac{x_4(k)}{x_3(k)+ x_4(k)}[c_2 x_1(k) + c_3 x_2(k) + c_4] + d_2 $$
I tried to simply these equations and represent these of form
$$x(k+1) M_1= x(k) M_2 + M_3$$
where, $x(k+1) = [x_1(k+1) \quad x_2(k+1) \quad x_3(k+1) \quad x_4(k+1)]$ and $M_1, M_2, M_3$ being three matrices. However, due to the presence of the division operation, I am not able to remove some $x_i(k)x_i(K+1)$ terms.
I tried simple arithmetic operations to obtain the followings. Here, $A_i$ and $B_i$, $\forall i \in[1,5]$ can be constructed from the above equations.
$$x_1(k+1) + x_2(k+1) + x_3(k+1) + x_4(k+1) = A_1 x_1(k+1) + A_2 x_2(k+1) + A_3 x_3(k+1) + A_4 x_4(k+1) + A_5$$ and $$x_1(k+1) + x_2(k+1) - x_3(k+1) - x_4(k+1) = B_1 x_1(k+1) + B_2 x_2(k+1) + B_3 x_3(k+1) + B_4 x_4(k+1) + B_5$$
Using these, we can derive $M_1 = \begin{bmatrix} 1 & 1 \\ 1 & 1 \\ 1 & -1 \\ 1 & -1 \end{bmatrix} $, $M_2 = \begin{bmatrix} A_1 & B_1 \\ A_2 & B_2 \\ A_3 & B_3 \\ A_4 & B_4 \end{bmatrix} $. However, this doesn't help to show the stability of the above system.
I tried finding examples related to these kind of non-linear systems, but failed to achieve any success. Any direction that could help to proof the stability of such systems will be very much helpful. Thanks.