I am trying to solving a linked recurrent relations.
$$ \left\{\begin{matrix} T_t &= &(a_{t}p_{n}+c_{t}+b_{t}p_r+d_{t})*(1-P_o)\\ a_{t+1} &= &T_t*f(T_t)+(1-p_n)*a_t\\ b_{t+1} &= &T_t*(1-f(T_t))+(1-p_r)*b_t\\ c_{t+1} &= &(a_tp_n+c_t)Po\\ d_{t+1} &= &(b_tp_r+d_t)Po\\ \end{matrix}\right. $$ where $f(T_t)$ is kind of probability function which is not linear function. $p_n$, $p_r$ and $P_o$ are less than 1 and larger than 0.
The initial values are $$ \left\{\begin{matrix} a_{1} &= & 100\\ b_{1} &= & 0\\ c_{1} &= & 0\\ d_{1} &= & 0\\ \end{matrix}\right. $$ Since $f(T_t)$ is in $a_t$ and $b_t$, I thought it can not be solved as matrix form.
How can I determine whether the linked recurrent relations converge or not? Is there an idea to solve the convergence value?
I put the initial values to the linked recurrent relations in MATLAB, and find out the relations converge.