Suppose we have the ODE system \begin{array}{ccc} \frac{dT}{dt} & = & f_{1}(T,C,F,D)\\ \frac{dC}{dt} & = & f_{2}(T,C,F,D)\\ \frac{dF}{dt} & = & f_{3}(T,C,F,D)\\ \frac{dD}{dt} & = & -\mu D \end{array}
definen on every interval: $n\tau<t<(n+1)\tau$ where $n$ is an integer. And at each $t=n\tau$, D will change abruptly $D(n\tau^+)=D(n\tau^-)+10^a$ where $n\tau^-$ and $n\tau^+$ is just a moment before and after $t=n\tau$ respectively. I want to study the behavior of the variable T (fix points, bifurcations, periodic orbits) after many intervals, not only inside the interval of definition. There's some standard techniques or theory to achieve this? Thanks in advance.
Let's start by looking at $D$. You have $D(n \tau^{-}) = \exp(-\mu \tau) D((n-1) \tau^{+})$ so $D(n \tau^{+}) = 10^{a} + \exp(-\mu \tau) D((n-1) \tau^{+})$. This is a linear recurrence for $v_n = D(n \tau^+)$, which will approach a fixed point $v^* = 10^a/(1 - \exp(-\mu \tau))$ as $n \to \infty$.
Next you want to look at the map $P: (T_0, C_0, F_0) \to (T(\tau), C(\tau), F(\tau))$ for the system $$ \eqalign{dT/dt &= f_1(T,C,F, v^* e^{-\mu t})\cr dF/dt &= f_2(T,C,F, v^* e^{-\mu t})\cr dC/dt &= f_3(T,C,F, v^* e^{-\mu t})\cr}$$ with initial conditions $(T,F,C)(0) = (T_0, F_0, C_0)$. Find its fixed points, periodic points, etc.