I have a discrete dynamical system in $[0,1]^n$. Specifically, I am studying the dynamics of a probability distribution under certain operator $\phi$ where $\mathbf{q}[t+1]=\phi(\mathbf{q}[t])$. The probability distribution is specified by $n$ parameters in the $n$-dimensional vector $\mathbf{q}$. What I need to know is whether a specific starting point $\mathbf{q}_0$ is in the basin of attraction of another specific point $\mathbf{q}'$. The only way that I have to check this is by running a computer program. I would like to know how to analyze mathematically this type of global behaviors of the system. More in general, studying the size of the basin of attraction of the point $\mathbf{q}'$ would be very important. What type of mathematical tools are needed to study this kind of issues?

Thanks in advance!