2 corrected q_0 and q' switch; added 5 characters in body

I don't think there is a magic theorem here which works all the time. What you want depends a lot on the specific $\phi$, ${\bf q}'$, ${\bf q}_0$ you are working with. What you said sounds a lot like a toy model for the renormalization group, which is a way to study central limit type theorems. For instance the classical central limit theorem can be interpreted as a convergence in the basin of attraction of the Gaussian $\mathcal{N}(0,1)$ law with respect to a map $\phi$ on probability distributions of centered variables with variance 1 given by $$Z\longrightarrow \frac{X+Y}{\sqrt{2}}\ .$$ Namely given a probability distribution for a random variable $Z$, one makes two independent copies $X$, $Y$, and looks at the probability distribution of $\frac{X+Y}{\sqrt{2}}$.

In the absence of a precise description of your setup I can only throw in some general ideas. The first thing to do is to identify the fixed points of your map $\phi$. I assume it is nonlinear, so this is not a trivial question. If you cannot solve it completely, you need to at least find some easy fixed points. I assume this is what your ${\bf q}_0$ q}'$is. Then you need to analyze the linearization of$\phi$at these easy fixed points. I am again assuming that you did that and found that this linearization$A$has all eigenvalues of modulus less than 1 so it is contractive. From this it follows easily that if you start near${\bf q}_0$q}'$ you will converge to ${\bf q}_0$q}'$. If your starting point${\bf q}'$q}_0$ is not in this neighborhood of ${\bf q}_0$q}'$, then indeed you might want to run some simulations on the computer not only for${\bf q}'$q}_0$ but also for many other starting points. In the optimistic situation where the simulation indicates that all points go to ${\bf q}_0$q}'$, then there could be a Lyapunov function for your map, see http://en.wikipedia.org/wiki/Lyapunov_function That's a function which changes monotonically under$\phi$. If you have such a function which controls the norm of${\bf q}$then it might tell you that after enough iterations you will be in the neighborhood of${\bf q}_0$q}'$ where you have the contraction property. Of course the difficult thing is to come up with the correct guess for this Lyapunov function if it exists. This guess depends on the specifics of your example. For the central limit theorem there is a notion of entropy which does the job.

Some references:

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I don't think there is a magic theorem here which works all the time. What you want depends a lot on the specific $\phi$, ${\bf q}'$, ${\bf q}_0$ you are working with. What you said sounds a lot like a toy model for the renormalization group, which is a way to study central limit type theorems. For instance the classical central limit theorem can be interpreted as a convergence in the basin of attraction of the Gaussian $\mathcal{N}(0,1)$ law with respect to a map $\phi$ on probability distributions of centered variables with variance 1 given by $$Z\longrightarrow \frac{X+Y}{\sqrt{2}}\ .$$ Namely given a probability distribution for a random variable $Z$, one makes two independent copies $X$, $Y$, and looks at the probability distribution of $\frac{X+Y}{\sqrt{2}}$.

In the absence of a precise description of your setup I can only throw in some general ideas. The first thing to do is to identify the fixed points of your map $\phi$. I assume it is nonlinear, so this is not a trivial question. If you cannot solve it completely, you need to at least find some easy fixed points. I assume this is what your ${\bf q}_0$ is. Then you need to analyze the linearization of $\phi$ at these easy fixed points. I am again assuming that you did that and found that this linearization $A$ has all eigenvalues of modulus less than 1 so it is contractive. From this it follows easily that if start near ${\bf q}_0$ you will converge to ${\bf q}_0$. If your starting point ${\bf q}'$ is not in this neighborhood of ${\bf q}_0$, then indeed you might want to run some simulations on the computer not only for ${\bf q}'$ but also for many other starting points. In the optimistic situation where the simulation indicates that all points go to ${\bf q}_0$, then there could be a Lyapunov function for your map, see http://en.wikipedia.org/wiki/Lyapunov_function That's a function which changes monotonically under $\phi$. If you have such a function which controls the norm of ${\bf q}$ then it might tell you that after enough iterations you will be in the neighborhood of ${\bf q}_0$ where you have the contraction property. Of course the difficult thing is to come up with the correct guess for this Lyapunov function if it exists. This guess depends on the specifics of your example. For the central limit theorem there is a notion of entropy which does the job.

Some references: