# Identification of (Centers of) Cycles in a Discrete Time Dynamical System

I am studying dynamics on nonlinear Discrete Time Dynamical System of the form

$$\vec{X}_{t+1} = D(\vec{X}_t),$$

where D is some nonlinear function. I was looking for a (relatively) quick algorithm for numerically identifying fixed points and their stability, and found a helpful pdf that claims a system of equations with $F(\vec{X})=\vec{0}$ can be solved by minimizing $\sum_i F^2_i(\vec{X})$ using a Steepest Decent technique.

So I set $F(\vec{X}^*)=D(\vec{X}^*)-\vec{X}^*$ and implemented an algorithm for minimization, and it seems to work well at finding fixed points. Stability analysis is as simple as checking that all the eigenvalues are within the unit circle.

However, in some cases the system in question produces limit cycles. Since I am mostly experienced with Continuous Time Dynamical Systems I guessed by analogy that the minimum found by the steepest decent algorithm found the 'center' of the cycle. In retrospect, this seems exceedingly unlikely.

My questions:

1.Is this intuition correct at least in some cases, i.e. can I expect some fixed point $\vec{X}_c$ that is the 'center' of the cycle? If so, in which cases?

2.If this is the case, can I identify it as a center (as opposed to some other stable or unstable fixed point) by the eigenvalues or by some other method?

3.If this is not the case, how can I identify cases in which the system cycles? Note that these cycles are way too large to permit checking for fixed points of iterations of the function.

If theorems, books or other resources can be cited in the process, it would be doubly appreciated!