# Linear stability analysis of a 2-cycle

In a discrete $$N$$-dimensional Hamiltonian map $$\mathbf{X}^{(n+1)}=f(\mathbf{X}^{(n)})$$, we often find a 2-cycle which shows oscillation between two points in phase space. In such a Hamiltonian map we analyze the stability of a fixed point from the eigenvalues of the jacobian matrix of the corresponding linearized equation. My question is, what is the procedure for the stability of a 2-cycle in such a system? Do I need to obtain $$\mathbf{X}^{(n+2)}=f(f(\mathbf{X}^{(n)}))$$, and then linearize it or anything else?

$$Y^{(n+1)}=f(Z^{(n)}),\\ Z^{(n+1)}=f(Y^{(n)}),$$ and analyze the stability of this two-dimensional system. This is indeed equivalent to analyzing the stable point of mapping $$Y^{(n+1)}=f(f(Y^{(n)}))$$