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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?

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

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