TITLE: Numerical search for periodic orbits
QUESTION [5 upvotes]: Suppose that $\phi$ is a map defined in all of $R^n$, with image also in $R^n$. Is there a known good algorithm to find the periodic points of $\phi$ of order $m$, (or fixed points of $\phi^m$) besides:

Applying Newton's method in some (random) initial $m$ points $x_0,x_1, ..., x_{m-1}$ and seeking for the solutions of the system $\phi(x_k)=x_{k+1}$;
Or applying Newton's method in a single (random) initial point $x$, and looking for the solution of $\phi^m(x)=x$ (thus having to compute the product of the Jacobians of $\phi$ on its first $m$ powers).

I would also appreciate any reference about this question. Thank you!

REPLY [2 votes]: A software toolbox like GAIO could help.