On r-periodic orbits of k-periodic maps

In this paper, we analyze r-periodic orbits of k-periodic difference equations, i.e. x(n+1) = F-n(x(n)), F-n = F-n (mod) (k), x(n) = x(n) (mod) (r), n is an element of N and their stability. These special orbits were introduced in S. Elaydi and R.J. Sacker (Global stability of periodic orbits of non-autonomous difference equations and population biology, J. Differ. Equ. 208(1) (2005), pp. 258-273). We discuss that, depending on the values of r and k, such orbits generically only occur in finite dimensional systems that depend on sufficiently many parameters, i.e. they have a large codimension in the sense of bifurcation theory. As an example, we consider the periodically forced Beverton-Holt model, for which explicit formulas for the globally attracting periodic orbit, having the minimal period k = r, can be derived. When r factors k the Beverton-Holt model with two time-variant parameters is an example that can be studied explicitly and that exhibits globally attracting r-periodic orbits. For arbitrarily chosen periods r and k, we develop an algorithm for the numerical approximation of an r-periodic orbit and of the associated parameter set, for which this orbit exists. We apply the algorithm to the generalized Beverton-Holt, the 2D stiletto model, and another example that exhibits periodic orbits with r and k relatively prime.