Preprint of the project: SFB 701: Spectral Structures and Topological Methods in Mathematics - Project A2

Numerical analysis of high-dimensional transfer operators

07-025 Wolf-Jürgen Beyn, Thorsten Hüls, Malte Samtenschnieder.
On r-periodic orbits of k-periodic maps

In this paper, we analyze r-periodic orbits of k-periodic difference equations, i.e.

and their stability. This notion was introduced in [7]. 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. The algorithm is applied to the generalized Beverton-Holt and the two-dimensional stiletto model.