On r-periodic orbits of k-periodic maps

Abstract

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

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.

Keywords:

• periodically forced discrete time dynamical systems
• periodic orbits
• stability analysis
• numerical approximation
• Beverton–Holt model
• population biology

Keywords:

• 39A11
• 37B55
• 37N25

Acknowledgements

The authors wish to thank the anonymous referees for several helpful suggestions that improved the first version of the paper.

Notes

1. Supported by CRC 701 ‘Spectral Structures and Topological Methods in Mathematics’.

2. [email protected]

3. [email protected]

Additional information

Notes on contributors

Wolf-Jürgen Beyn

1

Thorsten Hüls

2

Malte-Christopher Samtenschnieder

3