On r-periodic orbits of k-periodic maps

Beyn W-J, Hüls T, Samtenschnieder M-C (2008)
Journal of Difference Equations and Applications 14(8): 865-887.

Zeitschriftenaufsatz | Veröffentlicht | Englisch
 
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Abstract / Bemerkung
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.
Stichworte
Beverton-Holt model; stability analysis; periodically forced discrete time dynamical systems; periodic orbits; numerical approximation; population biology
Erscheinungsjahr
2008
Zeitschriftentitel
Journal of Difference Equations and Applications
Band
14
Ausgabe
8
Seite(n)
865-887
ISSN
1023-6198
Page URI
https://pub.uni-bielefeld.de/record/1587387

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Beyn W-J, Hüls T, Samtenschnieder M-C. On r-periodic orbits of k-periodic maps. Journal of Difference Equations and Applications. 2008;14(8):865-887.
Beyn, W. - J., Hüls, T., & Samtenschnieder, M. - C. (2008). On r-periodic orbits of k-periodic maps. Journal of Difference Equations and Applications, 14(8), 865-887. doi:10.1080/10236190801940010
Beyn, W. - J., Hüls, T., and Samtenschnieder, M. - C. (2008). On r-periodic orbits of k-periodic maps. Journal of Difference Equations and Applications 14, 865-887.
Beyn, W.-J., Hüls, T., & Samtenschnieder, M.-C., 2008. On r-periodic orbits of k-periodic maps. Journal of Difference Equations and Applications, 14(8), p 865-887.
W.-J. Beyn, T. Hüls, and M.-C. Samtenschnieder, “On r-periodic orbits of k-periodic maps”, Journal of Difference Equations and Applications, vol. 14, 2008, pp. 865-887.
Beyn, W.-J., Hüls, T., Samtenschnieder, M.-C.: On r-periodic orbits of k-periodic maps. Journal of Difference Equations and Applications. 14, 865-887 (2008).
Beyn, Wolf-Jürgen, Hüls, Thorsten, and Samtenschnieder, Malte-Christopher. “On r-periodic orbits of k-periodic maps”. Journal of Difference Equations and Applications 14.8 (2008): 865-887.