### Abstract

In this paper, we investigate the existence and stability of periodic orbits of the p-periodic difference equation with delays x(n) = f(n - 1, x(n-k)). We show that the periodic orbits of this equation depend on the periodic orbits of p autonomous equations when p divides k. When p is not a divisor of k, the periodic orbits depend on the periodic orbits of gcd(p, k) nonautonomous p/gcd(p, k)-periodic difference equations. We give formulas for calculating the number of different periodic orbits under certain conditions. In addition, when p and k are relatively prime integers, we introduce what we call the pk-Sharkovsky's ordering of the positive integers, and extend Sharkovsky's theorem to periodic difference equations with delays. Finally, we characterize global stability and show that the period of a globally asymptotically stable orbit must be divisible by p.

Original language | English |
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Pages (from-to) | 203-217 |

Number of pages | 15 |

Journal | International Journal of Bifurcation and Chaos |

Volume | 18 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jan 2008 |

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### Keywords

- Global stability
- Periodic difference equations
- Periodic orbits
- Sharkovsky's theorem

### ASJC Scopus subject areas

- General
- Applied Mathematics

### Cite this

*International Journal of Bifurcation and Chaos*,

*18*(1), 203-217. https://doi.org/10.1142/S0218127408020239