Existence and stability of periodic orbits of periodic difference equations with delays

Ziyad Alsharawi, James Angelos, Saber Elaydi

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5 Citations (Scopus)


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 languageEnglish
Pages (from-to)203-217
Number of pages15
JournalInternational Journal of Bifurcation and Chaos
Issue number1
Publication statusPublished - Jan 2008



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

ASJC Scopus subject areas

  • General
  • Applied Mathematics

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