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 |
Externally published | Yes |
Keywords
- Global stability
- Periodic difference equations
- Periodic orbits
- Sharkovsky's theorem
ASJC Scopus subject areas
- Modelling and Simulation
- Engineering (miscellaneous)
- General
- Applied Mathematics