In this paper, we investigate the existence and stability of periodic orbits of the p-periodic difference equation with delays xn = ƒ(n−1, xn−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 divide p.
World Scientific Publishing Co. Pte. Ltd.
Alsharawi, Z., Angelos, J., & Elaydi, S. (2008). Existence and stability of periodic orbits of periodic difference equations with delays. International Journal of Bifurcation and Chaos, 18(1), 203-217. doi:10.1142/S0218127408020239
International Journal of Bifurcation and Chaos