Existence and Stability of Periodic orbits of Periodic Difference Equations with Delays

In this paper, we investigate the existence and stability of periodic orbits of the p-periodic difference equation with delays xn = f(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.

• Type: Journal
• Year: 2008
• Where: IJBC
• Authors: Ziyad Alsharawi, James Angelos, Saber Elaydi
• Keywords: Difference Equations, IJBC 2008, P-periodic Difference Equation, Periodic Orbits