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 |
|---|---|
| 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
Existence and stability of periodic orbits of periodic difference equations with delays. / Alsharawi, Ziyad; Angelos, James; Elaydi, Saber.
In: International Journal of Bifurcation and Chaos, Vol. 18, No. 1, 01.2008, p. 203-217.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Existence and stability of periodic orbits of periodic difference equations with delays
AU - Alsharawi, Ziyad
AU - Angelos, James
AU - Elaydi, Saber
PY - 2008/1
Y1 - 2008/1
N2 - 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.
AB - 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.
KW - Global stability
KW - Periodic difference equations
KW - Periodic orbits
KW - Sharkovsky's theorem
UR - http://www.scopus.com/inward/record.url?scp=43949114849&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=43949114849&partnerID=8YFLogxK
U2 - 10.1142/S0218127408020239
DO - 10.1142/S0218127408020239
M3 - Article
AN - SCOPUS:43949114849
VL - 18
SP - 203
EP - 217
JO - International Journal of Bifurcation and Chaos in Applied Sciences and Engineering
JF - International Journal of Bifurcation and Chaos in Applied Sciences and Engineering
SN - 0218-1274
IS - 1
ER -