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

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    existence_stability_periodic_orbits.pdf (850.6Kb)
    Date
    2008
    Author
    Al-Sharawi, Ziyad
    Angelos, James
    Elaydi, Saber
    Advisor(s)
    Unknown advisor
    Type
    Peer-Reviewed
    Article
    Preprint
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    Abstract
    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 be divisible by p.
    DSpace URI
    http://hdl.handle.net/11073/16686
    External URI
    https://doi.org/10.1142/S0218127408020239
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