An extension of Sharkovsky's theorem to periodic difference equations

AlSharawi, Ziyad; Angelos, James; Elaydi, Saber; Rakesh, Leela

doi:10.1016/j.jmaa.2005.04.059

Cited by 48 publications

(52 citation statements)

References 27 publications

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“…Устойчивость равновесия (5) определяется значениями собственных чисел, удовлетво-ряющих характеристическому многочлену системы (8):…”

Section: модель рикера с периодическим мальтузианским параметромunclassified

“…определяет в параметрическом пространстве (α, ρ) границу области устойчивости ненуле-вых стационарных решений системы (8).…”

Section: модель рикера с периодическим мальтузианским параметромunclassified

“…Соответственно, ва-риация текущей численности популяции (начального условия) может привести к смене на-блюдаемого динамического режима. Следует отметить, что существование мультистабиль-ности в уравнении Рикера с периодическим параметром показано и теоретически доказано в целом ряде работ [3][4][5][6][7][8][9][10].…”

Section: Introductionunclassified

See 2 more Smart Citations

Dynamic modes of the Ricker model with periodic Malthusian parameter

Шлюфман¹,

Неверова²,

Фрисман³

2017

Nelin. Dinam.

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Section: модель рикера с периодическим мальтузианским параметромunclassified

Section: Introductionunclassified

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Dynamic modes of the Ricker model with periodic Malthusian parameter

Шлюфман¹,

Неверова²,

Фрисман³

2017

Nelin. Dinam.

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“…Stability notions for periodic difference equations have been investigated by many authors. Including, to cite few, AlSharawi and Angelos [3], AlSharawi, Angelos, Elaydi and Rakesh [4], Henson [11], Yakubu [15], Oliveira and D'Aniello [14], and Selgrado and Roberds [20].…”

Section: Stabilitymentioning

confidence: 99%

Non-autonomous periodic systems with Allee effects

Luís

Elaydi

Oliveira

2010

Journal of Difference Equations and Applications

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A new class of maps called unimodal Allee maps are introduced. Such maps arise in the study of population dynamics in which the population goes extinct if its size falls below a threshold value. A unimodal Allee map is thus a unimodal map with tree fixed points, a zero fixed point, a small positive fixed point, called threshold point, and a bigger positive fixed point, called the carrying capacity. In this paper the properties and stability of the three fixed points are studied in the setting of nonautonomous periodic dynamical systems or difference equations. Finally we investigate the bifurcation of periodic systems/difference equations when the system consists of two unimodal Allee maps.

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“…In [1] AlSharawi et al focused, among other things, on the extension of Sharkovsky's theorem to the periodic difference equation (1.4). Moreover, they were able to describe the combinatorial structure of the periodic orbits of equation (1.4).…”

Section: Introductionmentioning

confidence: 99%

Existence and Stability of Periodic Orbits of Periodic Difference Equations With Delays

AlSharawi

Angelos

Elaydi

2008

Int. J. Bifurcation Chaos

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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 divide p.

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An extension of Sharkovsky's theorem to periodic difference equations

Cited by 48 publications

References 27 publications

Dynamic modes of the Ricker model with periodic Malthusian parameter

Dynamic modes of the Ricker model with periodic Malthusian parameter

Non-autonomous periodic systems with Allee effects

Existence and Stability of Periodic Orbits of Periodic Difference Equations With Delays

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