# zbMATH — the first resource for mathematics

A new characterization of periodic oscillations in periodic difference equations. (English) Zbl 1271.39012
Summary: We characterize periodic solutions of $$p$$-periodic difference equations. We classify the periods into multiples of $$p$$ and nonmultiples of $$p$$. We show that the elements of the set of multiples of $$p$$ follow the well-known Sharkovsky’s ordering multiplied by $$p$$. On the other hand, we show that the elements of the set $$\Gamma_p$$ of nonmultiples of $$p$$ are independent in their existence. Moreover, we show the existence of a $$p$$-periodic difference equation with infinite $$\Gamma_p$$-set in which the maps are defined on a compact domain and agree exactly on a countable set. Based on the proposed classification, we give a refinement of A. N. Sharkovsky’s theorem [“Coexistence of cycles of a continuous map of the line into itself”, Int. J. Bifurcation Chaos Appl. Sci. Eng. 5, 1263–1273 (1995)] for periodic difference equations.

##### MSC:
 39A21 Oscillation theory for difference equations 39A23 Periodic solutions of difference equations
Full Text:
##### References:
 [1] Alsedà, L.; Llibre, J.; Misiurewicz, M., Combinatorial dynamics and entropy in dimension one, (1993), World Scientific Singapore · Zbl 0843.58034 [2] AlSharawi, Z., Periodic orbits in periodic discrete dynamics, Comput math appl, 56, 1966-1974, (2008) · Zbl 1165.37311 [3] AlSharawi, Z.; Angelos, J., On the periodic logistic equation, Appl math comput, 180, 342-352, (2006) · Zbl 1109.90010 [4] AlSharawi, Z.; Angelos, J.; Elaydi, S.; Rakesh, L., An extension of sharkovsky’s theorem to periodic difference equations, J math anal appl, 316, 128-141, (2006) · Zbl 1125.39001 [5] Alves, J.F., What we need to find out the periods of a periodic difference equation, J differ equ appl, 15, 833-847, (2009) · Zbl 1180.39021 [6] Alves, J.F., Odd periods of 2-periodic nonautonomous dynamical systems, Grazer math ber, 351, 21-26, (2007) · Zbl 1146.37015 [7] Burkart, U., Interval mapping graphs and periodic points of continuous functions, J combin theory ser B, 32, 57-68, (1982) · Zbl 0474.05032 [8] Cánovas, J.S.; Linero, A., Periodic structure of alternating continuous interval maps, J differ equ appl, 12, 847-858, (2006) · Zbl 1099.37028 [9] Clark, M.; Gross, L., Periodic solutions to nonautonomous difference equations, Math biosci, 102, 105-119, (1990) · Zbl 0712.39014 [10] Cushing, J.; Henson, S., Global dynamics of some periodically forced, monotone difference equations, J differ equ appl, 7, 859-872, (2001) · Zbl 1002.39003 [11] Cushing, J.; Henson, S., The effect of periodic habit fluctuations on a nonlinear insect population model, J math biol, 36, 201-226, (1997) · Zbl 0890.92023 [12] Elaydi, S.; Sacker, R., Global stability for periodic orbits of nonautonmous difference equations and population biology, J differ equ, 208, 258-273, (2005) · Zbl 1067.39003 [13] Elaydi, S.; Sacker, R., Periodic difference equations, population biology and the cushing – henson conjectures, Math biosci, 201, 195-207, (2006) · Zbl 1105.39006 [14] Franke, J.; Selgrade, J., Attractors for periodic dynamical systems, J math anal appl, 286, 64-79, (2003) · Zbl 1035.37020 [15] Henson, S., Multiple attractors and resonance in periodically forced population models, Physica D, 140, 33-49, (2000) · Zbl 0957.37018 [16] Ho, C.; Morris, C., A graph theoretic proof of sharkovsky’s theorem on the periodic points of continuous functions, Pacific J math, 96, 361-370, (1981) · Zbl 0428.26003 [17] Kon, R., A note on attenuant cycles of population models with periodic carrying capacity, J differ equ appl, 10, 8, 791-793, (2004) · Zbl 1056.92046 [18] Kon, R., Attenuant cycles of population models with periodic carrying capacity, J differ equ appl, 11, 423-430, (2005) · Zbl 1067.92048 [19] Li, T.; Yorke, J.A., Period three implies chaos, Am math mon, 82, 985-992, (1975) · Zbl 0351.92021 [20] Sharkovskii, A.N., Coexistence of cycles of a continuous map of the line into itself, Int J bifur chaos appl sci eng, 5, 1263-1273, (1995), [English translation of the original 1964] · Zbl 0890.58012 [21] Straffin, P.D., Periodic points of continuous functions, Math mag, 51, 99-105, (1978) · Zbl 0455.58022
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.