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An extension of Sharkovsky’s theorem to periodic difference equations. (English) Zbl 1125.39001
The following system is studied:
\[ x(n+1)=F(n,x(n))\tag{1} \] such that \(F(n+p,x)=F(n,x), p\) is a given positive integer. Geometric cycles of (1) are investigated. A simpler method for constructing \(p\)-periodic difference equations with \(r\)-periodic geometric cycles are given. A new ordering of positive integers (\(p\)-Sharkovsky ordering) is introduced. Both the Sharkovsky theorem and its converse are generalized to the system (1) [cf. A. N. Sharkovskij, Ukr. Math. Zh. 16, 61–71 (1964; Zbl 0122.17504)].

MSC:
39A11 Stability of difference equations (MSC2000)
39A12 Discrete version of topics in analysis
37C27 Periodic orbits of vector fields and flows
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[1] Block, L.S.; Coppel, W.A., Dynamics in one dimension, (1992), Springer
[2] Clark, M.E.; Gross, L.J., Periodic solutions to nonautonomous difference equations, Math. biosci., 102, 105-119, (1990) · Zbl 0712.39014
[3] Coleman, B.D., Nonautonomous logistic equations models of the adjustment of population to environmental changes, Math. biosci., 45, 159-173, (1979) · Zbl 0425.92013
[4] Cushing, J.M.; Henson, S.M., A periodically forced beverton – holt equation, J. differ. equations appl., 8, 1119-1120, (2002) · Zbl 1023.39013
[5] Cushing, J.M.; Henson, S.M., Global dynamics of some periodically forced, monotone difference equations, J. differ. equations appl., 7, 859-872, (2001) · Zbl 1002.39003
[6] Cushing, J.M.; Henson, S.M., The effect of periodic habit fluctuations on a nonlinear insect population model, J. math. biol., 36, 201-226, (1997) · Zbl 0890.92023
[7] Devaney, R., A first course in chaotic dynamical systems: theory and experiments, (1992), Addison-Wesley Reading, MA
[8] Elaydi, S., An introduction to difference equations, (2005), Springer New York · Zbl 1071.39001
[9] Elaydi, S., Discrete chaos, (2000), Chapman & Hall/CRC Boca Raton, FL · Zbl 0945.37010
[10] Elaydi, S., On a converse of Sharkovsky’s theorem, Amer. math. monthly, 103, 386-392, (1996) · Zbl 0893.58024
[11] Elaydi, S.; Sacker, R., Global stability of periodic orbits of nonautonomous difference equations and population biology, J. differential equations, 208, 258-273, (2005) · Zbl 1067.39003
[12] S. Elaydi, R. Sacker, Global stability of periodic orbits of nonautonomous difference equations in population biology and the Cushing-Henson conjectures, in: Proceedings of the 8th International Conference on Difference Equations, Brno, 2003 · Zbl 1087.39504
[13] S. Elaydi, R. Sacker, Nonautonomous Beverton-Holt equations and the Cushing-Henson conjectures, Trinity University Technical Report 91, J. Differ. Equations Appl., submitted for publication · Zbl 1084.39005
[14] El-Owaidy, H.; Mohamed, H.Y., The necessary and sufficient conditions of existence of periodic solutions of nonautonomous difference equations, Appl. math. comput., 136, 345-351, (2003) · Zbl 1025.39009
[15] Franke, J.E.; Selgrade, J.F., Attractor for periodic dynamical systems, J. math. anal. appl., 286, 64-79, (2003) · Zbl 1035.37020
[16] Franke, J.E.; Yakubu, A.-A., Multiple attractors via cusp bifurcation in periodically varying environments, J. differ. equations appl., (2005) · Zbl 1068.92038
[17] Grinfeld, M.; Knight, P.A.; Lamba, H., On the periodically perturbed logistic equation, J. phys. A, 29, 8035-8040, (1996) · Zbl 0898.58013
[18] Henson, S.M., Multiple attractors and resonance in periodically forced population, Phys. D, 140, 33-49, (2000) · Zbl 0957.37018
[19] Jillson, D., Insect populations respond to fluctuating environment, Nature, 288, 699-700, (1980)
[20] Kapral, R.; Mandel, P., Bifurcation structure of the nonautonomous quadratic map, Phys. rev. A, 32, 1076-1081, (1985)
[21] V.L. Kocic, A note on the nonautonomous Beverton-Holt model, J. Differential Equations, submitted for publication · Zbl 1084.39007
[22] Kincaid, D.; Cheney, W., Numerical analysis, (2002), Brooks/Cole
[23] Kon, R., A note on attenuant cycles of population models with periodic carrying capacity, J. differ. equations appl., 10, 791-793, (2004) · Zbl 1056.92046
[24] R. Kon, Attenuant cycles of population models with periodic carrying capacity, J. Differ. Equations Appl., submitted for publication · Zbl 1067.92048
[25] Li, J., Periodic solutions of population models in a periodically fluctuating environment, Math. biosci., 110, 17-25, (1992) · Zbl 0746.92022
[26] Li, T.-Y.; Yorke, J.A., Period three implies chaos, Amer. math. monthly, 82, 985-992, (1975) · Zbl 0351.92021
[27] Poincaré, H., Sur LES équations linéaires aux differentielles ordinaires et aux différences finies, Amer. J. math., 7, 203-258, (1885) · JFM 17.0290.01
[28] Selgrade, J.F.; Roberds, H.D., On the structure of attractors for discrete periodically forced systems with applications to population models, Phys. D, 158, 69-82, (2001) · Zbl 1018.37047
[29] Sell, G.R., Topological dynamics and ordinary differential equations, (1971), Van Nostrand-Reinhold London · Zbl 0212.29202
[30] Sharkovsky, A.N., Coexistence of cycles of a continuous transformation of a line into itself, Ukrain. math. zh., 16, 61-71, (1964), (in Russian)
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