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Folding and unfolding in periodic difference equations. (English) Zbl 1302.39024
Summary: Given a \(p\)-periodic difference equation \(x_{n+1}=f_{n\pmod p}(x_n)\), where each \(f_j\) is a continuous interval map, \(j=0,1,\dots,p-1\), we discuss the notion of folding and unfolding related to this type of non-autonomous equations. It is possible to glue certain maps of this equation to shorten its period, which we call folding. On the other hand, we can unfold the glued maps so the original structure can be recovered or understood. Here, we focus on the periodic structure under the effect of folding and unfolding. In particular, we analyze the relationship between the periods of periodic sequences of the \(p\)-periodic difference equation and the periods of the corresponding subsequences related to the folded systems.

MSC:
39A23 Periodic solutions of difference equations
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[1] Allison, A.; Abbott, D., Control systems with stochastic feedback, Chaos, 11, 715-724, (2001) · Zbl 0977.37049
[2] Al-Salman, A.; AlSharawi, Z., A new characterization of periodic oscillations in periodic difference equations, Chaos Solitons Fractals, 44, 921-928, (2011) · Zbl 1271.39012
[3] AlSharawi, Z., Periodic orbits in periodic discrete dynamics, Comput. Math. Appl., 56, 1966-1974, (2008) · Zbl 1165.37311
[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] Z. AlSharawi, J. Cánovas, A. Linero, Periodic structure of alternating maps, preprint. · Zbl 1302.39024
[6] Alves, J. F., What we need to find out the periods of a periodic difference equation, J. Difference Equ. Appl., 15, 833-847, (2009) · Zbl 1180.39021
[7] Buceta, J.; Escudero, C.; de la Rubia, F. J.; Lindenberg, K., Outbreaks of hantavirus induced by seasonality, Phys. Rev. E, 69, 177-184, (2004)
[8] Cánovas, J. S.; Linero, A., Periodic structure of alternating continuous interval maps, J. Difference Equ. Appl., 12, 847-858, (2006) · Zbl 1099.37028
[9] Cushing, J.; Henson, S., The effect of periodic habitat fluctuations on a nonlinear insect population model, J. Math. Biol., 36, 201-226, (1997) · Zbl 0890.92023
[10] Elaydi, S.; Sacker, R., Periodic difference equations, population biology and the cushing-henson conjectures, Math. Biosci., 201, 195-207, (2006) · Zbl 1105.39006
[11] Harmer, G. P.; Abbott, D., Losing strategies can win by Parrondo’s paradox, Nature, 402, 864, (1999)
[12] Jillson, D., Insect population respond to fluctuating environments, Nature, 288, 699-700, (1980)
[13] Parrondo, J. M.R.; Dinis, L., Brownian motion and gambling: from ratchets to paradoxical games, Contemp. Phys., 45, 147-157, (2004)
[14] Spurgin, R.; Tamarkin, M., Switching investments can be a bad idea when Parrondo’s paradox applies, J. Behav. Finance, 6, 15-18, (2005)
[15] Toral, R., Capital redistribution brings wealth by Parrondo’s paradox, Fluct. Noise Lett., 2, 305-311, (2002)
[16] Wolf, D. M.; Vazirani, V. V.; Arkin, A. A., Diversity in times of adversity: probabilistic strategies in microbial survival games, J. Theoret. Biol., 234, 227-253, (2005)
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