On the periodic logistic equation. (English) Zbl 1109.90010

AlSharawi, Ziyad; Angelos, James

Appl. Math. Comput. 180, No. 1, 342-352 (2006).

Summary: We show that the \(p\)-periodic logistic equation \(x_{n+1} = \mu _{n\,\text{mod\,} p}x_{n}(1 - x_{n})\) has cycles (periodic solutions) of minimal periods \(1, p, 2p, 3p, \dots\) Then we extend Singer’s theorem to periodic difference equations, and use it to show the \(p\)-periodic logistic equation has at most \(p\) stable cycles. Also, we present computational methods investigating the stable cycles in case \(p = 2\) and 3.

MSC:

90B06	Transportation, logistics and supply chain management
39A11	Stability of difference equations (MSC2000)
37C25	Fixed points and periodic points of dynamical systems; fixed-point index theory, local dynamics
37C70	Attractors and repellers of smooth dynamical systems and their topological structure

Keywords:

logistic map; non-autonomous; periodic solutions; Singer’s theorem; attractors

References:

[1]	AlSharawi, Z.; Angelos, J.; Elaydi, S.; Rakesh, L., An extension of sharkovsky’s theorem to periodic difference equations, J. math. anal. appl., 316, 129-141, (2006) · Zbl 1125.39001
[2]	Devaney, R., A first course in chaotic dynamical systems: theory and experiment, (1992), Addison-Wesley Reading, MA · Zbl 0768.58001
[3]	Elaydi, S., An introduction to difference equations, (2005), Springer-Verlag New York · Zbl 1071.39001
[4]	Elaydi, S., Discrete chaos, (2000), Chapman & Hall/CRC Boca Raton · Zbl 0945.37010
[5]	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
[6]	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
[7]	Franke, J.E.; Selgrade, J.F., Attractors for periodic dynamical systems, J. math. anal. appl., 286, 64-79, (2003) · Zbl 1035.37020
[8]	Gleick, J., Chaos: making a new science, (1987), Penguin · Zbl 0706.58002
[9]	Grinfeld, M.; Knight, P.A.; Lamba, H., On the periodically perturbed logistic equation, J. phys. A: math. gen., 29, 8035-8040, (1996) · Zbl 0898.58013
[10]	Hao, B.; Zheng, W., Applied symbolic dynamics and chaos, (1998), World Scientific · Zbl 0914.58017
[11]	Henson, S.M., Multiple attractors and resonance in periodically forced population, Physica D, 140, 33-49, (2000) · Zbl 0957.37018
[12]	Kot, M.; Schaffer, W.M., The effects of seasonality on discrete models of population grouth, Theor. population biol., 26, 340-360, (1984) · Zbl 0551.92014
[13]	Li, J., Periodic solutions of population models in a periodically fluctuating environment, Math. biosci., 110, 17-25, (1992) · Zbl 0746.92022
[14]	Martelli, M., Introduction to discrete dynamical systems and chaos, (1999), Wiley-Intersience · Zbl 1127.37300
[15]	May, R.M., Simple mathematical models with very complicated dynamics, Nature, 261, 459-467, (1976) · Zbl 1369.37088
[16]	Peitgen, H.; Jurgens, H.; Saupe, D., Fractals for the classroom, (1992), Springer-Verlag · Zbl 0746.58005
[17]	Selgrade, J.F.; Roberds, H.D., On the structure of attractors for discrete periodically forced systems with applications to population models, Physica D, 158, 69-82, (2001) · Zbl 1018.37047
[18]	Singer, D., Stable orbits and bifurcation of maps of the interval, SIAM J. appl. math., 35, 260-267, (1978) · Zbl 0391.58014