zbMATH — the first resource for mathematics

On the periodic logistic equation. (English) Zbl 1109.90010
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.

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
Full Text: DOI
[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
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.