A note on global stability in the periodic logistic map

Rafael Luís; Sandra Mendonça

doi:10.3934/dcdsb.2020094

Article Contents

2020, Volume 25, Issue 11: 4211-4220. Doi: 10.3934/dcdsb.2020094

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A note on global stability in the periodic logistic map

Rafael Luís^1, , and
Sandra Mendonça^2,

1.
Center for Mathematical Analysis, Geometry and Dynamical Systems, University of Lisbon, Portugal, University of Madeira, Funchal, Portugal
2.
University of Madeira, Funchal, Portugal, Center of Statistics and Applications, University of Lisbon, Portugal

^* Corresponding author: Rafael Luís
^* Corresponding author: Rafael Luís

Received: July 2019

Revised: October 2019

Early access: April 2020

Published: November 2020

The first and second authors are partially supported by FCT/Portugal through the projects UID/MAT/04459/2019 and UID/MAT/00006/2019, respectively

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Abstract

In this paper, the dynamics of the celebrated $ p- $periodic one-dimensional logistic map is explored. A result on the global stability of the origin is provided and, under certain conditions on the parameters, the local stability condition of the $ p- $periodic orbit is shown to imply its global stability.

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Mathematics Subject Classification: Primary: 37B55, 39A23; Secondary: 37C25, 37C75, 39A30.

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Figure 1. Regions of stability, in the parameter space $ r_{0}O r_1 $, of the fixed points of $ f_1\circ f_0 $, with $ f_i(x) = r_i x(1-x) $, $ i = 0, 1 $

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References

[1]	K. T. Alligood, T. D. Sauer, and J. A. Yorke, Chaos. An Introduction to Dynamical Systems, Textbooks in Mathematical Sciences, Springer-Verlag, New York, 1997. doi: 10.1007/978-3-642-59281-2.
[2]	Z. AlSharawi and J. Angelos, On the periodic logistic equation, Appl. Math. Comput., 180 (2006), 342-352. doi: 10.1016/j.amc.2005.12.016.
[3]	W. A. Coppel, The solution of equations by iteration, Proc. Cambridge Philos. Soc., 51 (1955), 41-43. doi: 10.1017/S030500410002990X.
[4]	R. L. Devaney, An Introduction to Chaotic Dynamical Systems, Studies in Nonlinearity, second edition, Westview Press, Boulder, CO, 2003.
[5]	H. A. El-Morshedy and V. J. López, Global attractors for difference equations dominated by one-dimensional maps, J. Difference Equ. Appl., 14 (2008), 391-410. doi: 10.1080/10236190701671632.
[6]	S. Elaydi, An Introduction to Difference Equations, Undergraduate Texts in Mathematics, third edition, Springer, New York, 2005.
[7]	S. N. Elaydi, Discrete Chaos. With Applications in Science and Engineering, Second edition, Chapman & Hall/CRC, Boca Raton, FL, 2008.
[8]	M. Grinfeld, P. A. Knight and H. Lamba, On the periodically perturbed logistic equation, J. Phys. A, 29 (1996), 8035-8040. doi: 10.1088/0305-4470/29/24/026.
[9]	R. B. Kellogg, T. Y. Li and J. Yorke, A constructive proof of the Brouwer fixed-point theorem and computational results, SIAM J. Numer. Anal., 13 (1976), 473-483. doi: 10.1137/0713041.
[10]	M. Kot and W. M. Schaffer, The effects of seasonality on discrete models of population growth, Theoret. Population Biol., 26 (1984), 340-360. doi: 10.1016/0040-5809(84)90038-8.
[11]	C. P. Li and M. Zhao, On the periodic logistic map, Acta Math. Sin. (Engl. Ser.), 34 (2018), 891-900. doi: 10.1007/s10114-017-6011-z.
[12]	J. Li, Periodic solutions of population models in a periodically fluctuating environment, Math. Biosci., 110 (1992), 17-25. doi: 10.1016/0025-5564(92)90012-L.
[13]	E. Liz, On the global stability of periodic Ricker maps, Electron. J. Qual. Theory Differ. Equ., 76 (2016), 8 pp. doi: 10.14232/ejqtde.2016.1.76.
[14]	R. M. May, Simple mathematical models with very complicated dynamics, Nature, 261 (1976), 459-467.
[15]	J. R. Munkres, Topology: A First Course, Prentice Hall, Inc., Englewood Cliffs, NJ, 1975.
[16]	R. J. Sacker and G. R. Sell, Lifting properties in skew-product flows with applications to differential equations, Mem. Amer. Math. Soc., 11 (1977). doi: 10.1090/memo/0190.
[17]	H. Sedaghat, Nonlinear Difference Equations. Theory with Applications to Social Models, Mathematical Modelling: Theory and applications, vol. 15, Kluwer Academic Publishers, Dordrecht, 2003. doi: 10.1007/978-94-017-0417-5.
[18]	D. Singer, Stable orbits and bifurcation maps of the interval, SIAM J. Appl. Math., 35 (1978), 260-267. doi: 10.1137/0135020.
[19]	P. F. Verhulst, Notice sur la loi que la population poursuit dans son accroissement, Correspondance Math. Phys., 10 (1838), 113-121.