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November  2020, 25(11): 4211-4220. doi: 10.3934/dcdsb.2020094

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

Received  July 2019 Revised  October 2019 Published  April 2020

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

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.

Keywords: Difference equations, periodic logistic map, periodic orbits, local stability, global stability.
Mathematics Subject Classification: Primary: 37B55, 39A23; Secondary: 37C25, 37C75, 39A30.
Citation: Rafael Luís, Sandra Mendonça. A note on global stability in the periodic logistic map. Discrete & Continuous Dynamical Systems - B, 2020, 25 (11) : 4211-4220. doi: 10.3934/dcdsb.2020094
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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

R. M. May, Simple mathematical models with very complicated dynamics, Nature, 261 (1976), 459-467.   Google Scholar

[15]

J. R. Munkres, Topology: A First Course, Prentice Hall, Inc., Englewood Cliffs, NJ, 1975.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

D. Singer, Stable orbits and bifurcation maps of the interval, SIAM J. Appl. Math., 35 (1978), 260-267.  doi: 10.1137/0135020.  Google Scholar

[19]

P. F. Verhulst, Notice sur la loi que la population poursuit dans son accroissement, Correspondance Math. Phys., 10 (1838), 113-121.   Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[3]

W. A. Coppel, The solution of equations by iteration, Proc. Cambridge Philos. Soc., 51 (1955), 41-43.  doi: 10.1017/S030500410002990X.  Google Scholar

[4]

R. L. Devaney, An Introduction to Chaotic Dynamical Systems, Studies in Nonlinearity, second edition, Westview Press, Boulder, CO, 2003.  Google Scholar

[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.  Google Scholar

[6]

S. Elaydi, An Introduction to Difference Equations, Undergraduate Texts in Mathematics, third edition, Springer, New York, 2005.  Google Scholar

[7]

S. N. Elaydi, Discrete Chaos. With Applications in Science and Engineering, Second edition, Chapman & Hall/CRC, Boca Raton, FL, 2008.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

R. M. May, Simple mathematical models with very complicated dynamics, Nature, 261 (1976), 459-467.   Google Scholar

[15]

J. R. Munkres, Topology: A First Course, Prentice Hall, Inc., Englewood Cliffs, NJ, 1975.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

D. Singer, Stable orbits and bifurcation maps of the interval, SIAM J. Appl. Math., 35 (1978), 260-267.  doi: 10.1137/0135020.  Google Scholar

[19]

P. F. Verhulst, Notice sur la loi que la population poursuit dans son accroissement, Correspondance Math. Phys., 10 (1838), 113-121.   Google Scholar

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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