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A note on global stability in the periodic logistic map

  • * Corresponding author: Rafael Luís

    * Corresponding author: Rafael Luís 
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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  • 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.

    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 $

  • [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. GrinfeldP. 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. KelloggT. 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. 
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