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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November
2020, 25(11): 4211-4220.
doi: 10.3934/dcdsb.2020094
A note on global stability in the periodic logistic map
| 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 |
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
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References:
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doi: 10.1007/978-3-642-59281-2. |
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Z. AlSharawi and J. Angelos,
On the periodic logistic equation, Appl. Math. Comput., 180 (2006), 342-352.
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W. A. Coppel,
The solution of equations by iteration, Proc. Cambridge Philos. Soc., 51 (1955), 41-43.
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R. L. Devaney, An Introduction to Chaotic Dynamical Systems, Studies in Nonlinearity,
second edition, Westview Press, Boulder, CO, 2003. |
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Global attractors for difference equations dominated by one-dimensional maps, J. Difference Equ. Appl., 14 (2008), 391-410.
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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. |
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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.
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The effects of seasonality on discrete models of population growth, Theoret. Population Biol., 26 (1984), 340-360.
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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. |
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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. |
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R. M. May, Simple mathematical models with very complicated dynamics, Nature, 261 (1976), 459-467. Google Scholar |
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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. |
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D. Singer,
Stable orbits and bifurcation maps of the interval, SIAM J. Appl. Math., 35 (1978), 260-267.
doi: 10.1137/0135020. |
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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. |
| [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. Google Scholar |
| [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. Google Scholar |
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