American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Bifurcation analysis and dynamic behavior to a predator-prey model with Beddington-DeAngelis functional response and protection zone
  • DCDS-B Home
  • This Issue
  • Next Article
    Spatial pattern formation in activator-inhibitor models with nonlocal dispersal
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, 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

[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

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 $
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Jan Sieber, Matthias Wolfrum, Mark Lichtner, Serhiy Yanchuk. On the stability of periodic orbits in delay equations with large delay. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 3109-3134. doi: 10.3934/dcds.2013.33.3109
[2]

Yoshihiro Hamaya. Stability properties and existence of almost periodic solutions of volterra difference equations. Conference Publications, 2009, 2009 (Special) : 315-321. doi: 10.3934/proc.2009.2009.315
[3]

Anna Cima, Armengol Gasull, Francesc Mañosas. Global linearization of periodic difference equations. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1575-1595. doi: 10.3934/dcds.2012.32.1575
[4]

Anatoli F. Ivanov, Sergei Trofimchuk. Periodic solutions and their stability of a differential-difference equation. Conference Publications, 2009, 2009 (Special) : 385-393. doi: 10.3934/proc.2009.2009.385
[5]

Răzvan M. Tudoran. On the control of stability of periodic orbits of completely integrable systems. Journal of Geometric Mechanics, 2015, 7 (1) : 109-124. doi: 10.3934/jgm.2015.7.109
[6]

Armengol Gasull, Héctor Giacomini, Maite Grau. On the stability of periodic orbits for differential systems in $\mathbb{R}^n$. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 495-509. doi: 10.3934/dcdsb.2008.10.495
[7]

Alessandra Celletti, Sara Di Ruzza. Periodic and quasi--periodic orbits of the dissipative standard map. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 151-171. doi: 10.3934/dcdsb.2011.16.151
[8]

Jifeng Chu, Zaitao Liang, Fangfang Liao, Shiping Lu. Existence and stability of periodic solutions for relativistic singular equations. Communications on Pure & Applied Analysis, 2017, 16 (2) : 591-609. doi: 10.3934/cpaa.2017029
[9]

Zigen Ouyang, Chunhua Ou. Global stability and convergence rate of traveling waves for a nonlocal model in periodic media. Discrete & Continuous Dynamical Systems - B, 2012, 17 (3) : 993-1007. doi: 10.3934/dcdsb.2012.17.993
[10]

E. Trofimchuk, Sergei Trofimchuk. Global stability in a regulated logistic growth model. Discrete & Continuous Dynamical Systems - B, 2005, 5 (2) : 461-468. doi: 10.3934/dcdsb.2005.5.461
[11]

Anatoly Neishtadt, Carles Simó, Dmitry Treschev, Alexei Vasiliev. Periodic orbits and stability islands in chaotic seas created by separatrix crossings in slow-fast systems. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 621-650. doi: 10.3934/dcdsb.2008.10.621
[12]

Qiang Li, Mei Wei. Existence and asymptotic stability of periodic solutions for neutral evolution equations with delay. Evolution Equations & Control Theory, 2020, 9 (3) : 753-772. doi: 10.3934/eect.2020032
[13]

Frédéric Mazenc, Michael Malisoff, Patrick D. Leenheer. On the stability of periodic solutions in the perturbed chemostat. Mathematical Biosciences & Engineering, 2007, 4 (2) : 319-338. doi: 10.3934/mbe.2007.4.319
[14]

Andrejs Reinfelds, Klara Janglajew. Reduction principle in the theory of stability of difference equations. Conference Publications, 2007, 2007 (Special) : 864-874. doi: 10.3934/proc.2007.2007.864
[15]

Nicola Guglielmi, Christian Lubich. Numerical periodic orbits of neutral delay differential equations. Discrete & Continuous Dynamical Systems - A, 2005, 13 (4) : 1057-1067. doi: 10.3934/dcds.2005.13.1057
[16]

E. Cabral Balreira, Saber Elaydi, Rafael Luís. Local stability implies global stability for the planar Ricker competition model. Discrete & Continuous Dynamical Systems - B, 2014, 19 (2) : 323-351. doi: 10.3934/dcdsb.2014.19.323
[17]

Jifeng Chu, Meirong Zhang. Rotation numbers and Lyapunov stability of elliptic periodic solutions. Discrete & Continuous Dynamical Systems - A, 2008, 21 (4) : 1071-1094. doi: 10.3934/dcds.2008.21.1071
[18]

Rafael Ortega. Stability and index of periodic solutions of a nonlinear telegraph equation. Communications on Pure & Applied Analysis, 2005, 4 (4) : 823-837. doi: 10.3934/cpaa.2005.4.823
[19]

Maria Carvalho, Alexander Lohse, Alexandre A. P. Rodrigues. Moduli of stability for heteroclinic cycles of periodic solutions. Discrete & Continuous Dynamical Systems - A, 2019, 39 (11) : 6541-6564. doi: 10.3934/dcds.2019284
[20]

Ana Cristina Mereu, Marco Antonio Teixeira. Reversibility and branching of periodic orbits. Discrete & Continuous Dynamical Systems - A, 2013, 33 (3) : 1177-1199. doi: 10.3934/dcds.2013.33.1177

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (47)
  • HTML views (137)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences