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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