A note on global stability in the periodic logistic map

Rafael Luís, ,Sandra Mendonça

doi:10.3934/dcdsb.2020094

Abstract

In this paper, the dynamics of the celebrated \begin{document}$ p- $\end{document} periodic one-dimensional logistic map is explored. A result on the global stability of the origin is provided and, under certain conditions on the parameters, the local stability condition of the \begin{document}$ p- $\end{document} periodic orbit is shown to imply its global stability.

Highlights

One of the most known and studied difference equation is the celebrated logistic map, a polynomial mapping of degree 2, often cited for being a simple non-linear dynamical equation with complex and chaotic dynamics.The logistic map was popularized by the biologist Robert May [14] in 1976, and may be seen as a discrete-time demographic model, with a continuous version given by the logistic equation, introduced by Pierre F
In this paper we show that the global stability condition may be extended for a larger set of parameters, as long as the parameters are such that p−1 i=0 ri and that the absolute value of the derivative along the periodic orbit is less or equal than 1
We show that this local stability condition implies global stability

Summary

Introduction

One of the most known and studied difference equation (or discrete dynamical system) is the celebrated logistic map, a polynomial mapping of degree 2, often cited for being a simple non-linear dynamical equation with complex and chaotic dynamics.The logistic map was popularized by the biologist Robert May [14] in 1976, and may be seen as a discrete-time demographic model, with a continuous version given by the logistic equation, introduced by Pierre F. It should be mentioned that in [11] the authors used different tools and studied the boundedness and the periodicity of non-autonomous periodic logistic map (3) when the sequence of parameters is periodic such that 0 < rn ≤ 2 for all n, and that sufficient conditions are given to support the existence of asymptotically stable and unstable p−periodic orbits, in this case. Coppel’s theorem ensures global stability only when the map f has a unique fixed point.

Results

Conclusion