Stability for one-dimensional discrete dynamical systems revisited

Daniel Franco; Juan Perán; Juan Segura

doi:10.3934/dcdsb.2019258

Article Contents

2020, Volume 25, Issue 2: 635-650. Doi: 10.3934/dcdsb.2019258

This issue Previous Article Existence and multiplicity results for second-order discontinuous problems via non-ordered lower and upper solutions Next Article Periodic orbits of discrete and continuous dynamical systems via Poincaré-Miranda theorem

Stability for one-dimensional discrete dynamical systems revisited

1.
Departamento de Matemática Aplicada, ETSI Industriales, Universidad Nacional de Educación a Distancia (UNED), c/ Juan del Rosal 12, 28040, Madrid, Spain
2.
Departament d'Economia i Empresa, Universitat Pompeu Fabra, c/ Ramón Trías Fargas 25-27, 08005, Barcelona, Spain

^* Corresponding author: Juan Perán
^* Corresponding author: Juan Perán

Dedicated to Prof. Juan J. Nieto on the occasion of his 60th birthday

Received: December 31, 2018

Revised: April 30, 2019

Early access: November 2019

Published: February 2020

This work was funded by grant MTM2017-85054-C2-2-P (AEI/FEDER, UE) and ETSII-UNED grant 2019-MAT11

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Abstract

We present a new method to study the stability of one-dimensional discrete-time models, which is based on studying the graph of a certain family of functions. The method is closely related to exponent analysis, which the authors introduced to study the global stability of certain intricate convex combinations of maps. We show that the new strategy presented here complements and extends some existing conditions for the global stability. In particular, we provide a global stability condition improving the condition of negative Schwarzian derivative. Besides, we study the relation between this new method and the enveloping technique.

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Mathematics Subject Classification: Primary: 37C05, 37C75; Secondary: 92D25.

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