descriptionPublicationkeyboard_double_arrow_right Article , Preprint 01 Jan 2020Embargo end date: 01 Jan 2018 English Publisher:American Institute of Mathematical Sciences (AIMS)Journal:Discrete & Continuous Dynamical Systems - B, volume 25, pages 651-670 (issn: 1553-524X,

Authors: Gasull Embid, Armengol; Mañosa Fernández, Víctor;

doi: 10.3934/dcdsb.2019259 , 10.48550/arxiv.1809.06208

arXiv: http://arxiv.org/abs/1809.06208

handle: 2117/173965

Periodic orbits of discrete and continuous dynamical systems via Poincaré-Miranda theorem

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Abstract

We present a systematic methodology to determine and locate analytically isolated periodic points of discrete and continuous dynamical systems with algebraic nature. We apply this method to a wide range of examples, including a one-parameter family of counterexamples to the discrete Markus-Yamabe conjecture (La Salle conjecture); the study of the low periods of a Lotka-Volterra-type map; the existence of three limit cycles for a piece-wise linear planar vector field; a new counterexample of Kouchnirenko's conjecture; and an alternative proof of the existence of a class of symmetric central configuration of the $(1+4)$-body problem.

26 pages, 7 figures

Related Organizations

Universitat Politècnica de Catalunya
Spain
Universitat Polite`cnica de Catalunya
Spain
Autonomous University of Barcelona
Spain

Keywords

Central configurations, Classificació AMS::34 Ordinary differential equations::34D Stability theory, Classificació AMS::39 Difference and functional equations::39A Difference equations, Dynamical Systems (math.DS), :37 Dynamical systems and ergodic theory::37C Smooth dynamical systems: general theory [Classificació AMS], FOS: Mathematics, Differentiable dynamical systems, Periodic orbits, Mathematics - Dynamical Systems, Classificació AMS::70 Mechanics of particles and systems, Kouchnirenko's conjecture, Àrees temàtiques de la UPC::Matemàtiques i estadística::Equacions diferencials i integrals::Sistemes dinàmics, :13 Commutative rings and algebras::13P Computational aspects of commutative algebra [Classificació AMS], :Matemàtiques i estadística::Equacions diferencials i integrals::Sistemes dinàmics [Àrees temàtiques de la UPC], :39 Difference and functional equations::39A Difference equations [Classificació AMS], Sistemes dinàmics diferenciables, Thue-Morse maps, Classificació AMS::13 Commutative rings and algebras::13P Computational aspects of commutative algebra, Classificació AMS::37 Dynamical systems and ergodic theory::37C Smooth dynamical systems: general theory, Discrete Markus-Yamabe conjecture, Lotka-Volterra maps, Limit cycles, Planar piecewise linear systems, Poincaré-Miranda theorem, :70 Mechanics of particles and systems [Classificació AMS], 37C25, 39A23 (Primary), 13P15, 34D23, 70F15, 70K05 (Secondary), Kouchnirenko conjecture, :34 Ordinary differential equations::34D Stability theory [Classificació AMS]

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Periodic orbits of discrete and continuous dynamical systems via Poincaré-Miranda theorem
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