ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

The Dynamical System Generated by the Floor Function λx

Journal of Applied Nonlinear Dynamics 5(2) (2016) 185--191 | DOI:10.5890/JAND.2016.06.005

U.A. Rozikov; I.A. Sattarov; J.B. Usmonov

Institute of Mathematics, 29, Do’rmon Yo’li str., 100125, Tashkent, Uzbekistan Namangan State University, Namangan, Uzbekistan

Abstract

We investigate the dynamical system generated by the floor function λx defined on R and with a parameter λ ∈ R. For each given m ∈ N we show that there exists a region of values of λ, where the floor function has exactly m fixed points (which are non-negative integers), also there is another region for λ , where there are exactly m+1 fixed points (which are non-positive integers). Moreover the full set Z of integer numbers is the set of fixed points iff λ = 1. We show that depending on λ and on the initial point x the limit of the forward orbit of the dynamical system may be one of the following possibilities: (i) a fixed point, (ii) a two-periodic orbit or (iii) ±ꝏ.

Acknowledgments

U.Rozikov thanks Aix-Marseille University Institute for Advanced Study IMéRA (Marseille, France) for support by a residency scheme. His work also partially supported by the Grant No.0251/GF3 of Education and Science Ministry of Republic of Kazakhstan.

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