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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

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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.

References

1. [1]	G. Teschl, Ordinary Differential Equations and Dynamical Systems, AMS, Graduate Studies in Mathematics, Volume 140, 2012.

2. [2]	U. A. Rozikov, A multi-dimensional-time dynamical system. Qual. Theory Dyn. Syst. 12(2), (2013) 361–375.

3. [3]	J.Benhabib, R.H. Day, Rational choice and erratic behaviour, Review of Economic Studies, 48, (1981) 459- 472.

4. [4]	R.H. Day, The emergence of chaos from classical economic growth, Quarterly Journal of Economics, 98, (1983) 201-213.

5. [5]	R.L. Devaney, An introduction to chaotic dynamical system, Westview Press, 2003.

6. [6]	R.U. Jensen, R. Urban, Chaotic price behaviour in a nonlinear cobweb model, Yale University. 1982.

7. [7]	P. Ribenboim, The New Book of Prime Number Records, New York: Springer, 1996.