On Existence of Period Three Orbit and Chaotic Nature of a Family of Mappings

Kulkarni Pramod Ramakant

doi:https://doi.org/10.14445/22315373/IJMTT-V66I12P513

International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 66 | Issue 12 | Year 2020 | Article Id. IJMTT-V66I12P513 | DOI : https://doi.org/10.14445/22315373/IJMTT-V66I12P513

On Existence of Period Three Orbit and Chaotic Nature of a Family of Mappings

Kulkarni Pramod Ramakant

Citation :

Kulkarni Pramod Ramakant, "On Existence of Period Three Orbit and Chaotic Nature of a Family of Mappings," International Journal of Mathematics Trends and Technology (IJMTT), vol. 66, no. 12, pp. 85-90, 2020. Crossref, https://doi.org/10.14445/22315373/IJMTT-V66I12P513

Abstract

The occurrence of dynamical systems is observed in all branches of sciences like differential equations, Biological sciences, Physical sciences Mechanics, Economics and many more. On a broad spectrum, a dynamical system can be classified in to two categories viz. continuous and the discrete. As a discrete dynamical system, many mathematicians have extensively studied the one parameter family of functions; specially the logistic family Fμ (x) = μx(1 – x), the Tent family, Quadratic family, etc. In last few decades, the chaotic nature of many non-linear phenomenon has been a topic of great interest for the researchers all over the world. The occurrence of chaotic regime for certain values of the parameter in case of the family of mappings fc (x) = x2 – x + c using the phenomenon of the period doubling has been proved by Kulkarni P. R. and Borkar V. C. In this paper, we have proved the existence of the period three orbit in the family of mappings fc (x) = x2 – x + c for the parameter value C = -1.5 which proves that it is a chaotic map.

Keywords

chaotic behavior, critical points, one parameter family of functions, periodic points, Sarkovskii's theorem

References

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