Some Results on Dynamics of Newton Differential Equation

Mehmet Özer

1 Istanbul Kultur University, Department of Physics 34191 Istanbul, Turkey

m.ozer@iku.edu.tr

Yaşar Polatoğlu

2 Istanbul Kultur University, Department of Mathematics and Computer, 34191 Istanbul, Turkey

y.polatoglu@iku.edu.tr

Gürsel Hacibekiroğlu

3 Aristotle University of Thessaloniki, Department of Informatics, 541 25 Thessaloniki Greece

antonisv@hotmail.com

Antonis Valaristos and Antonis N. Anagnostopoulos

4 Aristotle University of Thessaloniki, Department of Physics, 541 25 Thessaloniki Greece

anagnost@physics.auth.gr

Abstract

In this paper the case where the Newton differential equation defined by the Newton-Raphson iteration method equals a constant (namely 2) is investigated. The dynamics of these functions are examined by showing that the solution function can only be a Möbius transformation in the form of . The result obtained after the iterations carried out for the parameters a = 0.1 and b = 0.01 is the so-called “Fibonacci sequences”.

Keywords: dynamical system, Newton derivative, Möbius transformation, Fibonacci sequences