Some Results on Dynamics of Newton Differential Equation

Mehmet �zer

1 Istanbul Kultur University, Department of Physics 34191 Istanbul, Turkey
m.ozer@iku.edu.tr

Ya�ar Polato�lu

2 Istanbul Kultur University, Department of Mathematics and Computer, 34191 Istanbul, Turkey
y.polatoglu@iku.edu.tr

G�rsel Hacibekiro�lu

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 M�bius 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, M�bius transformation, Fibonacci sequences