A characterization of the dynamics of Newton’s derivative

Abstract

In the present report the dynamic behaviour of the one dimensional family of maps $F_{a,b,c} \left( x \right) = c\left[ {\left( {1 - a} \right)x - b} \right]^{\frac{1} {{1 - a}}}$ is examined, for different ranges of the control parametres a, b and c. These maps are of special interest, since they are solutions of N ^′_f (x) = a, where N ^′_f is the Newton’s method derivative. In literature only the case N ^′_f (x) = 2 has been completely examined. Simultaneously, they may be viewed as solutions of normal forms of second order homogeneous equations, F″(x)+p(x)F(x) = 0, with immense applications in mechanics and electronics. The reccurent form of these maps, $x_n = c\left[ {\left( {1 - a} \right)x_{n - 1} - b} \right]^{\frac{1} {{1 - a}}}$ , after excessive iterations, shows an oscillatory behaviour with amplitudes undergoing the period doubling route to chaos. This behaviour was confirmed by calculating the corresponding Lyapunov exponents.

This work was supported by the PYTHAGORAS II project of the Greek Ministry of National Education and Religious Affairs and NATO ICS.EAP.CLG 981947. M. Özer acknowledges financial support from the Semiconductor Physics Institute, Vilnius, Lithuania (by the EC project PRAMA, contract Nr.G5MA-CT-2002-04014).