A characterization of the dynamics of Newton’s derivative

Abstract

In the present report the dynamic behaviour of the one dimensional family of maps [...] 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,[....] , 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.