Dynamics on relaxed newton's method derivative

Abstract

In the present report the dynamic behaviour of the one dimensional family of maps f(x) = b(x + a)~ is examined, for representative values of the control parametres d, b and A . These maps arc of special interest, since they are solutions of -/Viy = 2, where Nr is the Relaxed Newton's method derivative. The maps f(x) are proved to be solutions of the non-linear differential equation, ——— — ft • [f(x) , where p = / • 6 . The reccurent form of dx these maps, X^ = b(x^ + d)~ , after excessive iterations, shows in a X^ vs. A plot, an initial exponential decay followed by a bifurcation. The value of 2 at which this bifurcation takes place, depends on the values of the parameters Q, b . This corresponds to a switch to an oscillatory behaviour with amplitudes of f(x) undergoing a period doubling. For values of d slightly higher than 1 and at higher A. 's a reverse bifurcation occurs and a bleb is formed. This behaviour is confirmed by calculating the corresponding Lyapunov exponent.