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 + ar}' is examined, for representative values of the control parametres a, b and A.. These maps are of special interest, since theyare solutions of N;j = 2 , where N;j is the Relaxed Newton's method derivative. The maps f(x) are proved to be solutions of the non-linear df(x) fJ[.r/..I](1+A)/A fJ 1 b-1/A differential equation, dx - '. J \Xi , where = /l. •. The reccurent form of these maps, Xn = b(xn_i + arA, after excessive iterations, shows in a Xn vs. A. plot, an initial exponential decay followed by a bifurcation. The value ofA. at which this bifurcation takes place, depends on the values of the parameters a, b. This corresponds to a switch to an oscillatory behaviour with amplitudes of f (X) undergoing a period doubling. For values of a slightly higher than i and at higher A.'s areverse bifurcation occurs and a bleb is formed. This behaviour is confirrned by calculating the corresponding Lyapunov exponent.