The relaxed Newton method derivative: Its dynamics and non-linear properties

Abstract

The dynamic behaviour of the one-dimensional family of maps f(x) = c(2)[(a - 1)x + c(1)](-lambda/(alpha-1)) is examined, for representative values of the control parameters a, c(1), c(2) and lambda. The maps under consideration are of special interest, since they are solutions of the relaxed Newton method derivative being equal to a constant a. The maps f(x) are also proved to be solutions of a non-linear differential equation with outstanding applications in the field of power electronics. The recurrent form of these maps, after excessive iterations, shows, in an x(n) versus lambda plot, an initial exponential decay followed by a bifurcation. The value of lambda at which this bifurcation takes place depends on the values of the parameters a, c(1) and c(2). This corresponds to a switch to an oscillatory behaviour with amplitudes of f(x) undergoing a period doubling. For values of a higher than 1 and at higher values of lambda a reverse bifurcation occurs. The corresponding branches converge and a bleb is formed for values of the parameter c(1) between 1 and 1.20. This behaviour is confirmed by calculating the corresponding Lyapunov exponents. (c) 2007 Elsevier Ltd. All rights reserved.