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

The dynamic behaviour of the one-dimensional family of maps f(x)=c2[(a−1)x+c1]−λ/(α−1)f(x)=c2[(a−1)x+c1]−λ/(α−1) is examined, for representative values of the control parameters a,c1a,c1, c2c2 and λλ. The maps under consideration are of special interest, since they are solutions of the relaxed Newton method derivative being equal to a constant aa. The maps f(x)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 xnxn versus λλ plot, an initial exponential decay followed by a bifurcation. The value of λλ at which this bifurcation takes place depends on the values of the parameters a,c1a,c1 and c2c2. This corresponds to a switch to an oscillatory behaviour with amplitudes of f(x)f(x) undergoing a period doubling. For values of aa higher than 1 and at higher values of λλ a reverse bifurcation occurs. The corresponding branches converge and a bleb is formed for values of the parameter c1c1 between 1 and 1.20. This behaviour is confirmed by calculating the corresponding Lyapunov exponents.