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 ] − λ / ( α − 1 ) is examined, for representative values of the control parameters a , c 1 , c 2 and λ . 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 λ plot, an initial exponential decay followed by a bifurcation. The value of λ 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 λ 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.