Dynamics on relaxed newton's method derivative
Data
2006Autorius
Özer, Mehmet
Hacibekiroglou, Gürsel
Valaristos, Antonios
Miliou, Amalia N.
Polatoglu, Yasar
Anagnostopoulos, Antonios N.
Čenys, Antanas
Metaduomenys
Rodyti detalų aprašąSantrauka
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.