Dynamics On Relaxed Newton's Method Derivative
Miliou, Amalia N.
Anagnostopoulos, Antonios N.
Üst veriTüm öğe kaydını göster
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.