Transformation of chaotic nonlinear polynomial difference systems through Newton iterations

Abstract

Chaotic sequences generated by nonlinear difference systems or 'maps' where the defining nonlinearities are polynomials, have been examined from the point of view of the sequential points seeking zeroes of an unknown function f following the rule of Newton iterations. Following such nonlinear transformation rule, alternative sequences have been constructed showing monotonie convergence. Evidently, these are maps of the original sequences. For second degree systems, another kind of possibly less chaotic sequences have been constructed by essentially the same method. Finally, it is shown that the original chaotic system can be decomposed into a fast monotonically convergent part and a principal oscillatory part showing sharp oscillations. The methods are exemplified by the well-known logistic map, delayed-logistic map and the Henon map.