DEFINING CHAOS IN THE LOGISTIC MAP BY SHARKOVSKII'S THEOREM

The fixed points of 3-cycle in the logistic map are obtained by solving a sextic polynomial analytically. Therewith, the domain of chaos is established by Sharkovskii's theorem. A fix-point spectrum is then constructed in the chaotic domain. By Sharkovskii's theorem, a chaotic trajectory is shown to be a superposition of all finite cycles, termed an aleph cycle. An aleph cycle means chaos and it defines chaos in the logistic map in an absolute sense. In particular, a trajectory which is ergodic is aleph-cyclic, hence it is also chaotic.