Renormalized entropy for one dimensional discrete maps: periodic and quasi-periodic route to chaos and their robustness

We apply renormalized entropy as a complexity measure to the logistic and sine-circle maps. In the case of logistic map, renormalized entropy decreases (increases) until the accumulation point (after the accumulation point up to the most chaotic state) as a sign of increasing (decreasing) degree of order in all the investigated periodic windows, namely, period-2, 3, and 5, thereby proving the robustness of this complexity measure. This observed change in the renormalized entropy is adequate, since the bifurcations are exhibited before the accumulation point, after which the band-merging, in opposition to the bifurcations, is exhibited. In addition to the precise detection of the accumulation points in all these windows, it is shown that the renormalized entropy can detect the self-similar windows in the chaotic regime by exhibiting abrupt changes in its values. Regarding the sine-circle map, we observe that the renormalized entropy detects also the quasi-periodic regimes by showing oscillatory behavior particularly in these regimes. Moreover, the oscillatory regime of the renormalized entropy corresponds to a larger interval of the nonlinearity parameter of the sine-circle map as the value of the frequency ratio parameter reaches the critical value, at which the winding ratio attains the golden mean