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We apply renormalized entropy as a complexity measure to the logistic andsine-circle maps. In the case of logistic map, renormalized entropy decreases(increases) until the accumulation point (after the accumulation point up tothe most chaotic state) as a sign of increasing (decreasing) degree of order inall the investigated periodic windows, namely, period-2, 3, and 5, therebyproving the robustness of this complexity measure. This observed change in therenormalized entropy is adequate, since the bifurcations are exhibited beforethe accumulation point, after which the band-merging, in opposition to thebifurcations, is exhibited. In addition to the precise detection of theaccumulation points in all these windows, it is shown that the renormalizedentropy can detect the self-similar windows in the chaotic regime by exhibitingabrupt changes in its values. Regarding the sine-circle map, we observe thatthe renormalized entropy detects also the quasi-periodic regimes by showingoscillatory behavior particularly in these regimes. Moreover, the oscillatoryregime of the renormalized entropy corresponds to a larger interval of thenonlinearity parameter of the sine-circle map as the value of the frequencyratio parameter reaches the critical value, at which the winding ratio attainsthe golden mean.

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