Complexities and entropies of periodic series with application to the transition to turbulence in the logistic map

Linear and circular complexities and entropies of periodic series are defined. These concepts are then used to discuss the transition to turbulence in the infinite cascade of subharmonic bifurcations of the logistic map. When the stochasticity parameter increases from a critical value to the next critical value, the entropies are constant, or nearly constant. However, they undergo a discontinuity equal to In2 at each transition, increasing by steps. For chaotic orbits in the chaotic domain, a trivial extension of the concepts leads to infinite entropies.