A New Fractal Method for Studying Logistic Series

Logistic series is an important chaos series. A new fractal method, limit dot counting dimension based on dot counting dimension was introduced and the relationship between limit dot counting dimension and logistic series was studied. The fluctuation of limit dot counting dimension of logistic series can be divided three states. In the first state, the limit dot counting dimension was fixed value and equaled to 1 which reflects the periodicity of data. In the second state, the limit dot counting dimension began to change and can reflect the sensitivity of initial value about logistic series. In the last state, the limit dot counting dimension fluctuated and had no relationship with initial value of logistic time which reflected the characters of chaos series. The limit dot counting dimension descended first and ascended later because of period-doubling bifurcation. The limit dot counting dimension increased as bifurcation increased. The limit dot counting dimension needs less data and has good linear correlation for fractal calculation which can reflect characters.