A study on the numerical convergence of the discrete logistic map

Clocking convergence is an important tool for investigating various aspects of iterative maps, especially chaotic maps. In this work, we revisit to the numerical convergence of the discrete logistic map x n + 1 = rx n ( 1 - x n ) gauged with a finite computational accuracy. Most of the previous studies of the discrete logistic map have been made for r ∈ [ 3 , 4 ] and r ∈ [ - 2 , - 1 ] due to the rich complexity of the map in these regions. In this work, we consider regions with simple fixed points, i.e. r = [ - 1 , 3 ] for which no particular geometric structures are known, as well as the period-doubling regions. We numerically investigate the speed of convergence in these regions to expose underlying complexity. The convergence speed is mapped to the phase space with different finite precisions. Patterns generated through this map are investigated over r. Numerical results show that there exists an interesting geometric pattern in r ∈ [ - 1 , 3 ] when convergence is gauged with a finite computational precision and also show that this pattern cascades into the period-doubling areas.