Dynamics and patterns of the least significant digits of the infinite-arithmetic precision logistic map orbits

Among the dynamics that the logistic map orbit can present, the chaotic behavior is highlighted due to its complexity. Despite extensive research on this type of dynamics, chaotic systems still exhibit divergence between analytical and numerical behavior due to the numerical precision limitations of computers. This limitation significantly impacts research on chaotic systems using a numerical approach. To address this issue, methods have been developed to compute the last digits of infinite-arithmetic precision orbits of the logistic map. As a result, fixed and periodic dynamics are found in the last digits of real logistic map orbits. Despite the evidence of such digits, their dynamic properties have not been extensively explored. This paper analyzed the stability, attraction, and classes of the fixed digits, as well as the stability dependency and frequency properties of periodic digits. Among the results are fixed digits with stable and unstable behavior, as well as the phenomenon of attraction. Additionally, frequency in the digits was found in the cycles generated by the periodic digits, along with a dependency relationship with the fixed digits exhibiting stable behavior. Our results reveal a diversity of properties and dynamics in the least significant digits of analytical logistic map orbits.