The Collatz map is investigated from a nonlinear-dynamics perspective with emphasis on the structure of its iterative orbits. By embedding integers within Sharkovsky’s ordering, odd initial values are shown to be sufficient for a complete characterization of dynamics. A “direction-phase” decomposition is introduced to separate iterative orbits into upward and downward phases, yielding a family of recursive functions parameterized by the number of upward phases. This formulation reveals a logarithmic scaling relation between the total iteration count and the initial value, confirming finite-time convergence to the period-three orbit. The Collatz dynamics is further shown to be dynamically equivalent to a binary shift map, whose ergodicity implies inevitable evolution toward attractors, thereby reinforcing convergence. Numerical analysis indicates that attraction basins follow a power-law distribution and display pronounced self-similarity. Moreover, odd integers grouped by upward-phase counts are found to follow Gamma statistics. Beyond its research implications, the framework provides a concise pedagogical case study illustrating how nonlinear dynamics, symbolic dynamics, and statistical characterization can be integrated to analyze a classical discrete problem.
Fu et al. (Wed,) studied this question.