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Related Experiment Videos

Double precision errors in the logistic map: statistical study and dynamical interpretation.

J A Oteo1, J Ros

  • 1Departament de Física Teórica, Universitat de València, 46100-Burjassot, València, Spain. oteo@uv.es

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|October 13, 2007
PubMed
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This study details round-off errors in double precision logistic map computations. Error statistics reveal jumps related to bifurcation diagram crossings, converging geometrically with Feigenbaum constants.

Area of Science:

  • Computational Mathematics
  • Dynamical Systems Theory
  • Numerical Analysis

Background:

  • The logistic map is a fundamental model in chaos theory, exhibiting complex behaviors.
  • Understanding numerical precision limitations is crucial for accurate simulations of dynamical systems.

Purpose of the Study:

  • To investigate the nature and statistical properties of round-off errors in double precision computations of the logistic map.
  • To analyze error behavior across different iterative regimes, including chaotic ones.

Main Methods:

  • Detailed examination of round-off errors in double precision logistic map calculations.
  • Analysis of error histograms across various iterative regimes.
  • Derivation of an explicit formula for errors in fully developed chaos.

Related Experiment Videos

  • Statistical analysis of the largest double precision error as a function of the map parameter.

    • Main Results:

    • Identified specific iterative regimes and their associated error behaviors.
    • Developed an explicit formula for round-off errors in chaotic regimes.
    • Characterized the statistics of the largest error by jumps linked to bifurcation diagram boundary crossings.
    • Observed geometric convergence of these jumps and their locations, related to Feigenbaum constants.

    Conclusions:

    • Round-off error statistics in the logistic map are non-trivial and parameter-dependent.
    • The observed convergence properties suggest a deep connection between numerical errors and universal scaling laws in chaos.
    • Further comparison with Benford's law may offer insights into leading digit distributions of computed trajectories.