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Central limit behavior of deterministic dynamical systems.

Ugur Tirnakli1, Christian Beck, Constantino Tsallis

  • 1Department of Physics, Faculty of Science, Ege University, 35100 Izmir, Turkey.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|May 16, 2007
PubMed
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We studied probability density in dynamical systems. For critical systems, we found a q-Gaussian distribution, generalizing the central limit theorem (CLT) for complex physical systems.

Area of Science:

  • Complex Systems Physics
  • Statistical Mechanics
  • Dynamical Systems Theory

Background:

  • Many complex physical systems involve dependent random variables.
  • Central Limit Theorem (CLT) describes the probability distribution of sums of independent random variables.
  • The applicability of CLT to deterministic dynamical systems with dependent iterates is an open question.

Purpose of the Study:

  • To investigate the probability density of rescaled sums of iterates in deterministic dynamical systems.
  • To analyze the validity and limitations of the Central Limit Theorem (CLT) in such systems.
  • To explore generalized forms of the CLT for critical dynamical systems.

Main Methods:

  • Analytical calculation of leading-order corrections to the CLT for logistic and cubic maps.

Related Experiment Videos

  • Numerical simulations to validate analytical findings.
  • Investigation of probability density convergence at the critical point of period doubling accumulation.

    • Main Results:

    • Excellent agreement between analytical calculations and numerical experiments for standard maps.
    • Demonstrated that CLT is not valid at the critical point due to strong temporal correlations.
    • Provided numerical evidence for convergence to a q-Gaussian distribution at the critical point.

    Conclusions:

    • The study reveals limitations of the standard CLT in certain deterministic dynamical systems.
    • A q-Gaussian distribution emerges at critical points, offering a power-law generalization of the CLT.
    • The observed behavior is universal for critical dynamical systems, independent of map order.