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Related Experiment Videos

Box-counting dimension without boxes: computing D0 from average expansion rates.

P So1, E Barreto, B R Hunt

  • 1Department of Physics and Astronomy and the Krasnow Institute for Advanced Study, George Mason University, Fairfax, Virginia 22030, USA.

Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics
|April 24, 2002
PubMed
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We developed a new method to calculate the box-counting dimension of chaotic attractors using average expansion rates. This links geometric properties of complex sets to their dynamical behavior, validated with maps.

Area of Science:

  • Dynamical Systems and Chaos Theory
  • Fractal Geometry
  • Nonlinear Dynamics

Background:

  • Calculating fractal dimensions of chaotic attractors is crucial for understanding complex systems.
  • Existing methods can be computationally intensive or lack direct connection to dynamical properties.
  • The Kaplan-Yorke conjecture established a link between information dimension and expansion rates.

Purpose of the Study:

  • To propose an efficient iterative scheme for computing the box-counting dimension (capacity dimension) of chaotic attractors.
  • To establish a connection between the geometric property (box-counting dimension) and dynamical properties (average expansion rates).
  • To validate the proposed scheme analytically and numerically.

Main Methods:

  • Development of an iterative computational scheme.

Related Experiment Videos

  • Analytical demonstration using an exactly solvable two-dimensional hyperbolic map.
  • Numerical validation using a higher-dimensional nonhyperbolic map.

    • Main Results:

    • The proposed iterative scheme efficiently calculates the box-counting dimension.
    • A direct relationship between the box-counting dimension and average expansion rates was demonstrated.
    • The scheme's validity was confirmed across different types of chaotic maps.

    Conclusions:

    • The developed iterative scheme provides an efficient method for determining the box-counting dimension of chaotic attractors.
    • This work extends the insights of the Kaplan-Yorke conjecture to the box-counting dimension, linking geometry and dynamics.
    • The findings are robust, as shown by analytical and numerical evidence on diverse dynamical systems.