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Age-dependent Dynamics of Locomotion in Caenorhabditis elegans: A Lyapunov Exponent Analysis
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Discrete scale invariance and stochastic Loewner evolution.

M Ghasemi Nezhadhaghighi1, M A Rajabpour

  • 1Department of Physics, Sharif University of Technology, Tehran, P.O. Box 11365-9161, Iran.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|January 15, 2011
PubMed
Summary
This summary is machine-generated.

This study introduces a new class of fractal curves with discrete scale invariance (DSI) using the Weierstrass-Mandelbrot function within the Loewner equation. These fractal curves offer a novel method for classifying complex systems with scale invariance properties.

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Area of Science:

  • Complex Systems Analysis
  • Fractal Geometry
  • Stochastic Processes

Background:

  • Scale invariance is crucial for classifying statistical properties in complex systems.
  • The Loewner equation in two dimensions classifies fractal curves.
  • Weierstrass-Mandelbrot (WM) functions exhibit fractal properties.

Purpose of the Study:

  • Introduce a new class of fractal curves with discrete scale invariance (DSI).
  • Develop a method to classify fractal curves with DSI.
  • Investigate the physical applicability of these curves.

Main Methods:

  • Utilized the Loewner equation with the Weierstrass-Mandelbrot (WM) function as the drift.
  • Analyzed the diffusion coefficient of the variance trend of the WM function.
  • Examined contour lines of the 2D WM function as a physical candidate.

Main Results:

  • Introduced a large class of fractal curves exhibiting discrete scale invariance (DSI).
  • Demonstrated that fractal dimension can be extracted from the diffusion coefficient of the WM function's variance.
  • Showed that WM functions mimic Brownian motion behavior for fractal dimension calculations.

Conclusions:

  • Established a novel framework for classifying fractal curves with DSI.
  • The contour lines of the 2D WM function are proposed as a physical realization of these stochastic curves.
  • This research provides a new pathway for understanding and classifying complex fractal systems.