Is a Fractal Dimension of 0.65 Possible in Nonlinear Oscillator Systems?

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Discussion Overview

The discussion revolves around the possibility of obtaining a fractal dimension of 0.65 in the context of nonlinear coupled spring oscillators, specifically through the analysis of a Poincaré map. Participants explore the implications of this finding and the methods used to calculate the fractal dimension.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant questions the validity of a fractal dimension below 1, suggesting that self-similarity in the structures within each box is necessary.
  • Another participant proposes an alternative approach to plotting the data, indicating a different method for calculating the fractal dimension.
  • The original poster explains their method of plotting log(1/e) against log(N(e)), noting that the slope approaches 0 at small box sizes due to limited resolution in their Poincaré map.

Areas of Agreement / Disagreement

Participants express differing views on the acceptability of a fractal dimension of 0.65, with some questioning its validity based on the need for self-similarity, while others focus on the methodology used to derive this value. The discussion remains unresolved regarding the implications of the fractal dimension obtained.

Contextual Notes

The discussion highlights limitations related to the resolution of the Poincaré map and the dependence on the number of data points taken, which may affect the accuracy of the fractal dimension calculation.

Joran
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First post!

I'm investigating chaos in non linear coupled spring oscillators. After generating a poincare' map of said system i wanted to see if the map was fractal. i proceeded to use a box counting method in order to calculate a fractal dimension.

I generated a plot of log(number of occupied boxes) versus log(total boxes) and the slope should be the fractal dimension, i think. However my slope was approximately .65

Is this acceptable? I've only seen fractal dimensions of values between 1 and 2. Which leads me to believe I've done something wrong. any help?

also if i posted in the wrong place i apologize in advance.
 
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should be more than 1. Dont you need to guarantee that the structures in each box are self-similar?
 
I think this is more appropriate for your data:

4ef00bf6711362601423ddefe26b6237.png


where e is the s box size
 
thanks for the reply

i plotted log(1/e) on the x-axis and log(N(e)) on the y-axis, making the slope D

in terms of taking the limit as the box approaches zero: my poincare' map has a limited resolution (number of data points taken) and at small box sizes the total number of occupied boxes will only approach the max number of data points.

my plot of log(1/e) on the x-axis and log(N(e)) on the y-axis shows this because at small box sizes the slope approaches 0.
 

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