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

[Joran]

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.

 

[Pythagorean]

should be more than 1. Dont you need to guarantee that the structures in each box are self-similar?

 

[Pythagorean]

I think this is more appropriate for your data:

where e is the s box size

 

[Joran]

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.

 

[pmacias]

Hello and welcome to the world of fractal dimensions! It sounds like you are on the right track in your investigation. The box counting method is a common way to estimate the fractal dimension of a system, so your approach is valid.

A slope of 0.65 is within the acceptable range for a fractal dimension. As you mentioned, fractal dimensions typically fall between 1 and 2, but there are exceptions. For example, the famous Mandelbrot set has a fractal dimension of approximately 2.1.

It is possible that you have done something wrong in your calculations, but it is also possible that your system has a fractal dimension of 0.65. I would recommend double checking your calculations and perhaps trying a different method to confirm your results. Additionally, it may be helpful to compare your results to other systems with known fractal dimensions to see if they align.

Overall, it seems like you are on the right track and with some further investigation, you should be able to determine the true fractal dimension of your system. Best of luck in your research!