Calculating Fractal Dimension for the Lorenz Strange Attractor

In summary, the conversation discusses the process of calculating the dimension of a fractal object, specifically the Lorenz strange attractor. The questioner asks if there are different ways to define fractal dimension and if they are equivalent. The respondent mentions an old citation that may be helpful in calculating the dimension from small data sets.
  • #1
broegger
257
0
Hi.

How can I "experimentally" (by way of computer simulation) calculate an approximate value for the dimension of a fractal object? The object in question is the Lorenz strange attractor, which has a dimension between 2 and 3.

Also, I know there is a number of different ways to define fractal dimension (Hausdorff dimension, Correlation dimension, Pointwise dimention etc.): are these equivalent or does it matter which one is used?

Thanks.
 
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  • #2
My notes have this old citation:

N. B. Abraham, A. M. Albano, B. Das, G. De Guzman, S. Yong, R. S.
et al , Calculating the dimension of attractors from small data sets, Phys. Lett. A 114 (1986) 217.

I no longer have the paper, but I believe it may help.
 
  • #3
I'll try to locate that one. Thank you!
 

1. What is a fractal object?

A fractal object is a geometric shape or pattern that exhibits self-similarity at different scales. This means that the object appears similar or identical when viewed from different magnifications or perspectives. Fractals are often used to describe natural phenomena such as coastlines, clouds, and trees, but they can also be created mathematically.

2. How is the dimension of a fractal object determined?

The dimension of a fractal object is determined using the concept of Hausdorff dimension, which takes into account the self-similarity of the object at different scales. This dimension is typically a non-integer value, unlike the familiar dimensions of 1, 2, and 3 for lines, surfaces, and volumes, respectively.

3. Can fractal objects exist in more than three dimensions?

Yes, fractal objects can exist in any number of dimensions. In fact, some fractals are defined in higher dimensions, such as the Mandelbrot set which exists in four dimensions. However, it is difficult for us to visualize these higher-dimensional fractals because our perception is limited to three dimensions.

4. How are fractal dimensions different from Euclidean dimensions?

Fractal dimensions and Euclidean dimensions are fundamentally different concepts. Euclidean dimensions describe the extent of an object in a specific direction, while fractal dimensions describe the complexity or irregularity of the object. Additionally, Euclidean dimensions are always whole numbers, while fractal dimensions can be non-integer values.

5. What are some real-world applications of fractal objects?

Fractal objects have many practical applications in fields such as biology, geography, and computer science. They are often used to model natural phenomena, such as the growth of plants and the distribution of galaxies. In computer graphics, fractal algorithms are used to generate realistic-looking terrain and landscapes. Fractals are also being studied for their potential use in data compression and encryption.

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