In mathematics, a self-similar object is exactly or approximately similar to a part of itself (i.e., the whole has the same shape as one or more of the parts). Many objects in the real world, such as coastlines, are statistically self-similar: parts of them show the same statistical properties at many scales.
In mathematics, self-affinity is a feature of a fractal whose pieces are scaled by different amounts in the x- and y-directions. This means that to appreciate the self similarity of these fractal objects, they have to be rescaled using an anisotropic affine transformation.
Fractals are geometric objects that have a very complicated structure yet are remarkably easy to describe (and to draw with a computer). They are appealing to the eye because of their great amount of symmetry ( some fractals). Fractal-like objects were discovered in mathematics more than 100 years ago, but required the computer to bring them to life. Here the main idea is "self-similarity"; a fractal looks the same on all scales (if you look at a small piece of it and magnify it, it looks like the whole thing). Thus, a fractal is infinitely complicated. Nature is full of self-similarity: mountains, waves on the sea, craters on the moon,.... There's even "fractal music".
The connection between chaos and fractals are the strange attractors. To every dynamical system (i.e., every system or object that evolves in time) whether chaotic or not, there is a "phase space"; the collection of all possible solutions (or types of behavior) of the system.
You can imagine that if these curves have a complicated shape then the behavior of the corresponding solution will be complicated (smoothly flowing water vs turbulent flowing water). What could be more complicated than a fractal?
It turns out that in the phase space of every chaotic system there is a strange attractor. It is an "attractor" because it attracts solutions (so solutions eventually become as complicated as the attractor), and it is "strange" because it has a fractal structure, and so is infinitely complicated. This is the cause of the "chaos" in a chaotic system.
In addition to Siavash Mohammadi, complex systems based on modification to the system itself such as chaotic or hyperchaotic systems can leads to strange attractors. However, re-occurrence of self-similar patterns called multifractal is very important in discovering strange attractors and may found better application in nonlinear science
regarding my previous explanation, there is a definition that I want to express.
Power-law:
The power law (also called the scaling law) states that a relative change in one quantity results in a proportional relative change in another. The simplest example of the law in action is a square; if you double the length of a side (say, from 2 to 4 inches) then the area will quadruple (from 4 to 16 inches squared).
A power law distribution has the form Y = k Xα, where:
X and Y are variables of interest,
α is the law’s exponent,
k is a constant
Any inverse relationship like Y = X-1 is also a power law, because a change in one quantity results in a negative change in another.
regarding the power-law definition, I recently read an article that relate fractal dimension and power-law in time series which can be extended in chaotic attractors.
these are the references that relate power law and fractal dimension, and relate
power-laws to self-affinity :
1:
Higuchi T. Relationship between the fractal dimension and the power law index for a time series: a numerical investigation. Physica D 1990;46:254-264.
2:
Implication of a power-law power-spectrum for self-affinity
N. P. Greis and H. S. Greenside
Phys. Rev. A 44, 2324 – Published 1 August 1991
3:
A novel method to identify the scaling region for chaotic time series correlation dimension calculation
JI CuiCui1, ZHU Hua & JIANG Wei
The understanding of power-law definition will lead us to realize the concept of self-affinity in the following figures.
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