7. Chaotic Systems

7.1. Chaotic Systems

Chaos theory is a branch of mathematics focusing on the behavior of dynamical systems that are highly sensitive to initial conditions. "Chaos" is an interdisciplinary theory stating that within the apparent randomness of chaotic complex systems, there are underlying patterns, constant feedback loops, repetition, self-similarity, fractals, self-organization, and reliance on programming at the initial point known as sensitive dependence on initial conditions. [1]

\begin{align} k = 3.9976543219876543210 \\ chaos(x) = x_{i+1} = x_i . k . (1 - x_i) \tag{1} \end{align}

It can be seen how a small variation in its initial conditions produces completely different results.

In the example above approximately from iteration number 25.

7.2. Chaotic Systems with Attractors

In the mathematical field of dynamical systems, an attractor is a set of numerical values toward which a system tends to evolve, for a wide variety of starting conditions of the system. System values that get close enough to the attractor values remain close even if slightly disturbed. [2]

7.3. Fractals

In mathematics, a fractal is a subset of a Euclidean space for which the Hausdorff dimension strictly exceeds the topological dimension. Fractals tend to appear nearly the same at different levels, as is illustrated here in the successively small magnifications of the Mandelbrot set; because of this, fractals are encountered ubiquitously in nature

Fractals exhibit similar patterns at increasingly small scales called self similarity, also known as expanding symmetry or unfolding symmetry; if this replication is exactly the same at every scale, as in the Menger sponge, it is called affine self-similar. Fractal geometry lies within the mathematical branch of topology. [3]

Properties:

7.3.1. Sierpiński triangle

The Sierpinski triangle, also called the Sierpinski gasket or Sierpinski sieve, is a fractal and attractive fixed set with the overall shape of an equilateral triangle, subdivided recursively into smaller equilateral triangles. Originally constructed as a curve, this is one of the basic examples of self-similar sets–that is, it is a mathematically generated pattern that is reproducible at any magnification or reduction. [4]

7.3.2. Julia fractal

7.3.3. Mandelbrot set

The Mandelbrot set is the set of complex numbers $c$ for which the function $ f_{c}(z)=z^{2}+c $ does not diverge when iterated from $ z=0 $, i.e., for which the sequence $ f_{c}(0), f_{c}(f_{c}(0)) $, etc., remains bounded in absolute value. [5]

Reference

[1] Wikipedia - Chaos Theory.
[2] Wikipedia - Attractor.
[3] Wikipedia - Fractal.
[4] Wikipedia - Sierpiński triangle.
[5] Wikipedia - Mandelbrot set.


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