Dynamical Systems
Fractals: Mandelbrot and Julia
Zoom into the Mandelbrot set boundary at −0.7269 + 0.1889i to a depth of 10⁻¹² reveals tiny copies of the entire set, nested inside spirals nested inside spirals. This is a mathematical object of infinite complexity, generated by three symbols: z² + c.
- **Medicine:** the fractal dimension of capillary networks and pulmonary alveoli correlates with health-d ≈ 1.7 is normal, deviations indicate pathology
- **Finance:** asset price time series exhibit statistical self-similarity-"roughness" is consistent across scales from minutes to years
- **Computer graphics:** procedural generation of mountains, clouds, and trees is based on fractal noise (Perlin, Brownian)
Предварительные знания
The Mandelbrot Set
**Take a point c in the complex plane. Run the iteration z_{n+1} = z_n² + c starting from z₀ = 0. If the orbit escapes to infinity, c does not belong to the Mandelbrot set. If it stays bounded, it does.** This simple rule generates one of the most complex mathematical objects ever described.
**The Mandelbrot set M** is the set of values c ∈ ℂ for which the iteration **z_{n+1} = z_n² + c** (with z₀ = 0) remains bounded. Key criterion: if |z_n| > 2 for some n, the orbit will escape to infinity. The boundary of M is a fractal with Hausdorff dimension 2, exhibiting infinite complexity at every zoom level.
| Region c | Behavior of the orbit z_{n+1} = z_n² + c |
|---|---|
| c = 0 | z_n = 0 for all n-fixed point |
| c = −2 | Orbit: 0 → −2 → 2 → 2 → 2...-limit cycle |
| c = −0.75 | Orbit converges to a 2-cycle |
| c = i | Orbit bounded-c ∈ M |
| c = 1 | Orbit 0 → 1 → 2 → 5 → ... escapes-c ∉ M |
Benoit Mandelbrot, 1980
Benoit Mandelbrot first visualized his set on an IBM computer in 1980. When he saw the image he was astonished by its complexity-it looked like the machine had made an error. He coined the term "fractal" himself, from the Latin fractus meaning "broken". His book "The Fractal Geometry of Nature" (1982) changed how scientists describe complex structures in nature.
Point c = 0.5 + 0.5i. After iterating z_{n+1} = z_n² + c from z₀ = 0 we find |z₁| = |c| = √0.5 ≈ 0.707. How can it be determined whether c belongs to the Mandelbrot set?
Julia Sets
**Mandelbrot and Julia are two views of the same iteration.** In the Mandelbrot set we fix z₀ = 0 and vary c. In a Julia set it is reversed: we fix c and ask which starting points z₀ produce bounded orbits. Each value of c corresponds to its own Julia set J_c.
**Julia set J_c** is the set of starting points z₀ ∈ ℂ for which the iteration **z_{n+1} = z_n² + c** (with fixed c) remains bounded. Its complement is the basin of attraction of infinity. The boundary of J_c is a fractal. Connection to Mandelbrot: if c ∈ M then J_c is connected; if c ∉ M then J_c is Cantor dust.
| c | J_c | Structure |
|---|---|---|
| c = 0 | Unit circle |z| = 1 | Perfect circle |
| c ≈ −0.12 + 0.74i (c ∈ M) | Connected fractal | Dragon-like |
| c = −0.5 + 0.5i (c ∈ M) | Connected with filaments | Snowflake |
| c = 2 (c ∉ M) | Cantor set | Totally disconnected |
**Dichotomy theorem:** for any c, either J_c is connected (and then c ∈ M), or J_c is totally disconnected Cantor dust (and then c ∉ M). The Mandelbrot set is precisely those values of c for which J_c is connected. The beauty of M follows from the beauty of this dichotomy.
If c ∉ M (outside the Mandelbrot set), the corresponding Julia set J_c:
Fractal Dimension
**How is the "complexity" of a fractal measured?** The coastline of Britain: the more detailed the map, the longer the coastline-it has no finite length. But it cannot be described by a two-dimensional area either. Fractal dimension is a non-integer number between 1 and 2, reflecting how much space an object fills.
**Hausdorff dimension**: a generalization of dimension to fractals. For a self-similar fractal consisting of N copies at scale r: **d_H = log(N) / log(1/r)**. Examples: Koch curve d = log(4)/log(3) ≈ 1.26, Sierpinski triangle d = log(3)/log(2) ≈ 1.58, Lorenz attractor d ≈ 2.06.
| Fractal | N copies | Scale r | d_H = log N / log(1/r) |
|---|---|---|---|
| Cantor set | 2 | 1/3 | log(2)/log(3) ≈ 0.631 |
| Koch curve | 4 | 1/3 | log(4)/log(3) ≈ 1.261 |
| Sierpinski triangle | 3 | 1/2 | log(3)/log(2) ≈ 1.585 |
| Lorenz attractor | - | - | ≈ 2.06 (Kaplan-Yorke) |
The Sierpinski triangle consists of 3 scaled copies at ratio 1/2. Its fractal dimension is:
Self-Similarity
**Self-similarity is the defining property of fractals.** Zoom into the boundary of the Mandelbrot set at any point reveals the same structures again and again: spirals, "elephants", "seahorses". This is not coincidence-it is a mathematical consequence of the iterative process.
**Self-similarity** comes in several varieties: **exact**: copies are identical to the original (Cantor, Sierpinski); **statistical**: identical statistical properties at different scales (coastlines, clouds); **quasi-similarity**: similar but not identical structures at different scales (boundary of M). Natural fractals are almost always statistically self-similar.
| Type of self-similarity | Example | Characteristic |
|---|---|---|
| Exact | Koch curve, Sierpinski | Copies identical to original |
| Statistical | Coastline, mountains | Same statistics at different scales |
| Quasi-similarity | Mandelbrot boundary | Similar structures, not identical |
| Affine | Barnsley fern | Linear transformations + self-similarity |
Lewis Richardson and the Coastline
In 1961 Lewis Richardson asked: how long is the coastline of Britain? He found that the answer depends on the scale of measurement: the finer the ruler, the longer the coastline. Mandelbrot formalized this observation in 1967, introducing the concept of fractal dimension. The coastline of Britain has d ≈ 1.25.
Fractals are just pretty computer pictures with no mathematical substance
Fractals are sets with non-integer Hausdorff dimension, self-similar across scales. They describe real objects: coastlines (d ≈ 1.2), clouds (d ≈ 2.35), capillary networks (d ≈ 1.7).
Mandelbrot's fractal geometry answers the question "why are natural objects so complex?". Classical geometry describes ideal spheres and planes. Nature creates mountains, trees, rivers-objects best described by fractals with non-integer dimensions.
The length of the Koch curve under infinite iteration:
Key Takeaways
- **Mandelbrot set M**: values c for which z_{n+1}=z_n²+c (z₀=0) stays bounded; its boundary is a fractal
- **Julia set J_c**: starting points with bounded orbits for fixed c; c ∈ M ↔ J_c connected
- **Hausdorff fractal dimension** d = log(N)/log(1/r)-a non-integer characterizing how much space is filled
- **Self-similarity**: structure repeats across scales: exact (Sierpinski), statistical (coastline), quasi-similar (Mandelbrot)
Related Topics
Fractals permeate all of nonlinear dynamics:
- Chaos and Strange Attractors — Strange attractors have fractal structure-chaos and fractals are inseparable
- Ergodic Theory — Invariant measures on fractals are central objects of ergodic theory
- Population Dynamics — Bifurcation diagrams of the logistic map have fractal structure
Вопросы для размышления
- A coastline is fractal-but a real coastline is made of atoms. At what scale does self-similarity break down? What does this say about fractals as models of reality?
- If c ∈ M is determined by the boundedness of the orbit from z₀=0, why is z₀=0 the "right" starting point for defining M?
- A vascular network in the heart has a fractal dimension. Would that dimension increase or decrease in heart failure?