Nikola Bozhinov

Lorenz attractor with Python

2020-09-12T00:00:00Z

Imagine a world where even the smallest butterfly can destroy the whole planet. It sounds ridiculous, right? However, there may be some truth in it. In the mid-twentieth century the mathematician Edward Lorenz developed a simple mathematical model of atmospheric convection. The model contains a system of three ordinary differential equations which we now know as the Lorenz equations:

\begin{aligned} \frac{dx}{dt} &= \sigma (y - x) \\ \frac{dy}{dt} &= x (\rho - z) - y \\ \frac{dz}{dt} &= x y - \beta z \end{aligned}

These equations show that if we change the initial conditions even a little bit, we get a different output. Here is the tricky part: the output is fundamentally different, not just slightly off. This behaviour is called chaotic and it is the reason why many physical systems are fundamentally unpredictable.

Solving the Lorenz system

There are many methods to solve this system. We will use the easiest one, the Euler method. First let us understand how it works on the equation

y' = y, \quad y(0) = 1

The formula is

y_{n+1} = y_n + h f(t_n, y_n)

where $h$ is the step. With $h = 1$ :

$y_1 = y_0 + h f(y_0) = 1 + 1 \cdot 1 = 2$ , $y_2 = 2 + 1 \cdot 2 = 4$ , $y_3 = 8$ , $y_4 = 16$ .

Now we apply the same idea to the Lorenz system:

import numpy as np
import matplotlib.pyplot as plt

sigma, rho, beta = 10, 28, 8 / 3
h, t = 0.01, 10000

x = np.empty(t + 1); y = np.empty(t + 1); z = np.empty(t + 1)
x[0], y[0], z[0] = 10, 10, 10

for i in range(t):
    x[i + 1] = x[i] + h * (sigma * (y[i] - x[i]))
    y[i + 1] = y[i] + h * (x[i] * (rho - z[i]) - y[i])
    z[i + 1] = z[i] + h * (x[i] * y[i] - beta * z[i])

fig = plt.figure(figsize=(12, 9))
ax = fig.add_subplot(projection="3d")
ax.plot(x, y, z, color="g", alpha=0.7, linewidth=0.5)
plt.show()

Sensitivity to initial conditions

Run the same loop for three starting points that differ by $10^{-5}$ and the trajectories separate after a few seconds of simulated time:

The divergence grows roughly exponentially. A simple chart of the distance between two such trajectories:

How the pieces fit together:

flowchart LR
  A[Initial conditions] --> B[Euler step]
  B --> C{t < T?}
  C -- yes --> B
  C -- no --> D[Plot with matplotlib]

The Seven Rila Lakes in one day

2026-09-27T00:00:00Z

The Seven Rila Lakes are the most visited place in the Rila mountains, and for a good reason: in a few hours the path climbs past seven glacial lakes, each on its own terrace. This is the route I walked, the map, the track and the photos.

The route

Approximate route past the lakes Open in OpenStreetMap GPX

GPX track of the walkDownload · GPX · 572 B

The coordinates are approximate. For a real hike use a track from a verified source and check the weather the evening before.

Photos

The Lower Lake
Sunset over the Kidney
The Eye from above

Video

A short video from the walk

What to take

Item	Why
Windproof jacket	Even in August the wind on the Lake Peak is cold
1.5 l of water	There are no springs above the Fish Lake
Cash	The chairlift does not always take cards

Musala: the highest point of the Balkans

2026-10-11T00:00:00Z

Musala is the highest peak of the Balkan peninsula and, thanks to the Yastrebets gondola from Borovets, one of the easiest 2900 m summits you can reach in a day.

Musala and the path from the Yastrebets gondola Open in OpenStreetMap

Part two of the series. The first part is the Seven Rila Lakes.

Everything a post can contain

2026-10-20T00:00:00Z

This post is the cheat sheet I use when writing. Every block below is written in plain Markdown with a few shortcodes; the source file is src/content/blog/writing-guide_en.md.

Callouts

Click to expand

Hidden content for long tables or spoilers.

Formulas

Inline $e^{i\pi} + 1 = 0$ and display:

\int_{-\infty}^{\infty} e^{-x^2}\,dx = \sqrt{\pi}

Diagrams and charts

sequenceDiagram
  participant M as Markdown
  participant E as Eleventy
  participant B as Browser
  M->>E: build
  E->>B: static HTML
  B->>B: mermaid.render() only here

Code

const greet = (name) => `Hello, ${name}!`;
console.log(greet("world"));

Keyboard: press Ctrl + K.

Images, figures and galleries

A figure with a caption. Click to open it full screen.

Video, YouTube and embeds

A YouTube video (loads on click)

Any iframe, in a responsive frame

Raw HTML works too

Cards

Lucide IconsBeautiful & consistent icons. Every icon on this site is from here. lucide.dev

A file to downloadDownload · GPX · 572 B

A button

Columns and tables

Left column text. Markdown works inside.

Right column text with a list:

Block	Syntax
Callout	`::: tip` … `:::`
Formula	$…$ and `$$…$$`
Diagram	```mermaid
Chart	```chart with JSON
Map	`{% map { … } %}`
Gallery	`{% gallery "images/folder" %}`

Nikola Bozhinov

Lorenz attractor with Python

Solving the Lorenz system #

Sensitivity to initial conditions #

The Seven Rila Lakes in one day

The route #

Photos #

Video #

What to take #

Musala: the highest point of the Balkans

Everything a post can contain

Callouts #

Formulas #

Diagrams and charts #

Code #

Images, figures and galleries #

Video, YouTube and embeds #

Cards #

Columns and tables #

Solving the Lorenz system

Sensitivity to initial conditions

The route

Photos

Video

What to take

Callouts

Formulas

Diagrams and charts

Code

Images, figures and galleries

Video, YouTube and embeds

Cards

Columns and tables