Lorenz Attractor

Edward Lorenz was an American meteorologist and mathematician, working on a model to predict the climate. He had 12 parameters (temperature, pressure, humidity, etc.) and 12 sets of equations and used a vacuum tube computer to run the model and printed out the result of each iteration as a row of 12 numbers and studied the parameters as it evolved over time.

When Lorenz re-ran the model using inputs from the old output, the two models were totally different, predicting an entirely different state of climate. This happened as the printer output was rounded of to 3 decimal points, while his vacuum tube computer ran the model with 6 decimal points. Thus a 1/1000th change in decimal had a massive impact on the results.

This gave birth to the most famous term “The Butterfly Effect” which states the flapping of butterflies in Brazil (as insignificant as they are) can lead to a tornado in Texas (thousands of miles away).

And is there a better time than right now to talk about chaos, as we are all in lockdown, waiting for the nightmare to be over. Underlying basis of chaos is a very sensitive dependence of initial condition. A small change in any of the initial condition can have very different consequence. The world is in chaos because someone in one corner of our world decided to give a firm handshake after a hearty meal, could have been very different if the one had waved goodbye.

“

Chaos isn’t a pit. Chaos is a ladder.

Lord Petyr Littlefinger Baelish, Game of Thrones

But the reason for this post is due to the fact that I was always fascinated by this system since I was 14, though not understanding it completely then (nor now), but I always loved the picture for its two-winged nature.

Embrace the Chaos

Dr. Sheldon Lee Cooper, The Big Bang Theory

So I decided to create a 3D plot of the Lorenz system and here is my attempt at it through programming of R…

Lorenz System

The 12 sets of equation of the atmospheric model was simplified to the following 3 equations,

3D Plot

Swipe, zoom and pan around the plot to discover order in the chaotic system.
Isn’t it beautiful!!!

library(viridis) # Default Color Maps from matplotlib
library(data.table) # Extension of data.frame
library(dplyr) # A Grammar of Data Manipulation
library(plotly) # Create Interactive Web Graphics via plotly.js

# Defining the equation in a function
lorenz_attractor <- function(s = 20, dt = 0.01,
                             r = 28, b = 8/3,
                             x = -1, y = 1,
                             z = -1, n = 100) {
  # Setting up the initial conditions
  x[1] <- x
  y[1] <- y
  z[1] <- z

  for (i in 2:n) {
    x[i] <- x[i-1] + (s * (y[i-1] - x[i-1])) * dt
    y[i] <- y[i-1] + (x[i-1] * (r - z[i-1]) - y[i-1]) * dt
    z[i] <- z[i-1] + (x[i-1] * y[i-1] - b * z[i-1]) * dt
  }

  df <- data.table(x = x, y = y, z = z)
  return(df)
}

# Creating a dataset with 10000 data points
# The more datapoints you have, more chaotic system
la_df <- lorenz_attractor(n = 10000)

plot_ly(la_df, x = ~x, y = ~y, z = ~z,
        type = 'scatter3d', mode = 'markers',
        marker = list(
          size = 1,
          color = ~z,
          colorscale = 'Viridis'
        ),
        hoverinfo = 'skip') %>%
  layout(scene = list(
    xaxis = list(title = ' ', showticklabels = FALSE,
                 showspikes = FALSE, showgrid = FALSE, zeroline = FALSE),
    yaxis = list(title = ' ', showticklabels = FALSE,
                 showspikes = FALSE, showgrid = FALSE, zeroline = FALSE),
    zaxis = list(title = ' ', showticklabels = FALSE,
                 showspikes = FALSE, showgrid = FALSE, zeroline = FALSE)
  ), paper_bgcolor = 'black') %>%
  config(displayModeBar = FALSE)

2D Plot

Alternatively, we can also plot the system in 2D using Plotly.

plot_ly(la_df, x = ~x, y = ~z,
        name = ' ',
        type = 'scatter',
        mode = 'lines',
        line = list(width = 1),
        hoverinfo = 'skip') %>%
  layout(
    xaxis = list(title = ' ', showticklabels = FALSE,
                 showspikes = FALSE, showgrid = FALSE,
                 zeroline = FALSE, fixedrange = TRUE),
    yaxis = list(title = ' ', showticklabels = FALSE,
                 showspikes = FALSE, showgrid = FALSE,
                 zeroline = FALSE, fixedrange = TRUE),
    plot_bgcolor = 'black',
    paper_bgcolor = 'black'
  ) %>%
  config(displayModeBar = FALSE)

Life Lesson

An insignificant choice can have a massive impact on life.
That is the essence of chaos.