A BRIEF INTRODUCTION INTO THE COMPUTATIONAL MATHEMATICS OF NON-LINEARITY WITHIN SYSTEMS:

FEATURING THE METEOROLOGY PROFESSOR EDWARD LORENZ (1)

A BRIEF INTRODUCTION INTO THE COMPUTATIONAL MATHEMATICS OF NON-LINEARITY WITHIN SYSTEMS, FEATURING THE METEOROLOGY PROFESSOR EDWARD LORENZ

INTRODUCTION

Quoted from: Russell, Rupert.  Price Wars: How the Commodities Markets Made Our Chaotic World.  (New York, NY: Doubleday, 2022).  Pp. 15-

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A chain of events was coming into view (in 2015-16): the Arab spring revolutions, the outbreak of civil wars, the rise of ISIS, the global refugee crisis, and the populist explosion.  This all seemed like a butterfly effect.  But when I looked into the science behind this famous metaphor, I discovered that it is far more than mere linking of disparate events.  It’s a powerful mathematical theory that describes not just their connection, but their explosiveness too.

The theory began on a routine winter day in 1961 at the Massachusetts Institute of Technology (MIT).  A climatology professor named Edward Lorenz stood by his Royal McBee as the 113 vacuum tubes inside whirred and rattled.  Enormous, slow and noisy, it was one of the first factory-made computers and Lorenz had programmed one of the first 

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climate simulations.  He watched a mechanical typewriter print line after line of numbers.  Each line predicted how the elements—pressure, temperature, rain, etc.—would combine to produce the weather.  Looking at the printout, he wondered what would happen a few more months into the future.  But rather than start the program over from the beginning, he entered numbers from a line in the middle of the printout.  He set the Royal McBee to work and left it to get himself a fresh cup of coffee.  When he returned an hour later, he thought the computer had malfunctioned.  The results didn’t make any sense.  “The numbers being printed were nothing like the old ones,” he later wrote.  “I immediately suspected a weak vacuum tube or some other computer trouble.”

The Royal McBee wasn’t broken.  The vacuum tubes were working just fine.  Lorenz found there was a tiny difference in how each one started.  When he restarted the simulation, he had rounded the original number 0.506127 down to 0.506.  “The initial round-off errors were the culprits,” he discovered.  “They were steadily amplifying until they dominated the solution.”  A thousandth of a degree Celsius should have no impact.  Satellites couldn’t even measure a difference that small.  Ever since Isaac Newton, physicists had assumed that cause and effect were proportional.  Small forces had small effects.  Measurements only needed to be approximate.  But Lorenz’s printouts suggested something different.  Small forces could have big effects.  But how?

There was something unusual about the equations Lorenz was using.  He was trying to capture how the weather today could impact the weather tomorrow, and how tomorrow’s weather would impact the day after tomorrow’s weather, and so forth.  His equations had to capture this feedback.  He had to use a “non-linear” function to do so.  Before computers, non-linear functions were hard to calculate.  Each new day would require a new set of calculations that would have to be done by hand.  It was cumbersome and impractical.  So the world of feedback was largely ignored, and its scientific importance dismissed.  Computers made this world of feedback suddenly accessible.  And almost as soon as the first computers were processing these non-linear equations, discoveries were made.

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Lorenz’s was one of the first.  He found that systems filled with feedback were highly sensitive.  Small changes in temperature or pressure could be amplified over time.  A gust of wind would become turbulence, the turbulence could gather into a storm and a storm could grow into a hurricane.  Feedback amplifies, it turns something small into something big.  Lorenz called the power of these small starting points the “sensitivity to initial conditions.”  He described it with a metaphor: a seagull flapping its wings in Brazil causing a tornado in Texas.

In 1972, he was ready to present his big idea to an academic conference in Washington, D.C.  But before giving the talk, he received a suggestion from the conference organizer, Philip Merilees.  Why not swap the seagull for a Butterly?  Lorenz doesn’t know why Merilees made this suggestion.  He thought he might have been inspired by Ray Bradbury’s short story “A Sound of Thunder,” where the death of a prehistoric butterfly sets in motion a series of events that alters the result of a presidential election.  But Merilees said he’d never heard of it.  “[T]he butterfly, with its seeming frailty and lack of power,” Lorenz later reasoned, “is a natural choice for a symbol of the small that can produce the great.”  His talk, “Does the Flap of a Butterfly’s Wings in Brazil Set Off a Tornado in Texas?,” sparked a revolution that spread through meteorology, mathematics, the natural sciences, and even philosophy and popular culture.  The chance change to the title of his talk—swapping out a seagull for a butterfly—was a testament to the power of sensitivity that he had discovered.  Without it the revolution may never have occurred.

The popular telling of the butterfly effect emphasises the importance of chance encounters in sparking a chain reaction.  But Lorenz’s point was quite different.  Sensitivity is not a universal feature of the world with its own causal power.  It is instead a feature of a system, a system that at its heart is an amplifying engine: something which grows small things into big things.

BIOGRAPHY OF EDWARD LORENZ

Edward Norton Lorenz was an influential American mathematician and meteorologist known primarily for his pioneering work in chaos theory and the concept of the "butterfly effect." Born on May 23, 1917, in West Hartford, Connecticut, Lorenz's academic journey took him through Dartmouth College, Harvard University, and eventually to MIT, where he spent the majority of his career.  

Edward Lorenz served in the U.S. Army Air Corps during World War II. His involvement in the military directly influenced his career in meteorology. Initially, Lorenz trained as a weather forecaster, a role that was crucial for military operations. This experience piqued his interest in atmospheric sciences and led him to further his studies in meteorology after the war.

Lorenz's military service involved applying his mathematical skills to improve weather forecasting techniques, a necessity for the military's strategic planning. This period laid the foundation for his later groundbreaking work in chaos theory and numerical weather prediction as he studied atmospheric dynamics and numerical weather prediction. In the 1960s, while using a computer to model weather patterns, he discovered deterministic chaos. This insight came about when he found that tiny differences in initial conditions could lead to vastly different outcomes in weather models, a phenomenon later popularized as the "butterfly effect." His groundbreaking paper "Deterministic Nonperiodic Flow," published in 1963, laid the foundation for modern chaos theory.

Throughout his career, Lorenz's contributions extended beyond meteorology, influencing a wide range of scientific fields. He received numerous awards, including the Crafoord Prize in 1983 and the Kyoto Prize in 1991, for his significant impact on understanding atmospheric processes and climate.

VIDEO RESOURCES

Chaos: The Science of The Butterfly Effect (and Chaos Theory; Deterministic Chaos)

By Veritasium  

What everyone gets wrong about the butterfly effect

By Simon Clark 

Chaos theory is weirder than you think. Try everything Brilliant has to offer for free for 30 days: https://www.brilliant.org/simonclark

When people talk about the butterfly effect, they miss a large part of what makes chaos theory so incredible. Lets talk about that, and what it tells us about Earth's atmosphere.

My book Firmament: https://geni.us/firmament

Strogatz textbook on chaos: https://geni.us/strogatz

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Music by Epidemic Sound: http://nebula.tv/epidemic

Some stock footage courtesy of Getty.

Edited by Luke Negus.

What does the butterfly effect say about chaos theory? What is the science of the butterfly effect? Where does chaos theory come from? Is the butterfly effect real? Who discovered the butterfly effect? All this and more in this science video essay about the butterfly effect and how chaos theory is weird. Weirder than you think. I made it through the whole video without making a single Warhammer reference!

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A Simple Guide to Chaos Theory—BBC World Service

Weather and Chaos: The World Service

382,399 views  Jul 2, 2024  #WorldService #chaostheory #BBCWorldService

According to classical physics and the laws of Isaac Newton, it should be easy to predict the behaviour of objects throughout the universe with relative ease.

Click here to subscribe to our channel 👉🏽 https://bbc.in/3VyyriM

But in 1961, a meteorologist unwittingly discovered this was not the case – that there was a lot more uncertainty around us. Here's how chaos theory and its butterfly effect, has changed the way we think about our Universe.

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Weather and Chaos: The Work of Edward N. Lorenz

By Josh Kastorf

498 views  Apr 8, 2024

Why is it that we can predict a solar eclipse centuries in advance, but we can only predict the weather about a week or two in advance? Can a small-scale movement, like the flap of a butterfly’s wing, influence large-scale systems like hurricanes? In the 1960s, an MIT meteorologist exploring these questions with the help of early digital computers made a discovery that would change the way we understand not only weather but nearly everything in our universe. His work suggested there are certain systems we may never be able to predict, not because they are too complex, but because “chaos” is built into their underlying math. Years later, this idea would enter popular culture as “the butterfly effect.”

This is the first film about Edward N. Lorenz and his role in Chaos Theory produced with the participation of scientists who worked alongside him. With their help we take a closer look at what the “butterfly effect” actually meant in the context of Lorenz’s work, and why it should make all of us rethink our understanding of our universe.

Official Selection: American Documentary Film Festival, Philip K. Dick Film Festival, Academia Film Oulomouc, Raw Science Film Festival, Lift-Off Global Network Online Documentary Film Festival, Sci-On! Film Festival, LabMeCrazy Film Festival, China International Conference of Science and Education Producers.

Written, Produced, and Edited by 

Josh Kastorf

Narrated by

William Marshall Cline

Music by

Rob Jaret

Interview Videography by 

Josh Kastorf & Neale Brown

Audio Effects and Mix by

Ben Templeton

Interviewees:

Prof. Kerrry Emanuel

Cecil & Ida Green Professor of Atmospheric Science

MIT Dept. of Earth, Atmospheric & Planetary Sciences

Prof. Daniel Rothman

Professor of Geophysics

MIT Dept. of Earth, Atmospheric & Planetary Sciences

Special Footage:

Handmade Orrery

Courtesy Ken Condal (zeamon.com)

Butterfly Flow

Courtesy Prof. Haibo Dong (U. Virginia) 

and Dr. Chengyu Li (Villanova U.)

Restored Royal McBee LGP-30

Courtesy Cory Heisterkamp (radar58.com/LGP30)

LGP-30 Teletype Lorenz Data Set

Courtesy William McKenna (MIT)

Atmospheric Model with Balloon Trajectories

Courtesy Dr. Lodovica Illari and Dr. Glenn Flierl (MIT)

Lorenz Attractors

Made with Processing (processing.org)

Double Pendulum

Courtesy of Prof. Jack Wisdom (MIT)

Solar System Model

Made with Universe Sandbox (universesandbox.com)

Ed Lorenz at MIT 2004

Courtesy Merry Caston

Additional Footage and Images Courtesy of:

The Library of Congress

NASA

NOAA

The Lorenz Family

MIT Museum

Inamori Foundation

Prof. Howard Bluestein

Hal Bergstrand

J. Oishi, B. Brown, K. Burns, S. Clark, G Vasil, D. Lecoanet

 via APS Gallery of Fluid Motion

Rubens Machado, Smootheye, 

GreenShortz DIY, helmut satzger, Scott R, footageonline

via Youtube

Special Thanks:

Angela Ellis

Lauren Hinkel 

Faith Zhang

Christine Maglio

Darius Colazzo

Prof. Brian Evans

Dr. Amanda Bosch

Dr. Meg Rosenburg

Dr. Elizabeth Cavicchi

Ariel Weinberg

Ellen Gille

Niraja Lorenz

Cheryl Lorenz

Dr. Richard D. Rosen

Prof. Paola Malanotte-Rizzoli

Prof. Alan Robock (Rutgers U.)

Dr. Rebecca E. Morss (NCAR)

Prof. Dale R. Durran (U. Washington)

Prof. Tim Palmer (U. Oxford)

Prof. Jagadish Shukla (George Mason U.)

Made possible in part by:

Prof. John Marshall

Cecil and Ida Green Professor of Oceanography

MIT Department of Earth, Atmospheric, and Planetary Sciences

Prof. Daniel Rothman and Prof. Kerry Emanuel

Co-Directors, The Lorenz Center

MIT Department of Earth, Atmospheric, and Planetary Sciences

©2018 Josh Kastorf

All Rights Reserved