## Key Messages

### How Chaos Theory Revolutionized Science And Explained The Order Hiding Behind Randomness

For a very long time, scientists had viewed the world as a place that ran like clockwork, organized in a few simple rules.

Any signs of randomness or disorder were disregarded and dismissed as mere flukes.

That is, until the 1970s when a select group of researchers decided to take these peculiar instances seriously.

Thanks to new computer technology developments available at the time, chaotic behavior began to be found in all sorts of places – from weather patterns to the beat of our hearts.

Chaos theory revolutionized science when it was realized that there was an underlying order even beneath chaotic behavior.

These sections tell the story of this miraculous breakthrough and helps explain why chaos may actually be the ordering principle behind life itself!

Learn why something so insignificant as a butterfly in Peking can lead to storms halfway around the world in New York or why Britain’s coastline appears infinite and you’ll even find ways on how best to jetlag pesky mosquitos!

Uncovering Chaos Theory revealed much more than we ever thought possible and opened up doors to knowledge that mankind had previously thought was unattainable.

### The Butterfly Effect: A Tiny Difference Can Greatly Impact Complex Physical Systems

Meteorologist Edward Lorenz made a groundbreaking discovery in 1961 when he accidentally stumbled upon the chaotic and unpredictable nature of weather.

While running a simulation on his computer, he noticed that a tiny difference in variables could cause massive incongruency in results over time.

This led him to dub it the “butterfly effect”, which is also known as “sensitive dependence on initial conditions”.

This revelation became the cornerstone of building chaos theory, a field of study Lorenz became an intellectual father of.

In fact, his groundbreaking discovery highlighted how fragile and unstable physical systems like weather can really be.

### The Butterfly Effect Shows Us That Even Simple Systems Can Produce Complex Behaviors

The discovery that simple, nonlinear systems can produce incredibly complex behavior was a revelation for Lorenz.

He realized that nonlinear equations, in which the output value isn’t necessarily proportional to the input value, can lead to unpredictable outcomes and chaotic behavior.

He found evidence of this when he studied the motion of weather patterns, as well as with a waterwheel experiment.

In both cases, tiny fluctuations and changes could lead to outsized effects over time.

By plotting his three nonlinear equations on a graph, Lorenz discovered the double spiral shape later known as the butterfly effect.

This graceful wingspan represented a system in which repeating patterns of behavior almost occurred – yet never quite exactly repeated themselves.

It only took these three simple equations to demonstrate how experiments such as these could show incredibly chaotic results – yet still have an underlying order grooved into them.

This presented powerful proof that complex behavior stemming from simple systems could be found throughout nature and our everyday lives.

### The Unconventional, Chaotic Discoveries Of 1970s Science Challenges Common Understandings

The 1970s ushered in a new shift for physicists and mathematicians as they began to take a closer look at nonlinear systems.

This was a departure from the linear theories that had been previously adhered to.

For example, Galileo had taken to studying pendulums in his research, believing that no matter how wildly it swings, its motion remains predictable.

We now know this isn’t true due to factors such as friction and air resistance that cause the pendulum to enter into a nonlinear dynamical system, whose motion can easily become chaotic.

Mathematician Stephen Smale of UC Berkeley took a geometric approach and was one of the first people to take chaos seriously, even before he had heard about Lorenz’s work on chaos theory.

He studied oscillating electronic circuits using topology, which he used to create powerful visual analogies for the behavior of nonlinear systems.

Through this work he realized chaotic systems can actually be far more stable than their linear counterparts when it comes to average behavior despite outside noise or disturbances.

The discoveries made by both Lorenz and Smale paved the way for future generations of scientists who found interest and complexity in simple yet deterministic systems.

### How Nonlinear Dynamical Systems Showed Us The Chaos Of Animal Populations

Animal populations don’t simply grow and grow; their behavior is far more complex than that.

For instance, limited food resources, predators and environmental factors can all influence population growth in unpredictable ways.

This is why Animal populations typically exhibit nonlinear dynamical behaviour – meaning that their growth rate changes in unpredictable ways over time.

In order to understand the behaviour of animal populations, a branch of science called ecology was developed which studies how these populations change over time.

It was one of the first fields to link its findings to chaos theory thanks to the work of pioneering scientists like Robert May, James Yorke and Edward Lorenz.

One way biologists analyse population dynamics is by using difference equations, which measure population change year by year instead of smoothly like with differential equations (which are much harder to calculate).

A simple equation used here is the logistic differential equation – but this only works if it is able to restrain the populations’s growth after a certain point (otherwise it continues growing exponentially).

### Benoit Mandelbrot’S Fractal Geometry Reveals The Infinite Intricacy Of Nature

Benoit Mandelbrot’s revolutionary fractal geometry made a huge splash in the mathematical and scientific community when it was unveiled.

Fractal geometry is the study of infinitely intricate patterns found in complex dynamical systems and their properties.

Mandelbrot first noticed these patterns while studying fluctuations in cotton prices in 19th century, finding that there were small trends nested inside bigger trends and so on.

He also found this same symmetry of scale in nature’s structures, like mountains and clouds; these were what he called self-similar or fractals.

This infinity accounts for the rugged, scattered and fragmented nature of our world, something traditional Euclidean geometry could not account for.

The implications of Mandelbrot’s discovery are far reaching – from physics to economics to art and even philosophy – so its no wonder why his work continues to be celebrated today!

### How Strange Attractors Helped Unravel The Mystery Of Turbulence

Turbulence is notoriously difficult for physicists to study, but with the introduction of chaos theory and strange attractors, a light was shone on this complex behavior.

Strange attractors are points in phase space that describe the motion of particles in chaotic flows such as turbulence.

By plotting these points, physicists can track their movement and get hints at the underlying structure of turbulence and other chaotic systems.

This approach was first introduced by Belgian physicist David Ruelle when he developed an alternative to Lev D.

Landau’s theory on fluid motion.

Harry Swinney and Jerry Gollub then provided further evidence by constructing a system of two cylinders, one rotating inside the other, with a liquid flowing in the space between them.

This showed that as the rotation speed was increased, turbulent regions began to emerge alongside smooth areas – though not in a gradual manner.

Edward Lorenz had already discovered this mathematical phenomenon before Ruelle published his paper – having discovered a repeating butterfly pattern while plotting his first nonlinear system – which is known today as the Lorenz attractor.

With further studies by Michel Hénon on globular clusters showing similar results, it became clear that strange attractors were helping scientists understand these complicated motions of turbulence better than ever before.

### The Feigenbaum Constant: The Unifying Principle That Gave Chaos Theory Its Credibility

Mitchell Feigenbaum was a mathematical physicist whose work at Los Alamos National Laboratory was groundbreaking.

He was researching phase transitions in nonlinear equations, much like Robert May did with animal populations.

It was this research that ultimately led to the discovery of the Feigenbaum constants – universal features of nonlinear systems that remain constant regardless of system or equation.

Feigenbaum’s theory united the fledgling field of chaos theory and gave it greater credibility among traditional scientists.

His discovery provided chaotic systems with an underlying order, showing that there is predictability within complex patterns.

Over time, he identified more features of nonlinear systems and further advanced our understanding of chaos theory.

In 1979 his theory was proved mathematically, and since then Feigenbaum’s work has helped influence a wide range of disciplines from computer engineering to climate prediction to biology, psychology, economics and more.

Thanks to him, chaos theory is now a highly respected area of study with strong influences across a variety fields.

### The Dynamical Systems Collective At The University Of California Santa Cruz Played A Pivotal Role In Popularizing Chaos Theory

In 1977, a group of young mathematicians at the new Santa Cruz campos of the University of California led by Robert Stetson Shaw came together to popularize chaos theory and lay the foundation for what we now call “chaos studies.” This group, known as the Dynamical System Collective, revolutionized how chaos theory was perceived.

One way they achieved this was by using computer visuals created using an analog computer to help explore and visualize nonlinear systems which were chaotic in nature.

For example, they would graph equations related to Lorenz attractors and use knobs to adjust different variables.

Their goal was to demonstrate how strange attractors can lead to higher entropy, thus creating interesting and unpredictable behaviors.

The Dynamical Systems Collective also wanted to prove that chaos theory could be applied in everyday scenarios.

They used ordinary phenomena such as car fenders rattling and flags snapping in the wind as examples of chaotic behavior in order to make chaos understandable for people who weren’t experts in physics or mathematics.

In one famous experiment, Shaw even demonstrated that a dripping faucet could produce chaotic results!

### Chaos Theory Reveals The Loaded Dice Of Life: How Nonlinear Dynamics Found Their Way Into Medicine

The idea of chaos theory has found its way into scientific mainstream and researchers have discovered nonlinear dynamical systems in many different places.

An example is the tiny box experiment conducted by French physician Albert Libchaber, where he heated two plates containing liquid helium and observed it organizing itself into rolling cylinders first before bifurcations occurred with increasing temperature.

Through his experiment, Libchaber came to the realization that nonlinearity can be a beneficial defense against noise, glitches, and errors.

He noticed that when a linear system receives a small push off its natural course it will stay off track forever; whereas if a nonlinear system gets the same nudge, it tends to find its way back to its original state.

His realization about chaos was further fueled by Bernardo Huberman’s presentation on how it applies in human biology such as eye movement in schizophrenia patients and patterns of heartbeat in medical conditions such as ventricular fibrillation.

Indeed, doctors today have identified more dynamical diseases where nonlinear dynamics are key factors.

This just further proves that nonlinear dynamical systems can be found everywhere in nature and they play an important role in our own biology.

All these findings challenge Einstein’s famous claim of “God does not play dice with the universe”, giving us reason to believe that evolution has enabled us to better understand why certain behavior occur in unexpected ways – what motivates them and how could we deal with them better amid chaotic motes in our environment.

## Wrap Up

In conclusion, Edward Lorenz’s discovery in the 1960s proved that even with simple physical systems, there is hidden chaos.

This chaos has manifest all around us: in weather phenomena, population levels, and even our own heartbeat.

Mathematicians like Benoit Mandelbrot and physicists like Mitchell Feigenbaum further revealed the strange and beautiful order behind this chaos.

To test this principle for yourself, try the Chaos Game invented by British mathematician Michael Barnsley.

All you need is a coin to flip, a piece of paper, and a pen.

Make up rules for where to place each coin flip with your points to eventually result in a distinct shape–the chaotic system produces its own mathematical pattern thanks to random processes!