The "Butterfly Effect" is often ascribed to Lorenz. In a paper in 1963 given to the New York Academy of Sciences he remarks:

By the time of his talk at the December 1972 meeting of the American Association for the Advancement of Science in Washington, D.C. the sea gull had evolved into the more poetic butterfly - the title of his talk was* :

*One meteorologist remarked that if the theory were correct, one flap of a seagull's wings would be enough to alter the course of the weather forever.*By the time of his talk at the December 1972 meeting of the American Association for the Advancement of Science in Washington, D.C. the sea gull had evolved into the more poetic butterfly - the title of his talk was* :

*Predictability: Does the Flap of a Butterfly's Wings in Brazil set off a Tornado in Texas?*

Lorenz Definition of Chaos

"When the present determines the future, but the approximate present does not approximately determine the future."Lorenz Definition of Chaos

## Lorenz and the Butterfly Effect.

Meterologist Ed Lorenz MIT

Created Modern Chaos Theory

Model of weather patterns he developed

1) Dynamical Forecasting in the Spirit of Newton

Inch weather forward on instant at a time..

Too hard

2) Pattern Matching Forecasting - common sense based on previous experiences

3) Statisical Forecasting - numbers adjusting .... curve fitting

Lorenz- Tested the statistical methods.

Used a computer for thought experiments (telescope of the mind).

Computer centers whole rooms

Too simple at first... Either stationary state or oscillating.

Deterministic system of differential equations but non-periodic

Found a system that was deterministic but did not repeat itself (non-periodic).

In studying his artificial weather model what he found was

Butterfly extreme sensitivity of a chaotic system to tiny changes in initial conditions.

Image of butterfly flapping its wings in Brazil creating a tornado in texas.

Goes down the hall gets his cup of coffee and looks nothing like the weather he did the first time.

(he entered the numbers and put them back in the equations).

Really puzzled....

Where in computer program got the

they agreed perfectly at first

Noticed the error growed ... doubled every four days of his simulation...

The initial conditions he used was three digit numbers in printout, but internally computer was used

Round off errors in fourth decimal place...

Created Modern Chaos Theory

Model of weather patterns he developed

1) Dynamical Forecasting in the Spirit of Newton

Inch weather forward on instant at a time..

Too hard

2) Pattern Matching Forecasting - common sense based on previous experiences

3) Statisical Forecasting - numbers adjusting .... curve fitting

Lorenz- Tested the statistical methods.

Used a computer for thought experiments (telescope of the mind).

Computer centers whole rooms

Too simple at first... Either stationary state or oscillating.

Deterministic system of differential equations but non-periodic

Found a system that was deterministic but did not repeat itself (non-periodic).

In studying his artificial weather model what he found was

Butterfly extreme sensitivity of a chaotic system to tiny changes in initial conditions.

Image of butterfly flapping its wings in Brazil creating a tornado in texas.

Goes down the hall gets his cup of coffee and looks nothing like the weather he did the first time.

(he entered the numbers and put them back in the equations).

Really puzzled....

Where in computer program got the

they agreed perfectly at first

Noticed the error growed ... doubled every four days of his simulation...

The initial conditions he used was three digit numbers in printout, but internally computer was used

Round off errors in fourth decimal place...

## Lorenz Equation

Butterfly effect is signature of chaos

Any tiny error gets amplified infinitely fast.

Unable to predict long term...

The current state of the atmosphere.... cannot measure precisely.

Non periodic system that was deterministic... keenly vigilant.

Any tiny error gets amplified infinitely fast.

Unable to predict long term...

The current state of the atmosphere.... cannot measure precisely.

Non periodic system that was deterministic... keenly vigilant.

## Meaning of the Butterfly Effect

Term probably originated from a Lecture Lorenz gave back in 1972

"Predictability: Does the flap of a Butterfly's Wings in Brazil Set off a tornado in texas?"

Jurassic Park ... Jeff Goldbloom puts a drop on her hand and asks which

Too complex, you cannot predict it.

Sliding doors... She plays a character

2 trajectories of a persons life... How whole outcome of your life can be changed

**Arcadia - Tom Stoppard**

Everyone knows little things can make a big difference.. Ancient idea and concept.

Storm 1941

George Stewart

Meteorologist character

A man sneezing in China can set off a blizzard in NY

Part of the culture of meteorology... Slightest change could affect the weather.

A sound of thunder... A man goes back in time and steps on a butterfly.

Small effects can cascade and create big

**John Gower 1390**

For Want of a Nail...

*For want of a nail the shoe was lost,*

*for want of a shoe the horse was lost,*

*for want of a horse the knight was lost,*

*for want of a knight the battle was lost,*

*for want of a battle the kingdom was lost.*

*So a kingdom was lost—all for want of a nail.*

There was one nail that led to the loss of a Kingdom.

**Small changes can make a big difference in outcomes by a cascading of effects.**So what is so novel.

What is new is that the same kind of sensitivity to tiny changes can afflict even the simplest systems.

Double pendulum for example... Poincares 3 body problem or Lorenzs simple weather problem, systems like that you would not think chaos would manifest.

**Because we know ALL the variables and we know the laws that predict their behavior.**So many complexities in our life, but simple deterministic systems also display chaos.

We know we cannot predict our fate, or wars, etc.. Too complex.

## Poincare Quote

Poincare - Nightmarish tangle... How two states of a system that are indistinguishably close could lead to different fates for the 3 body problem.

Address the Question how chance can emerge in a deterministic world.

Exact

Exactly the Laws and Exactly the situation of the Universe (initial states).

You can never know the state of the universe exactly.

Poincare really understood the butterfly effect.

Address the Question how chance can emerge in a deterministic world.

Exact

Exactly the Laws and Exactly the situation of the Universe (initial states).

*"Even if we know the laws of nature EXACTLY we could still only know the initial situation approximately!"*You can never know the state of the universe exactly.

*"But it is not always so; it may happen that the small differences in the initial conditions produce very great ones in the final phenomena."*

When a small change blows up into a big change.*"A small error in the former will produce an enormous error in the latter. Prediction becomes impossible and we have a fortuitous phenomenon."*

That is how chance arises as a fortuitous one, from small uncertainties cascading into enormous ones.Poincare really understood the butterfly effect.

**Lyapunov exponent - a measure of the butterfly effect.**

Lyapunov exponent - a measure of the butterfly effect.

Lee-app-uh-noff

Rate of exponential growth.

Lee-app-uh-noff

Rate of exponential growth.

## Sir James Lighthill

"The recently recognized failure of predictability in Newtonian Dynamics." (1986)

Sir James Lighthill

We thought Newtons laws gave us total predictability.

Why were scientists so slow to recognize the butterfly effect.

Distasteful because threatening to science itself.

Sciences need to put systems in a box.

Scientists hate "everything effects everything else.." its a matter of degree.

The things you thought were negligible can have big effects.

Butterfly effect does not apply to

Does not occur in systems gently relaxing to some equilibrium state (particle rolling down a curved surface).

Two metronomes oscillating and flick it... errors grow linearly.

Errors do not snowball

Tides very predictable - huge fluid mechanical system. Weather is too... yet tides predictable.

One is periodic, and one is not.

Halleys Comet

Timing of Eclipses

All of these things very periodic which means they are predictable.

Tiny errors do not mushroom

It is by putting Non-periodicity into the weather that created chaos and the butterfly effect.

Two double pendulums initially are in sync, but then become unpredictable.

It is the TIME it takes for these tiny errors to double in size.

What is the length of the Horizon of predictability.

Predictibility Horizon demo.. 3 time units yellow and blue not the same (Lorenz computer animate)

Decorrelate and become totally different.

Solar System is Chaotic

But 5 million year Horizon of Predictability.

The solar system is chaotic but predictable because 5 million years is so long on the timescale of a human life.

You cannot predict beyond the predictability Horizon no matter how good your instruments are.

Trying to figure out the position of the planets 4 billion years ago is meaningless.

What if my instruments give me no error, but are perfect.

You cannot do that because it would require an infinite number of decimal places.

There is always some errors somewhere.

Even with a thousand decimal places, because errors grow exponentially the answers will be wrong at some point.

You need exponential accuracy to get a linear increase in approximation time.

10 times to get one unit time, 100 times to get 2 more units of time, 1000 times to get 3 more units of time...etc.

So far we have emphasized the unpredictable and disorderly side of chaos.

Chaos also has a secret order hidden within and the patterns will amaze you.

Sir James Lighthill

We thought Newtons laws gave us total predictability.

Why were scientists so slow to recognize the butterfly effect.

Distasteful because threatening to science itself.

Sciences need to put systems in a box.

Scientists hate "everything effects everything else.." its a matter of degree.

The things you thought were negligible can have big effects.

Butterfly effect does not apply to

Does not occur in systems gently relaxing to some equilibrium state (particle rolling down a curved surface).

Two metronomes oscillating and flick it... errors grow linearly.

Errors do not snowball

Tides very predictable - huge fluid mechanical system. Weather is too... yet tides predictable.

One is periodic, and one is not.

Halleys Comet

Timing of Eclipses

All of these things very periodic which means they are predictable.

Tiny errors do not mushroom

**It is Not Determinism, it is Determinism PLUS Periodicity that gives predictability.**It is by putting Non-periodicity into the weather that created chaos and the butterfly effect.

Two double pendulums initially are in sync, but then become unpredictable.

*****Horizon of predictability**is a phrase the Lighthill introduced.It is the TIME it takes for these tiny errors to double in size.

What is the length of the Horizon of predictability.

Predictibility Horizon demo.. 3 time units yellow and blue not the same (Lorenz computer animate)

Decorrelate and become totally different.

Solar System is Chaotic

But 5 million year Horizon of Predictability.

The solar system is chaotic but predictable because 5 million years is so long on the timescale of a human life.

You cannot predict beyond the predictability Horizon no matter how good your instruments are.

Trying to figure out the position of the planets 4 billion years ago is meaningless.

What if my instruments give me no error, but are perfect.

You cannot do that because it would require an infinite number of decimal places.

There is always some errors somewhere.

Even with a thousand decimal places, because errors grow exponentially the answers will be wrong at some point.

You need exponential accuracy to get a linear increase in approximation time.

10 times to get one unit time, 100 times to get 2 more units of time, 1000 times to get 3 more units of time...etc.

So far we have emphasized the unpredictable and disorderly side of chaos.

Chaos also has a secret order hidden within and the patterns will amaze you.