Commercial Tom Kreitinger

MAT 305 – 9/28/2009

Can an insect create a tornado?

Edward Norton Lorenz& Chaos theory

Commercial Overview:

Part I - Who was Edward Norton Lorenz and what did he do?

Part II - What is Chaos theory and what are some examples?

Part I

Who was Edward Norton Lorenz and what did he do?

Edward Norton Lorenz1917- 2008

American Mathematician & Meteorologist

1948 - Postdoctoral work at MIT Department of Meteorology

1955 - Director of statistical weather forecasting (pioneered by his department)

1961 Computer Weather Models

Lorenz studied the relationships between temperature, pressure and wind speed using computer models.

One day he takes a short cut and restarts a program half way through its run by manually entering initial values.

1961 Computer Weather Models

Enters a value .506.506 for .506127 (from a printout)

The differences was only .000127 and was thought to be inconsequential.

He gets a cup of coffee while the program runs

Tiny initial change major impact

Lorenz finds the tiny difference of .000127 (.506.506 and .506127) generates a weather pattern nowhere near the original.

The end result: Two completely different weather results!

Lorenz Conclusions

Lorenz realized that a minute differences in the initial conditions - like a puff of wind could prove catastrophic to weather models.

Lorenz: “It implies that two states differing by imperceptible amounts may eventually evolve into two considerably different states, If, then, there is any error whatever in observing the present state – and in any real system such errors seem inevitable – an acceptable predication of the instantaneous state in the distant future may well be impossible.”

Butterfly Effect

1972 – Lorenz presents a paper at a conference in Washington, entitled “Does the Flap of a Butterfly's Wings in Brazil Set Off a Tornado in Texas?”

Lorenz coins the phrase ”Butterfly Effect” (Lorenz Attractor)

Lorenz – main idea

The “butterfly effect” shows how slightly different initial conditions can produce an actual result very different from previous.

Plinko – Try as you might dropping a puck produces a different result

Part II

What is Chaos theory and what are some examples?

Some Terms: System

Any entity that changes with time is called a ”system”

Time prevents everything from happening at once.

All systems have variables (e.g. temperature, pressure and wind speed)

Another example: population of fish

System Feedback

Feedback is a characteristic of any system in which the output or result affects the input of the system

Example: Feedback from a public address amplifier system

Robert May

fish population: x(next)=rx(1-x)

Periodic System

A variable in a periodic system exactly repeats its past behavior after the passage of a fixed interval of time

Example: A swinging pendulum is periodic and deterministic

Aperiodic System

Aperiodic behavior occurs when no variable affecting the state of the system undergoes a completely regular repetition of values

Example: Water as it goes down a drain

Unstable Aperiodic System

Unstable aperiodic behavior is highly complex. It never repeats itself and continues to show the effects of any small change to the system. This makes exact predications impossible and appear random

Double Pendulum

A Double Pendulum is a pendulum with an extra pivot joint added

Unstable Aperiodic Example

The motion of a double pendulum is seemingly random

A characteristic of chaos is that a deterministic system may appear to generate random behavior.

Chaos (definitions)

”The ability of simple models, without inbuilt random features, to generate highly irregular behavior”

”Apparently random recurrent behavior in a simple deterministic (clock-work-like) system”

”The qualitative study of unstable aperiodic behavior in deterministic nonlinear dynamic systems”

Summary

Edward Norton Lorenz's “butterfly effect” illustrates how slight changes in the initial conditions of a system can greatly affect outcome.

Chaos is the occurrence of aperiodic, apparently random events in a deterministic system

Sources

Book: “Introducing Chaos” - Ziauddin Sardar and Iwona Abrams ISBN 1-84046-078-4

Book: “50 mathematical ideas you really need to know” - Tony Crilly ISBN 1-84724-147-6

Double Pendulum - http://en.wikipedia.org/wiki/Double_pendulum

http://en.wikipedia.org/wiki/File:Double-compound-pendulum.gif

Chaos Theory - http://en.wikipedia.org/wiki/Chaos_theory

Edward Lorenz - http://en.wikipedia.org/wiki/Edward_Norton_Lorenz

Attractor - http://en.wikipedia.org/wiki/Attractor

Butterfly Effect - http://en.wikipedia.org/wiki/Butterfly_effect

The End

Can an insect create a tornado?

Thank you

Tom Kreitinger