What was that?

The Mandelbrot Set is a beautiful example of Chaos Theory.

Chaos theory is a discipline of pure mathematics that has many applications in medicine, physics,

engineering , biology, psychology etc.

Chaotic behavior can be witnessed in nature.

Mathematicians attempt to explain such behavior through rigid analysis of mathematical models.

More specifically, Chaos theory studies the behaviors of dynamical systems.

These “dynamical systems” exhibit sensitivity to initial conditions .

This sensitivity is commonly referred to as the butterfly effect.

When a butterfly flaps its wings in one part of the world

it can cause a hurricane  in another part of the world.  

Author Unknown

Coined the term “Butterfly Effect”

Discovered that even a tiny alteration in initial conditions (.506127 to .506), can transform a long term forecast.

Butterfly effect is based on his 1972 paper “ Predictability: Does the flap of a butterfly’s wings in Brazil set off a tornado in Texas?

EDWARD LORENZ

Contributions 1917-2008 Mathematician & Meteorologist

It is just the concept that small events can have large, widespread

consequences.

Is the “Butterfly Effect” REAL?

Yes or No?

YES!The event that a butterfly flaps its wings, represents a small change in the initial condition of the system. This small change causes a chain of events that lead to huge large-scale alterations.

In other words, If the butterfly decided NOT TO flap its wings, the end result may have been different!

RECAP: Chaotic behavior can be witnessed in nature.

Mathematicians attempt to explain such behavior through rigid analysis

of mathematical models.

Let’s take a look at one such model….

- By Rivkah Singh

COLONIAL GROWTH OF SINGLE CELL ORGANISMS

FACTIF A METEORITE CARRYING AN ALIEN COLONY OF SINGLE

CELL ORGANISMS CRASHES ON EARTH, IT WILL ONLY TAKE 63.1 DAYS TO COVER THE ENTIRE SURFACE OF THE UNITED

STATES.

Assuming that the 48 states have an area of 1019 square mm and in n days the area of the mold doubles to 2n times its starting area.

QUESTIONHow long will it take for the

organisms to inhabit the entire planet?

Assume the total area of the surface of the earth is 5 x 1020 mm2 , and the organisms double in size every day . Or rather, in n days they have grown to 2n times their beginning area.

Let’s assume that the mold colonybegins as a one-square mm blob and it’s area doubles in size every day. Since we know that the entire surface of the earth is 5 x 1020 mm2,  Log(2n ) = log (5 x 1020 )N x log 2 = log 5 + 20 log 10N = (log5 + 20log10)/log2= 0.6989 + 20 Thus, n = = 68.7 

Therefore, it will take a mere 68.7 days for the mold to inhabit the entire surface of theEarth!

However, Are we certain that the world is doomed in 68.7 days?

IF OUR ALIEN MOLD IS HARMFUL TO HUMAN LIFE, IT WOULD APPEAR THAT WE ARE

DOOMED.

WHY?

ARE YOU SURE?

Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions, an effect which is

popularly referred to as the butterfly effect.

Butterfly effect- In chaos theory, the butterfly effect is the sensitive dependence on initial conditions, where a small

change at one place in a deterministic nonlinear system can result in large differences to a later state..

LET’S TAKE A LOOK AT SOME BASIC WIKIPEDIA DEFINITIONS

IN OTHER WORDS,

IF OUR SYSTEM IS BOTH NON-LINEAR AND DYNAMIC, WE CAN STUDY IT EVEN FURTHER………………………………………………….

Days Area63.1

64.1 2 x 65.1 4 x 66.1 8 x 67.1 16 x 68.1 32 x 69.1 64 x

Our alien colony exhibits exponential growth.

Dynamical systems have the following requirements:

• 1. The state of the system must be defined.

• 2. It must include a set rule of change. This rule is referred to as a dynamic. (by dynamic it is inferred that it includes all sorts of deterministic change)

Single Cell Organisms have the following characteristics:

• 1.They reproduce within regular intervals. These intervals are commonly refereed to as generations.

• 2.They exhibit fecundity. Fecundity are the fixed rates of birth and death

SINGLE CELL COLONIAL GROWTH IS AN EXAMPLE OF A DYNAMICAL SYSTEM

OK, SO OUR ALIEN SYSTEM IS BOTH NONLINEAR AND DYNAMICAL….. NOW WHAT?

NOW WE FIND A FORMULA……

Since we know that the colonial growth of single cell organisms is an example of a dynamical system we can

set up a formula

In order to set up a formula, we must first decide on the variable of the populace after n generations and the fecundityWhere:Pn = the quantity of cells inherent within the population after the nth generationF = fecundity : f= 1 +( birthrate) – (death rate)

ThusPn+1 = f * Pn

Is our formula that will determine the size of our populace for every generation

WITH A FORMULA, WE CAN MAKE A MATHEMATICAL MODEL.

THE BEST WAY TO DEVELOP A MATHEMATICAL MODEL IS WITH THE USE OF COMPUTERS

THIS MODEL WAS DEVELOPED WITH SOFTWARE CALLED GEOGEBRA

GeoGebra Tube

Pn+1 = *Pn is equivalent to the equation of a straight line y=a*x. This line is called the compounding line.The addition of two lines f(x) and g(x)(the 45 degree line) enable us to find the graphical iteration more easily.

SINGLE CELL GROWTH EXHIBITING EXPONENTIAL GROWTH, DECAY, AND

BIFURCATION

Time series for periodic cycling

ARE WE DOOMED YET ?

Are We Doomed Yet?

The tent map is the simplest dynamical system.

Function[If[x ≤ 0.5, a x, a - a x], 0, 1

How is the Tent Map related to our example of colonial growth?

It is related because if we assume our population is governed by the equation:

Xn+1 = Xn + b*Xn – d*XnWhere b = births, d = deaths, and n=generation.Then after time, when space is limited it starts to control the birth rate.

Xn+1 =Xn+b*(1-Xn) – d*XnHere 1 = the total space or the universal set and Xn is the space already occupied by the previous generation. Thus, 1-Xn is the fraction of space still available.So if we measure time in units of 1/d (rescale time so that d=1). We have our tent.Which is:

Xn+1=a*Xn If Xn is less than 0.5Xn+1=a*(1-Xn) otherwise

When a <1, 0 is an attractor.

An example of a few iterations when a>1

a=1.99

Chaos

A=1.6

a = 1.2

The a values between 1 and 2 never exactly repeat. Hence, there is deterministic chaos..

We can therefore conclude that since our time series acts erratically and chaotic. It exhibits deterministic chaos. Hence, we are not necessarily “doomed” .

We are saved by the knowledge that small differences in our initial conditions (however minute) give us diverging outcomes; therefore, prediction as precise as ours in the beginning of this presentation is nearly impossible. We can safely assume that even if our system is deterministic – it does not automatically make it predictable.