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Lesson 10: Chaos TheoryLesson 10: Chaos TheoryDr. Michael J. PiersonDr. Michael J. Pierson

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All systems can be All systems can be influenced by chaos.influenced by chaos.

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Chaos

SystemSystem

The Influence Of Chaos On A SystemThe Influence Of Chaos On A System

Leadership

Input Process Output

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Traditional Definition Of ChaosTraditional Definition Of Chaos

A system whose behavior is A system whose behavior is unpredictable & out-of-control.unpredictable & out-of-control.

It produces uncertainty, change & It produces uncertainty, change & crisis.crisis.

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What does chaos look What does chaos look like?like?

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When One Variable Is Plotted, When One Variable Is Plotted, Chaos Looks Like...Chaos Looks Like...

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The fluctuations of this systemThe fluctuations of this system look chaotic-totally unpredictable!look chaotic-totally unpredictable!

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We Need A New Definition Of We Need A New Definition Of ChaosChaos

What happens when we observe the What happens when we observe the same chaotic system three-same chaotic system three-dimensionally over time?dimensionally over time?

We see a pattern or order that emerges.We see a pattern or order that emerges. The order is without predictability.The order is without predictability. The available order is not through the The available order is not through the

control of people, processes, or events.control of people, processes, or events.

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When Three Variables Are Plotted When Three Variables Are Plotted Concurrently,Chaos Looks Like...Concurrently,Chaos Looks Like...

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pattern emerge.

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You have watched the history of a You have watched the history of a system emerge on a computer screen.system emerge on a computer screen.

Initially, the system moves violently & Initially, the system moves violently & unpredictability.unpredictability.

Later, the chaotic movements form a Later, the chaotic movements form a pattern & order emerges.pattern & order emerges.

The shape of this order is called a The shape of this order is called a strange attractor or Lorenze Butterfly.strange attractor or Lorenze Butterfly.

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Chaos Is The Natural Route To Chaos Is The Natural Route To Order Order

Chaos is a necessary stage to order.Chaos is a necessary stage to order. All natural & man-made systems have All natural & man-made systems have

an innate ability to reorganize & an innate ability to reorganize & establish order.establish order.

Structures can self-regulate processes Structures can self-regulate processes in such a way that the structure can in such a way that the structure can renew & survive.renew & survive.

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We try to build systems We try to build systems that are devoid of chaos. that are devoid of chaos. They are closed systems.They are closed systems.

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This Is The Result!This Is The Result!

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Task 1Task 1 Task 2Task 2

Task 3Task 3Equilibrium

A closed A closed system does system does something something

withoutwithoutdisturbingdisturbingequilibriumequilibrium

or changing.or changing.

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Systems that are open to Systems that are open to chaos are different!chaos are different!

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Or the systemOr the systemcan reorganizecan reorganize

How Does Chaos Impact An Open How Does Chaos Impact An Open System?System?

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System wantsSystem wantsequilibriumequilibriumChaosChaos

intrudesintrudes

Bifurcationoccurs

System canexperience

death (entropy)

Order Order re-establishedre-established

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Nature has self-Nature has self-organizing structures. organizing structures.

Clouds can change Clouds can change instantly with the input of instantly with the input of

atmospheric energy.atmospheric energy.

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They can go from this...They can go from this...

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to a hurricane!to a hurricane!

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Nature also has repetitive Nature also has repetitive patterns of order in patterns of order in

fractals. Fractal objects fractals. Fractal objects repeat a pattern at ever repeat a pattern at ever smaller levels of scale.smaller levels of scale.

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The Fractal Nature Of FernsThe Fractal Nature Of Ferns

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Another example is a Julia Set. Another example is a Julia Set. These fractal images These fractal images

represent a simple non-linear represent a simple non-linear formula (formula (ZZn+1n+1=Z=Znn

22+c+c) over ) over

millions of iterations. Color millions of iterations. Color values were set to correspond values were set to correspond

to different numeric values to different numeric values created by the equation.created by the equation.

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View The Following Julia Set Fractal View The Following Julia Set Fractal Images At Various Levels Of Images At Various Levels Of

Magnification.Magnification.

Initial FractalInitial Fractal256 X256 XI Million XI Million X1 Billion X1 Billion X40 Billion X1 Trillion X

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Another example of Another example of nature’s inherent order is nature’s inherent order is

the Fibonacci Ratio.the Fibonacci Ratio.

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Leonardo Pisano Fibonacci Leonardo Pisano Fibonacci

Circa 1175-1240.Circa 1175-1240. Grew up in North Grew up in North

Africa.Africa. Father was customs Father was customs

official.official. Well-educated.Well-educated. Published Published Liber Abaci.

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Fibonacci never really studied the Fibonacci never really studied the sequence bearing his name.sequence bearing his name.

The Fibonacci sequence was simply an The Fibonacci sequence was simply an answer to a question posed in one of answer to a question posed in one of his books.his books.

Edward Lucas gave the sequence Edward Lucas gave the sequence Fibonacci’s name after studying the Fibonacci’s name after studying the sequence.sequence.

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Fibonacci’s Rabbit ProblemFibonacci’s Rabbit Problem

There is one pair of rabbits in an enclosure on There is one pair of rabbits in an enclosure on January 1.January 1.

This pair produces another pair on February 1 This pair produces another pair on February 1 and on the first day of every month thereafter.and on the first day of every month thereafter.

Each new pair matures for one month and then Each new pair matures for one month and then produces a new pair on the first day of the produces a new pair on the first day of the third month of its life and every month third month of its life and every month thereafter.thereafter.

How many rabbits are in the enclosure on How many rabbits are in the enclosure on January 1 of the following year?January 1 of the following year?

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What Did Edward Lucas Find?What Did Edward Lucas Find?

While studying the breeding of rabbits, While studying the breeding of rabbits, he discovered that a definite he discovered that a definite progression appeared if one lists the progression appeared if one lists the number of pair rabbits present at the number of pair rabbits present at the end of each month (1-2-3-5-8-13-21-34-end of each month (1-2-3-5-8-13-21-34-55-89-144-233).55-89-144-233).

Each number was the sum of the two Each number was the sum of the two preceding numbers (1+2=3, 2+3=5, preceding numbers (1+2=3, 2+3=5, 3+5=8, 5+8=13, etc.).3+5=8, 5+8=13, etc.).

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If you divide any number in the series If you divide any number in the series by the next number you always get a by the next number you always get a number that is very close to .62 number that is very close to .62 (.618034).(.618034).

This number has been called the This number has been called the Golden Ratio.Golden Ratio.

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Golden Ratio In ArtGolden Ratio In Art

Fechner and Wundt found that people Fechner and Wundt found that people unconsciously favor “Golden Mean unconsciously favor “Golden Mean dimensions” when selecting rectangular dimensions” when selecting rectangular objects.objects.

The Golden Ratio dominates The Golden Ratio dominates Renaissance sculptures, paintings, and Renaissance sculptures, paintings, and architecture.architecture.

Leonardo Da Vinci used it in many of his Leonardo Da Vinci used it in many of his works.works.

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St. JeromeSt. Jerome By Da Vinci By Da Vinci

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BathersBathers By Seurat By Seurat

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Architecture & The Golden RatioArchitecture & The Golden Ratio

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The Parthenon

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Now Let’s Divide A Rectangle With Now Let’s Divide A Rectangle With This Ratio This Ratio

First, divide the rectangle so First, divide the rectangle so you have a perfect square “A”.you have a perfect square “A”.

AA

““B”, the remainder has that sameB”, the remainder has that samepleasing ratio (.618034).pleasing ratio (.618034).BB

If you keep dividing the rectanglesIf you keep dividing the rectanglesyou will continue to get the same you will continue to get the same ratio.ratio.

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If You Connect All Of The Squares’ If You Connect All Of The Squares’ Centers You Will Form A SpiralCenters You Will Form A Spiral

This spiral is called Hambridge’s This spiral is called Hambridge’s Whirling Squares. Any section Whirling Squares. Any section of the spiral is .618034 as large of the spiral is .618034 as large as the remainder of the spiral.as the remainder of the spiral.

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Nearly every living & Nearly every living & many non-living creatures many non-living creatures fit this spiral that contains fit this spiral that contains

the Fibonacci Ratio.the Fibonacci Ratio.

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Some ExamplesSome Examples

Spiral arm of the galaxies in space.Spiral arm of the galaxies in space. Curve of comet tails.Curve of comet tails. Curve of the surf.Curve of the surf. Curve of snail shells.Curve of snail shells. Curve of elephant tusks.Curve of elephant tusks. Curve of the sabre-toothed tiger’s tooth.Curve of the sabre-toothed tiger’s tooth. Curve of the ram’s horn.Curve of the ram’s horn.

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Curve of the roots of human teeth.Curve of the roots of human teeth. Curve of a spider’s web.Curve of a spider’s web. Curve of bacteria growth.Curve of bacteria growth. Curve of a parrot's beak.Curve of a parrot's beak. Curve of the inner ears spiral in Curve of the inner ears spiral in

mammals.mammals.

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Example Of Phyllotaxis Example Of Phyllotaxis (arrangement of leaves on a stem)(arrangement of leaves on a stem)

leaf 5leaf 4

leaf 3

leaf 2

leaf 0leaf 1 leaf 1

leaf 5

leaf 7

leaf 6

leaf 4 leaf 3

leaf 2

leaf 0

leaf 8

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2 Turns2 TurnsRatio: 2/5Ratio: 2/5

3 TurnsRatio: 3/8

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Fibonacci Spirals In A Pine ConeFibonacci Spirals In A Pine Cone

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2

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5 6

7

8

11

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7 8

3 9

10

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111213

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Fibonacci sequence is a Fibonacci sequence is a fascinating tendency that fascinating tendency that occurs far too often to be occurs far too often to be

discounted as chance.discounted as chance.Nature has order? Nature has order?

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The Bottom LineThe Bottom Line

Nature has orderly patterns, but we Nature has orderly patterns, but we cannot define order as the lack of change.cannot define order as the lack of change.

The permanent state of a system is The permanent state of a system is disequilibrium resulting from chaos.disequilibrium resulting from chaos.

The temporary state of a system is The temporary state of a system is equilibrium (stasis).equilibrium (stasis).

Chaos has shape & will never exceed the Chaos has shape & will never exceed the bounds of its strange attractor.bounds of its strange attractor.

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What we once thought was chaos is What we once thought was chaos is non-chaotic.non-chaotic.

Order comes from chaos.Order comes from chaos. You can never predict where a system You can never predict where a system

is headed until you have observed it is headed until you have observed it over time.over time.

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Leaders Must Change Their Leaders Must Change Their Approach To ChaosApproach To Chaos

There is order in chaos.There is order in chaos. We can’t reorganize & reorder a system We can’t reorganize & reorder a system

without going through chaos.without going through chaos. Normally, we rush to closure to get Normally, we rush to closure to get

something done & try to minimize the something done & try to minimize the impact of chaos.impact of chaos.

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If we want break-through thinking, we If we want break-through thinking, we need people to be overwhelmed & need people to be overwhelmed & confused- not forever, but at key times.confused- not forever, but at key times.

We need to develop open rather than We need to develop open rather than closed systems.closed systems.

We must work with chaos & not avoid We must work with chaos & not avoid it.it.

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What Have You Learned?What Have You Learned?

1. Strange attractors pull a system into an 1. Strange attractors pull a system into an orderly state.orderly state.

2. Effective leadership allows chaos to 2. Effective leadership allows chaos to invade a structure.invade a structure.

3. The temporary state of a system is 3. The temporary state of a system is chaos.chaos.

4. Fractals are examples of natural 4. Fractals are examples of natural repetitive patterns.repetitive patterns.

DirectionsDirections: Read each question & click on “T” (true) or “F” (false).: Read each question & click on “T” (true) or “F” (false).

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What Have You Learned?What Have You Learned?

5. The Fibonacci Ratio is an example of an 5. The Fibonacci Ratio is an example of an orderly world. orderly world.

6. Hambridge’s Whirling Squares forms a 6. Hambridge’s Whirling Squares forms a circle.circle.

DirectionsDirections: Read each question & click on “T” (true) or “F” (false).: Read each question & click on “T” (true) or “F” (false).

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AssignmentsAssignments

Use PowerPoint to develop a concept Use PowerPoint to develop a concept map of Lesson 10 and e-mail it to your map of Lesson 10 and e-mail it to your professor.professor.

Read chapter 7, "Chaos and the Strange Read chapter 7, "Chaos and the Strange Attractor of Meaning", in Margaret Attractor of Meaning", in Margaret Wheatley's Wheatley's Leadership and the New Leadership and the New Science.Science.

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