Lynn S. FichterDept Geology/Environmental ScienceJames Madison University

1410hAN: ED23C-03

1. Complex systems are not just simple 1. Complex systems are not just simple systems with a lot more parts.systems with a lot more parts.

PremisesPremises

2. Complex systems have their own 2. Complex systems have their own properties, behaviors, and terminology.properties, behaviors, and terminology.

3. Our students enter our classes with 3. Our students enter our classes with virtually no familiarity with these ideas.virtually no familiarity with these ideas.

Therefore, we must build these Therefore, we must build these concepts for our students from the concepts for our students from the bottom-up – just like for any other new bottom-up – just like for any other new subject.subject.

Bifurcation Bifurcation

Self-similarity Self-similarity

Fractal Fractal

Agents Agents

Self-Self-organized organized criticalitycriticality

AvalanchesAvalanches

Power LawsPower Laws

Strange Strange attractors attractors EmergenceEmergence

The Language of The Language of Complexity Complexity

We can introduce the basic We can introduce the basic concepts in 3 - 5 one-hour concepts in 3 - 5 one-hour classes . . . classes . . .

Teaching Timing ?Teaching Timing ?

. . . I use them in at least 5 different . . . I use them in at least 5 different classes, classes, . . . and, can pull out and develop . . . and, can pull out and develop

specific concepts for specific specific concepts for specific purposes . . . purposes . . .

. . . depending on the depth we want, . . . depending on the depth we want,

All the programs and All the programs and supporting materials are supporting materials are

available on line.available on line.

jmu.edu/geology/ComplexEvolutionarySystems/jmu.edu/geology/ComplexEvolutionarySystems/

Learning Outcomes for understanding chaos theory.

ChaosTheory

Studies why and how the behavior of simple systems—simple algorithms—becomes more complex and unpredictable as the

energy/information the system dissipates increases.Xnext = rX (1-X)

System evolves to equilibrium

The logistic system

A random sampling of logistic curves pulled from Google\images\

logistic curve

ChaosTheory

Studies why and how the behavior of simple systems—simple algorithms—becomes more

complex and unpredictable as the energy/information the system dissipates

increases.Xnext = rX (1-X)

System evolves to equilibrium System evolves to complexity

The logistic system

But, if we push the system

harder

Its behavior evolves into

this.

X next = rX (1-X)

Valueof X

(Populatonsize)

1.0

0.0

0.5

Number of Equation Iterations

r = 2.7

.02.05

.13

.35

.58

.65

.60

.64

.61

.62

X = .02 and r = 2.7 X next = rX (1-X)

X next = (2.7) (.02) (1-.02 = .98)

X next = .0529

Iteration X Value

0 0.0200000

1 0.0529200

2 0.1353226

3 0.3159280

4 0.5835173

5 0.6561671

6 0.6091519

7 0.6428318

8 0.6199175

9 0.6361734

10 0.6249333

11 0.6328575

12 0.6273420

13 0.6312168

14 0.6285118

15 0.6304087

16 0.6290826

17 0.6300117

18 0.6293618

44 0.6296296

45 0.6296296

46 0.6296296

47 0.6296296

48 0.6296296

49 0.6296296

50 0.6296296

.05

.13

.35

.58

.65

.60

.64

.61

.62

X = .02 and r = 2.7 X next = rX (1-X)

X next = (2.7) (.02) (1-.02 = .98)

X next = .0529

.62Equilibrium state

All these systems can be modeled in a All these systems can be modeled in a computer, in class, in real time.computer, in class, in real time.

20 generations

But, what about these irregularities?Are they just meaningless noise, or do they mean something?

r = 2.7

r = 2.9

r = 3.0

r = 3.1

r = 3.5

r = 3.7

r = 4.0

r = 4.1

r = 2.7r = 2.9

X = .629 X = .655

r Value

PopulationSize = X

This axis was a time series, but becomes . . .

Converting a Time Series Diagram into a Converting a Time Series Diagram into a Bifurcation DiagramBifurcation Diagram

r = 3.3 r = 3.5X = .48 & .82 X = .50, .87, .38, .82

split

split

split

r = 3.8

0.877682831619863 0.407951579058487 0.917802935168261 0.286674687986186 0.777070782765993 0.658280769082272 0.854799352927153 0.471646192817398 0.946945034149357 0.190912518507075 0.58696672938057

0.921259794327218 0.275652705596884 0.758739507677205 0.695604695234438 0.804607452168521 0.597414340316927 0.913939695942348 0.298884926867995 0.796300363964611 0.616383158394868

Bifurcation diagram showing behavior of system at all values

of r

2.5 2.9 3.0 3.3

GreatStability at

1 value

IncreasingInstability

Vibrating so hardIt flies apart

Return tostability, but with

2 stable points

L.P. 11 - Change is always accompanied by increasing instability

Properties of Complex Evolutionary Systems

Sensitive Dependenceon Initial Conditions:

Xnext

r = 4.000001

r = 4.000002

Universality

These two runs differ by a millionth of an ‘r’

Complex SystemsTheory

ChaosTheory

Is imbedded within . . .

Complex SystemsTheory

. . . studies how systems with many “agents” that are already at high

energy/information dissipation interact and behave.

ChaosTheory

Agent:the individual units that are

interacting, like . . .

Agents

Agents = units of friction along a

fault zone

Agents = sand

grains in a

migrating ripple

Complex SystemsTheory

Complex systems theory studies how systems with many “agents” that are already at high energy/information dissipation interact and behave.

ChaosTheory

How does complex system theory say the

agents behave?The central dogma is The central dogma is complex systems are complex systems are

Self-OrganizingSelf-Organizing

Self-Organized CriticalityCellular

Automata

BoidsOscillatin

gReactions

Cellular Automata and Self Organization

Survival Rules – 2/3 a live cell survives to the next generation if at least 2 but no more than three of the surrounding 8 cells are alive. Less than 2 and it dies of loneliness; more than 3 and it dies of over crowding.-Birth Rules – 3/3 a dead cells comes alive the next generation if 3, any 3, of the surrounding 8 cells are also alive.

Local Rules/Global Behavior

1 2 3

4

567

8

1 2 3

4

567

8 ?

jmu.edu/geology/ComplexEvolutionarySystems/