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Lynn S. Fichter Dept Geology/Environmental Science James Madison University 1410h AN: ED23C-03

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1. Complex systems are not just simple systems with a lot more parts. Premises 2. Complex systems have their own properties, behaviors, and terminology. 3. Our students enter our classes with virtually no familiarity with these ideas. Therefore, we must build these concepts for our students from the bottom-up – just like for any other new subject.

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Bifurcation Self-similarity Fractal Agents Self-organized criticality Avalanches Power Laws Strange attractors Emergence The Language of Complexity

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We can introduce the basic concepts in 3 - 5 one-hour classes... Teaching Timing ?... I use them in at least 5 different classes,... and, can pull out and develop specific concepts for specific purposes...... depending on the depth we want, All the programs and supporting materials are available on line.

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jmu.edu/geology/ComplexEvolutionarySystems/

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Learning Outcomes for understanding chaos theory.

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Studies why and how the behavior of simple systems— simple algorithms—becomes more complex and unpredictable as the energy/information the system dissipates increases. X next = rX (1-X) System evolves to equilibrium The logistic system

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

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Studies why and how the behavior of simple systems— simple algorithms—becomes more complex and unpredictable as the energy/information the system dissipates increases. X next = rX (1-X) System evolves to equilibriumSystem evolves to complexity The logistic system But, if we push the system harder Its behavior evolves into this.

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X next = rX (1-X) 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 =.0529.62 Equilibrium state All these systems can be modeled in a computer, in class, in real time.

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

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r = 2.7

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r = 2.9

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r = 3.0

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r = 3.1

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r = 3.5

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r = 3.7

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r = 4.0

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r = 4.1

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r = 2.7 r = 2.9 X =.629 X =.655 r Value Population Size = X This axis was a time series, but becomes... Converting a Time Series Diagram into a Bifurcation Diagram

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

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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

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Great Stability at 1 value Increasing Instability Vibrating so hard It flies apart Return to stability, but with 2 stable points L.P. 11 - Change is always accompanied by increasing instability

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Properties of Complex Evolutionary Systems Sensitive Dependence on Initial Conditions: X next r = 4.000001 r = 4.000002 Universality These two runs differ by a millionth of an ‘r’

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Is imbedded within...

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... studies how systems with many “agents” that are already at high energy/information dissipation interact and behave. Agent: the individual units that are interacting, like...

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Agents

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Agents = units of friction along a fault zone

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Agents = sand grains in a migrating ripple

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Complex systems theory studies how systems with many “agents” that are already at high energy/information dissipation interact and behave. How does complex system theory say the agents behave? The central dogma is complex systems are Self- Organizing Self-Organized Criticality Cellular Automata Boids Oscillating Reactions

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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 123 4 567 8 123 4 567 8 ?

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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 123 4 567 8 123 4 567 8 ?

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jmu.edu/geology/ComplexEvolutionarySystems/

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Self Organization. A naturally occurring group of interacting, interrelated, or interdependent elements, forming a complex whole, existing far from equilibrium,

Self Organization. A naturally occurring group of interacting, interrelated, or interdependent elements, forming a complex whole, existing far from equilibrium,

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