1. CMSC/BIOL 361: Emergence 2/5/08
Random versus Chaotic
CMSC/BIOL 361: Emergence
2/5/08

2. ∞ ? Mechanism? Chance Determinism Data x(t) t d(phase space set) 
d(phase space set)  low

3. Phase Space Phase: a stage in a process (state)
Phase Space: space (or diagram) in which all possible states are represented

4. Dependent Variable
Population Size
Independent Variable
Time

5. Population Size
Time

6. R1, C1
R2, C2
Prey Population Size
R3, C3
Predator Population Size

7. Pendulum Motion

8. Pendulum Motion

9. Dimension
Parameters required to describe the position of any object in space
Topological Dimension: 1-D, 2-D, 3-D
Minimum number of real parameters needed to describe all points in the system
Covering: a collection of subdivisions of the object whose union contains all possible points contained within the object
Refinement: further subdivisions of the object until each new subdivision is contained within exactly one covering
Smallest number of refinements required

12. Fractal Dimension A measure of how completely a fractal fills a space
Deviation from topological dimension

13. Fractal Dimension

14. Chaotic Phase Space Dimensionality determined by number of parameters
Low fractal dimension

15. R1, C1
R2, C2
Prey Population Size
R3, C3
Predator Population Size

16. Stochastic Phase Space
Factors determining dimensionality are unpredictable
High fractal dimension