CMSC/BIOL 361: Emergence 2/5/08

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Presentation transcript:

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

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

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

Dependent Variable Population Size Independent Variable Time

Population Size Time

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

Pendulum Motion

Pendulum Motion

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

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

Fractal Dimension

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

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

Stochastic Phase Space Factors determining dimensionality are unpredictable High fractal dimension http://www.esrf.eu/UsersAndScience/Experiments/CRG/BM25/BeamLine/SourceCharacteristics