Santa Fe Institute Complex Systems Summer School 2003

Summer school activities Lectures on ‘foundation’ topics: nonlinear dynamics, information theory, statistical mechanics, computational mechanics, agent-based modelling, adaptive computation Lectures on specific application areas: RNA folding, economic game theory, emergent engineering Experimental laboratory

A Belmonte et al (2001) Physical Review Letters, 87, A. Belmonte et al (1997) Journal de Physique II 7, Belousov-Zhabotinsky Reaction Motion of a shaken hanging chain

Faraday experiment Foam coarsening

Chaos you can play in: the Malkus Waterwheel Aaron Clauset, Nicky Grigg, May Tan Lim, Erin Miller Santa Fe Institute Complex Systems Summer School June 2003

Lorenz equations

Periodic and strange attractors

Malkus waterwheel

Equations of Motion Mass change in each cup: Torque balance for wheel: Angle change for each cup:

Simulated mass time series

Angular velocity Lorenz equations time series Waterwheel equations time series

Model-data comparison

Phase space reconstruction Delay coordinate embedding requires a delay time (  ) and an embedding dimension (d E ) Delay time from first minimum in average mutual information function Embedding dimension from false nearest neighbours analysis

Reconstructed waterwheel attractors (simulation data) Reconstructed Lorenz attractors

Reconstructed attractors from model and measured time series

Acknowledgments Andrew Belmonte, Department of Mathematics, Pennsylvania State University Ray Goldstein, Physics Department, University of Arizona