Structure & Strangeness


Title: Chaos You Can Play In
Authors: Aaron Clauset, Nicky Grigg, May Lim, Erin Miller

Real world instances of complex systems often exhibit complex dynamical behavior that is not well characterized as either periodic or random. Indeed, many such systems exhibit chaotic or quasi-periodic behavior. Systems which exhibit chaotic dynamics may be broken into mathematical models, many of which like the logistic map or the Lorenz equations are well understood, and real world systems. Chaos in real world systems is significantly more difficult to both model mathematically and explore experimentally. This is due not only to the inherent variability which chaotic behavior creates, but also the inherent difficulties in acquiring good time series measurements. Noise and low sampling rates present problems, as does the fact that there may only be a single stream of data that can be measured in a real system, while in a mathematical model, all variables of interest can be analysed.

Willem Malkus and Lou Howard Strogatz in the 1970s at MIT improvised the waterwheel, a mechanical analogue of the Lorenz equations. Surprisingly, the waterwheel has remained a largely unexplored system, with what little work has been done focusing on simplifications of the mathematical model. The beauty of the waterwheel is in its simplicity. Water is poured into the system at a steady rate from the top of the tilted wheel. Each cup has a hole drilled in the bottom which allows water to leak out of the system. Some damping is introduced into the rotation of the wheel. By varying only two parameters, the inflow rate of water and friction applied to the wheel, one can cause the wheel to exhibit simple periodic behavior (either unidirectional behavior where the wheel rotates continuously in one direction, or bi-directional behavior in which the wheel reverses direction periodically) or unpredictable transitions between these two simple behaviors. In this paper we describe an experimental and modeling study of the Malkus waterwheel system.

"Chaos You Can Play In" published in the SFI CSSS 2003 proceedings

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