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Making Chaos at Home

Geodesic flow on (negatively curved) surfaces are basic examples of chaotic flows in ergodic theorey. However, it is not at all easy to visualize them and their dynamics. However, it is possible to construct simple mechanical models (linkages) whose behaviour is governed by the same equations of motion as those of a geodesic flow on a suitable surface. Moreover these models can easily be easily constructed using simple household materials. The instructions are here.

Model

The configurations of the system are determined by the positions of the three (black) weights. The totality of all such positions determines a two dimensional surface sitting inside, say, a six dimensional space given by the three pairs of coordinates of the weights. To introduce some dynamics, consider giving the system a gentle push and then the configuration evolves with time in such a way as to correspond to moving along a geodesic on this embedded surface. Since with suitable choices (on the lengths of the straws, say) the curvature of the surface can be made predominantly negative, the geodesic flow can be made chaotic.

The movie of this chaotic behaviour is here.