Zeeman's Catastrophe Machine in Flash by Daniel J. Cross
The control parameters of this system are given by the position of the green dot that starts on the -v-axis. Click and drag this dot around to see how the shape of the potential and corresponding equilibrium state changes.
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This flash animation is an interactive rendering of Zeeman's Catastrophe Machine (ZCM), which is a simple mechanical device that represents the cusp catastrophe. The setup consists of a wheel, free to rotate about its axis, with one end of two springs connected at a point on its rim. The other end of the first spring is fixed on the -u-axis, while the other end of the second spring is freely movable. The state of the system is the angle, theta, between the point of attachment of the two springs and the -u-axis, while the control parameters are the u and v coordinates of the free end of the second spring. The angle represents the equilibrium position of the wheel given by the minimum of the potential energy function which changes shape based on the values of the control parameters.
Outside the central star-shaped region the potential possesses an unique equilibrium state, and smooth changes in the control parameters result in a smooth change of that equilibrium state. Inside the star-shaped region, however, the potential develops a second minimum. It is then possible, upon exiting the star-shaped region, for the equilibrium state the system is in to disappear (just as the additional equilibrium state had appeared) forcing the system to undergo an abrupt change of state due to a smooth change of state variables! This discontinuous behavior is known as a catastrophe.
The potential energy of the system is given by
for a wheel diameter of 1, an equilibrium spring lengths of 1, a distance between the end of the top spring and the wheel center of 2, and where a denotes the distance of the wheel center from the origin. Now, in a neighborhood of the origin there exists diffeomorphisms x,y, and z and a smooth function w,such that, in that neighborhood, the potential has the functional form
where the function F is given by
which is called the cusp and is one of Thom's seven elementary catastrophes. Here, x is the state variable, while y and z are the control parameters.
The catastrophe surface of the cusp is given by the set of points (x,y,z) that are critical points of F for constant control parameters (that is, the extrema of the potential function) and is given by
which is the following surface in 3-space:
The castrophe set is the subset of the above surface which consists of those critical points that are degenerate, that is, where the second derivative also vanishes. These points define where local minima are created and destroyed and thus dictate the "interesting" behavior of the potential as control parameters vary. This set is given by
Finally, there is the bifuraction set, which is the projection of the catastrophe set onto the control parameters. It is given simply by
and its shape is the reason for calling this catastrophe the cusp.
The cusp is such an interesting catastrophe because it possesses the features of bimodality and hysteresis, that is, there are two equilibrium positions, but which state the system ends up in depends on the path taken through parameter space. What is also interesting to note is that the system is all the time governed by a smooth potential function. The catastrophe surface and set are also smooth. However, the projection of a smooth object down onto the plane (the bifurcation set) develops singularities which drive the interesting behavior of the system. This is a common theme throughout nonlinear dynamics.
Going back to the ZCM there are actually 4 connected cusps as suggested by the shape of the boundary of multiple-valued region, so that the behavior of this system is quite complicated (and interesting). However, the left and right cusps are actually dual cusps, so that there are two distinct maximum states rather than minimums. There is no interesting quasistatic behavior in a neighborhood of these points since there is always an unique minimum there. A rough sketch of the actual castrophe surface for the cusp is given below.
Finally, the ZCM also gives rise to interesting dynamical behavior when periodically driven, rather than moved quasistatically. In fact, the driven ZCM displays a period doubling route to chaos and a chaotic attractor based on the parameters of the driving term. A good discussion of this chaotic dynamical behavior can be found in the article Litherland, T. J. and Siahmakoun, A. Chaotic Behavior of the Zeeman Catastrophe Macine, Am. J. Phys. 63(5), May 1995, 426-431.
Daniel J. Cross, Ryan Michaluk, and R. Gilmore
R. Gilmore, Jean Marc Ginoux, Timothy Jones, C. Letellier, and U. S. Freitas
Daniel J. Cross and R. Gilmore