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A chaotic situation arises in the experiment of swinging a magnetic pendulum over a plane containing two or more attractive magnets: which magnet the pendulum ultimately rests over is highly dependent on the starting position and velocity of the pendulum, so that small variations in initial conditions lead to different results.The situation with three magnets can be modeled with the following pair of differential equations:

,where the

xand_{i}yare the coordinates of the positions of the three magnets;_{i}x(t) andy(t) are the positions of the pendulum at timet;dis the distance between the pendulum and the plane containing the magnets; andRandCare friction and spring (self-restoring force) parameters.The situation can be modeled easily in

MathematicausingNDSolve(so I won't type in the code); in each of the following graphs, the pendulum is given a slightly different starting position (with identical initial velocity, etc.), and in each picture the pendulum ends up over a different magnet.

x(0) = -0.95,y(0) = 0.09:

x(0) = -0.95,y(0) = 0.095:

x(0) = -0.95,y(0) = 0.1:

(Sorry about the repulsive coloring scheme; I don't know what I was thinking. The red dots represent the three magnets, and the yellow squiggle is the path of the pendulum. If such things interest you, I arbitrarily chose

R= 0.2,C= 0.5,d= 0.25, andx'(0) =y'(0) = 0. For all I know, it will even interest you to know that the basins of attraction formed by the magnetic-pendulum model are very similar in character to the basins of attraction formed by using Newton's method to find complex roots to an equation.)Designed and rendered using

Mathematicaversion 3.0 for the Apple Macintosh.Equations pinched from Peitgen

et al, Chaos and Fractals, Springer-Verlag, 1992.

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