HMC

Magnetic Pendulum

A key concept in the theory of dynamical systems is that of "sensitive dependence on initial conditions". Intuitively, deterministic systems (e.g., equations based on Newton's laws) should possess the property that if the initial data is close, then the resulting solutions are close. Surprisingly for many systems this is just not true. This has important consequences in terms of predictive modeling, since there is intrinsically error in measuring initial data, how can one accurately predict what state the system will ultimately be in? The magnetic pendulum is a simple toy that demonstrates this. A metallic mass is attached at the end of a pendulum bob that can move in all directions. If you place several attracting magnets equidistant from the pendulum's equilibrium position and start the pendulum in a nonequilibrium position, where will it ultimately end up? Watch the movie below for a sample with 3 attracting magnets.

Assuming the pendulum mass evolves in a plane above the attracting magnets with attracting force proportional to the inverse of the distance squared from the magnet one can write down a system of differential equations for the motion of the mass. The figure below shows what happens to each initial state. That is, all green points represent initial states that are ultimately attracted to the mass at the center of the large green region, and likewise for the blue and red points. This amazing figure shows the fractal basin boundaries of attraction for this deterministic problem. Of course, this picture is for a specific set of parameter values (related to the damping and attracting force parameters).
Pretty amazing!