Spring in Field of Two Magnets
This Demonstration shows the solution for the following system. An iron mass is constrained to slide on the axis. The mass is attached to a spring anchored above the axis and subject to an external force with amplitude (red arrow), conforming to Hooke's law. The mass is also attracted to two long bar magnets with their poles shown as orange squares, placed at a distance below the axis. To a reasonable approximation, each magnet attracts the mass with a force proportional to the inverse square of its distance to the pole. Different types of qualitative behavior, including chaotic motions, can be seen in the phase portraits.
The equation of motion is
 P. Blanchard, R. L. Devaney, and G. R. Hall, Differential Equations, 4th ed., Boston: Brooks/Cole, 2012, pp. 452–453.