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A uniformly magnetized sphere moves without friction in a plane in response to the field of a second, identical, fixed sphere and makes elastic hard-sphere collisions with this sphere. Numerical simulations of the threshold energies and periods of periodic finite-amplitude nonlinear bouncing modes agree with small-amplitude closed-form mathematical results, which are used to identify scaling parameters that govern the entire amplitude range, including power-law scaling at large amplitudes. Scaling parameters are combinations of the bouncing number, the rocking number, the phase, and numerical factors. Discontinuities in the scaling functions are found when viewing the threshold energy and period as separate functions of the scaling parameters, for which large-amplitude scaling exponents are obtained from fits to the data. These discontinuities disappear when the threshold energy is viewed as a function of the threshold period, for which the large-amplitude scaling exponent is obtained analytically and for which scaling applies to both in-phase and out-of-phase modes.

The purpose of this work is to investigate the scaling relationships between the threshold energy, the threshold period, the bouncing number, the rocking number, and the phase of 1497 periodic modes found previously for the motion of a uniformly magnetized sphere subject to the field of a second, identical, fixed sphere. This large dataset offers the opportunity to identify scaling relationships to high precision for this highly nonlinear problem. Such scaling relationships recall techniques used in studying phase transitions and fractals and invite the search for universal scaling laws that may also apply to other systems. This work is motivated by our interest in the properties of collections of small neodymium magnet spheres that are used to create beautiful magnetic sculptures and are used both in and out of the classroom to teach principles of mathematics, physics, chemistry, biology, and engineering.

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