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Periodic bouncing modes for two uniformly magnetized spheres. II. Scaling
Boyd F Edwards1, Bo A Johnson1, John M Edwards2
1Department of Physics, Utah State University, Logan, Utah 84322, USA.
Chaos (Woodbury, N.Y.)
|February 5, 2020
Summary
This study models two magnetized spheres colliding elastically. Numerical simulations reveal scaling laws for energy and period, unifying small and large amplitude bouncing modes.
Area of Science:
- Physics
- Magnetohydrodynamics
- Nonlinear Dynamics
Background:
- Investigates the complex dynamics of interacting magnetized spheres.
- Focuses on elastic collisions and nonlinear bouncing modes in a frictionless system.
Purpose of the Study:
- To numerically simulate and analyze the threshold energies and periods of nonlinear bouncing modes.
- To identify universal scaling parameters governing the system's behavior across different amplitudes.
- To reconcile small-amplitude analytical results with large-amplitude numerical findings.
Main Methods:
- Employs numerical simulations to model the motion and collisions of two identical, uniformly magnetized spheres.
- Utilizes closed-form mathematical results for small-amplitude regimes to inform large-amplitude analysis.
- Performs data fitting to determine scaling exponents and analyze discontinuities.
Main Results:
- Numerical simulations of threshold energies and periods align with small-amplitude analytical results.
- Identifies scaling parameters (bouncing number, rocking number, phase) that govern the entire amplitude range.
- Discontinuities in scaling functions for energy and period separately are resolved when energy is plotted against period.
Conclusions:
- A unified scaling framework is established for nonlinear bouncing modes of magnetized spheres.
- Power-law scaling is observed at large amplitudes, with analytical exponents derived for energy versus period.
- The study demonstrates universal scaling behavior for both in-phase and out-of-phase collision modes.
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