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Determing The Chaotic Nature Of Periodic Orbits, Bo Johnson
Determing The Chaotic Nature Of Periodic Orbits, Bo Johnson
Physics Capstone Projects
The determination of the long-term behavior of periodic orbits is considered. Different numerical techniques, including the Lyapunov Exponent, the Smaller Alignment Index, and the Generalized Alignment Index are used. Because of the discontinuous nature of the system under consideration, these methods are found to be insufficient and a more simplistic approach is utilized. The simplistic approach determines long-term behavior up to 500 periods of an orbit. It is found that in-phase periodic modes result in the largest amount of stable modes. Future work will look at the common characteristics of the in-phase modes to better understand why they are more …
Periodic Bouncing Modes For Two Uniformly Magnetized Spheres. I. Trajectories, Boyd F. Edwards, Bo A. Johnson, John M. Edwards
Periodic Bouncing Modes For Two Uniformly Magnetized Spheres. I. Trajectories, Boyd F. Edwards, Bo A. Johnson, John M. Edwards
All Physics Faculty Publications
We consider a uniformly magnetized sphere that moves without friction in a plane in response to the field of a second, identical, fixed sphere, making elastic hard-sphere collisions with this sphere. We seek periodic solutions to the associated nonlinear equations of motion. We find closed-form mathematical solutions for small-amplitude modes and use these to characterize and validate our large-amplitude modes, which we find numerically. Our Runge-Kutta integration approach allows us to find 1243 distinct periodic modes with the free sphere located initially at its stable equilibrium position. Each of these modes bifurcates from the finite-amplitude radial bouncing mode with infinitesimal-amplitude …
Periodic Bouncing Modes For Two Uniformly Magnetized Spheres. Ii. Scaling, Boyd F. Edwards, Bo A. Johnson, John M. Edwards
Periodic Bouncing Modes For Two Uniformly Magnetized Spheres. Ii. Scaling, Boyd F. Edwards, Bo A. Johnson, John M. Edwards
All Physics Faculty Publications
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 …