COUPLED OSCILLATORS NEAR RESONANCE

Lance Eric Arsenault, Ph.D.
Department of Physics
University of Illinois at Urbana-Champaign, 1996

Advisor: Alfred Hubler

Committee on Final Examination:

E. Atlee Jackson (Chairperson), Bob Eisenstein, Tony M. Liss, Paul T. Debevec

thesis abstract

We study the dynamics of two conservative librating oscillators with perturbations from a linear displacement coupling and non-Hamiltonian forces such as damping. We examine the dynamics of these systems when they are near a resonance using secular perturbation theory. We show that near resonance a large class of driven oscillators and two coupled oscillators can be transformed to the same ordinary differential equations (ODEs). We consider two types of resonances: accidental and intrinsic. For an accidental resonance, we find that the dynamics near a resonance is a generalization of the standard Hamiltonian dynamics of two coupled conservative oscillators, which we call the standard equation. For an intrinsic resonance, we show that a primary resonance island can fill all of the available phase space. We derive expressions for the parameters in these ODEs. From a fixed-point analysis of these ODEs, we show that hard oscillators lock in-phase and soft oscillators lock out-of-phase. We develop a novel method for calculating accurate response curves for driven strongly nonlinear oscillators, where no existing method can give accurate results. We present a method for finding the steady state frequency of two coupled oscillators. We compare our theoretical predictions with computer simulations of many examples including: a sinusoidally driven highly nonlinear Duffing oscillator, and two coupled van der Pol oscillators with a highly nonlinear Duffing force.

download thesis as compressed postscript file: Note that some figures are incomplete in this version. At the time I wrote this I had to complete some of the figures using paper. I didn't have a good figure editor program.