(1) Department of Theoretical & Applied Mechanics, Cornell University, Ithaca, NY, 14853, U.S.A.;(2) Department of Meteorology and GFDI, Florida State University, Tallahassee, FL, 32306, U.S.A.
Abstract:
We investigate the dynamics of a system consisting of a simple harmonic oscillator with small nonlinearity, small damping and small parametric forcing in the neighborhood of 2:1 resonance. We assume that the unforced system exhibits the birth of a stable limit cycle as the damping changes sign from positive to negative (a supercritical Hopf bifurcation). Using perturbation methods and numerical integration, we investigate the changes which occur in long-time behavior as the damping parameter is varied. We show that for large positive damping, the origin is stable, whereas for large negative damping a quasi-periodic behavior occurs. These two steady states are connected by a complicated series of bifurcations which occur as the damping is varied.
