Splitting Frequencies for Resonant Solutions in Central Fields

Abstract

The behavior of a mass point moving in a plane under the effect of a central field and an ex-ternal periodic excitation in resonance with the natural frequency is studied. The asymptotic perturbation method is used in order to determine the nonlinear modulation equations for the amplitude and the phase of the oscillation. Firstly, we calculate the second order approximate solution of the unforced system. It is well known that generally the solution is two period quasiperiodic, but we find some new cases of periodic solutions. If appropriate Diophantine equations are satisfied, the motion is periodic with a frequency depending on the nonlinear terms. Subsequently, the forced system is considered and external force-response curves are shown and moreover jump phenomena are also observed. In certain cases we observe a frequency splitting and a third frequency appears in addition to the forcing frequency Stable three period quasi-periodic motions are present with amplitudes depending on the initial conditions.