Bifurcations in Dynamical Systems With Internal Resonance

Abstract

Dynamical systems with quadratic nonlinearities and exhibiting internal resonance under periodic excitations are studied. Two types of transition from stable to unstable motions are shown to occur. One kind are shown to be associated with jump phenomena while the other kind are shown to be associated with Hopf bifurcations of the averaged system of equations. In the case of the latter, the motions are shown to be amplitude modulated motions at the excitation frequency with the amplitude of modulation determined by the motion of a point on a torus.