Qualitative analysis of classes of motion for multiresonant systems II. A geometrical method

Abstract

Classes of motion of general multiresonant systems are derived through a geometrical algorithm based on a set representation. First, the elementary classes existing under simple resonance conditions are evaluated; rules governing the interaction between elementary classes belonging to different resonance conditions are then drawn up as applications of a unique theorem. Illustrative examples are given. The method also permits a hierarchical ordering of the amplitudes of the resonant modes, according to their participation in the classes; it also makes it possible to ascertain in advance the existence of a standard form for the amplitude modulation equations. The stability analysis of incomplete steady solutions is then addressed. Three classes of perturbation are distinguished, namely: in-class perturbations, out-of-class resonant perturbations, and out-of-class nonresonant perturbations. The structure of the Jacobian variational matrix is studied. The Jacobian matrix is shown to comprise three diagonal blocks associated with the three perturbation classes, so that the stability equations are uncoupled. Further possible uncouplings of one of the blocks are analyzed in relation to some of the geometrical properties of the classes.