Multi-frequency oscillations in self-excited systems

A self-excited three-mass chain system is considered here. For a self-excitation of van der Pol type, the possibility of multi-frequency oscillations is investigated. Both analytical approximate solutions and numerical simulation are used. The averaging method is used to establish existence and stability of the normal modes, the two-frequency modes as well as the three-frequency oscillations solutions. We found at first that the single mode seems to prevail. However a three-frequency solution can be stabilised by adapting the system slightly. A generic bifurcation diagram is given where all the possible phase portraits are sketched. The flow turns out to be quite predictable. There is no “room” for chaos or strange attractors. This behaviour is not typical for systems of coupled oscillators but turns out to be partly related to the involved symmetries as well as the particular choice of the system parameters.