Circularly distributed multipliers with deterministic moduli assessing the stability of quasiperiodic response

The stability and bifurcation of a periodic solution of a dynamical system can be handled well by using the Floquet multipliers of the perturbed system with periodic coefficients. However, for a quasiperiodic (QP) response as a natural extension of a periodic one, it is much more difficult to do it quantitatively. Therefore, proposed here is an approach for defining and obtaining effective multipliers for QP stability. The proposed approach is based on a series of auxiliary variables via which the perturbed system with QP coefficients is transformed approximately into a constant one, whereupon the multipliers are obtained efficiently by performing eigenvalue analysis on the constant coefficient matrix. The major finding involves circularly distributed multipliers with deterministic moduli, with the QP response being stable if all the moduli are less than or equal to unity; otherwise it is unstable. When the QP response degenerates to periodic due to the reducibility of fundamental frequencies, the proposed approach exactly provides the Floquet multipliers for the periodic solution. From this respect, the obtained multipliers can be considered to some extent as being a generalization for QP response of the Floquet multipliers for a periodic solution.