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.

Figure 1

Time histories of perturbation state $Y$ provided by perturbed and auxiliary systems with $k_1=0.08$ and using (a) all basis functions or (b) only those with $|{\alpha }_n|\ge {10}^{-6}$.

Figure 2

(a) All multipliers (${\lambda }_n$) provided by auxiliary system (7) for QP response of system (9) with $k_1=0.080$; (b) moduli of coefficients (${\alpha }_n$) vs $n$; (c) moduli of multipliers truncated with $|{\alpha }_n|\ge {10}^{-6}$ vs $n$; (d) retained multipliers in complex plane.

Figure 3

Time histories of perturbation state $Y$ provided by perturbed and auxiliary systems with (a) $k_1=0.081$ and (b) $k_1=0.079$. The basis functions with $|{\alpha }_n|\ge {10}^{-6}$ are retained.

Figure 4

Discrete weight function $W\left(\right|\lambda \left|\right)$ with $▵\left|\lambda \right|={10}^{-5}$ vs moduli of multipliers provided by auxiliary system (7) for system (9) with $k_1=0.080$.

Figure 5

Discrete weight function vs moduli close to unity for system (9) with (a) $k_1=0.081$, (b) $k_1=0.080$, and (c) $k_1=0.079$.

Figure 6

The phase plane of the QP responses obtained by the RK method for system (1) with $k_1=0.081$ (a), the Poincare sections (c), and the FFT spectrum (e); and their counterparts for $k_1=0.0799$ are provided in panels (b), (d), and (f), respectively.

Figure 7

(a) Amplitudes of QP responses of Duffing system vs ${\omega }_1$; (b) deterministic moduli provided by weight function.

Figure 8

(a) QP responses of Duffing system for ${\omega }_1=0.8$ (top row) and ${\omega }_1=1.0$ (middle and bottom rows) obtained by [(a1)–(c1)] IHB method and [(a2)–(c2)] RK method. [(a3)–(c3)] Moduli of corresponding multipliers provided by weight function in comparison with the unit cycle (i.e., $\left|\lambda \right|=1$).

Figure 9

Multipliers with deterministic moduli provided by weight function for Duffing system with (a) irreducible (${\omega }_1=0.8$, ${\omega }_2=\sqrt{5}/2$) or (b) reducible (${\omega }_1=0.8$, ${\omega }_2=1$) frequencies. Red triangles, ${\lambda }_n=exp({\eta }_{n}2\pi /{\omega }_1)$; black squares, ${\lambda }_n=exp({\eta }_{n}2\pi /{\omega }_2)$; blue cycles, ${\lambda }_n=exp({\eta }_{n}2\pi /{\omega }_{12})$ with ${\omega }_{12}=0.2$ as common divisor of ${\omega }_1=0.8$ and ${\omega }_2=1$. Red crosses, Floquet multipliers of corresponding periodic response.