Stability of a Parametrically Driven, Coupled Oscillator System: An Auxillary Function Method Approach

Abstract

Coupled, nonlinear oscillators are often studied in applied biology, physics, fluids, and many other disciplines. In this paper, we study a parametrically driven, coupled oscillator system where the individual oscillators are subjected to varying frequency and phase with a focus on the influence of the damping and coupling parameters away from parametric resonance frequencies. In particular, we study the key long-term statistics of the oscillator system's trajectories and stability. We present a novel, robust and computationally efficient method come to be known as an auxillary function method for long-time averages, and we pair this method with classical, perturbative-asymptotic analysis to corroborate the results of this auxillary function method. These paired methods are then used to compute the regions of stability for a coupled oscillator system. The objective is to explore the influence of higher order, coupling effects on the stability boundary across a broad range of modulation frequencies, including frequencies away from parametric resonances. We show that both simplified and more general asymptotic methods can be dangerously un-conservative in predicting the true regions of stability due to high order effects caused by coupling parameters. The differences between the true stability boundary and the approximate stability boundary can occur at physically relevant parameter values in regions away from parametric resonance. The differences between the solutions depends on the specific parameters of the system, as explained in the results section. As an alternative to asymptotic methods, we show that the auxillary function method for long-time averages is an efficient and robust means of computing true regions of stability across all possible initial conditions.

Figures (5)

• Figure 1: The simplified, uncoupled asymptotic solution to Equation (\ref{['matrix3']}) (shown with colored, solid contour lines) and the more general asymptotic solution to Equation (\ref{['matrix2']}) (shown with dashed, gray contour lines). The markers $\text{I}_\text{S}$, $\text{I}_\text{U}$, $\text{II}_\text{S}$, and $\text{II}_\text{U}$ denote the stable (S) or unstable (U) regions of the simplified (I) and general (II) solutions, respectively. The parameter values are $r=.2$, $g=.01$, and $\phi=0$.
• Figure 2: There is strong agreement between our auxillary function method (shown with a thick, solid black line) with the theoretical predictions of the simplified, uncoupled asymptotic analysis (shown with colored, solid contour lines) and the more general, higher order asymptotic analysis (shown with dashed, gray contour lines) for $r=.2$, $g=.01$, and $\phi=0$. The open blue circles and red crosses indicate where direct numerical simulation results are shown in Figure \ref{['Fig3']} for stable and unstable points, respectively.
• Figure 3: The time histories predicted by DNS for the four points marked in Figure \ref{['Fig2']}. The left two plots correspond to the two blue, open circles just outside the region of instability for Figure \ref{['Fig2']}, and the two right plots correspond to the two red crosses just inside the region of instability for Figure \ref{['Fig2']}. In the above plot, an initial condition vector IC: [a,b,c,d] corresponds to $x_1(0)=a$, $x_2(0)=b$, $y_1(0)=c$, and $y_2(0)=d$ for Equation (17) with $r=0.2$, $g=0.01$, and $\phi=0$.
• Figure 4: The stability boundary as predicted by the auxillary function method is shown with a thick, solid black line and a thick, dotted gray line for $r=.2$ and $r=.4$, respectively, while the asymptotic analysis boundary is shown with a thin, purple dashed line and a thin, blue dotted line for $r=.2$ and $r=.4$, respectively, with $g=.01$ and $\phi=0$.
• Figure 5: The stability boundary as predicted by the auxillary function method for $\phi=0$ (shown with a thick, solid black line), $\phi=\frac{\pi}{2}$ (shown with a thin, solid blue line), and $\phi=\pi$ (shown with a thin, dashed gray line).