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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.

We investigate the effect of piezoelectric (PZT) material on the flutter speed, vibration mode and frequency, and energy harvesting power and efficiency of a flexible flag in various fluids. We develop a fully coupled fluid-solid-electric model by combining the inviscid vortex sheet model with a linear electro-mechanical coupling model. A resistance only circuit and a resonant resistance-inductance (RL) circuit are considered. For a purely resistive circuit, an increased electro-mechanical coupling factor results in an increased flutter speed, vibration frequency, averaged electric power and efficiency. A consistent optimal resistance is found that maximizes the flutter speed and the energy harvesting power. For a resonant RL circuit, by tuning the inductance to match the circuit frequency to the flag's vibration frequency, the flutter speed can be greatly decreased, and a larger averaged power and efficiency are obtained. We also consider a model scale set-up with several commonly used commercial materials for operating in air and water. Typical ranges of dimensionless parameters are obtained for four types of material that span a wide range of solid density and rigidity values. We find that the resistance only circuit is more effective when the flag is placed in a lighter fluid (e.g. air), while the RL circuit is able to reduce the flutter speed when the flag is placed in a heavier fluid (e.g. water).