This work concerns the dynamics of towed array lifting devices (LFDs), which help to maintain the position of a tow line being dragged behind a vessel. The LFD satisfies the non-dimensional quadratically damped Mathieu equation ẍ+ (δ + 2 cos t)x+ ẋ |ẋ| = 0. Numerical study of this equation shows a wide array of dynamical features. The dynamical features of the system are exploited to obtain a c...