水肺潜水员豚跳与失控上浮的理想化时滞微分模型
An Idealized Delay-Differential Model of Scuba Diver Porpoising and Runaway Ascent
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中文总结 AI 辅助
本研究建立水肺潜水员豚跳与失控上浮的理想化时滞微分模型，确定霍普夫边界，发现失控上浮由调节器饱和触发，给出转变的早期预警方法。
中文摘要 AI 辅助
保持恒定深度的水肺潜水员在不稳定平衡状态下维持平衡：潜水服和浮力调节器携带的气体随深度压缩，因此浮力随潜水员下沉而下降，随潜水员上浮而上升。我们将潜水员表示为比例-微分控制器，该控制器在有限反应延迟后调节这种可压缩浮力鞍点，并从垂直力平衡和等温气体定律推导控制时滞微分方程，将其简化为阻尼比、两个控制增益和一个无量纲延迟。通过半群生成元的伪谱配点法获得特征谱，并与特征方程的直接牛顿解法比对，确定分隔稳定悬停与持续豚跳的霍普夫边界；对于基准潜水员，临界反应延迟为3.36秒， onset 周期为28.8秒。该分岔为超临界分岔，且由于饱和力为二次流体动力阻力，极限环振幅与延迟过量成正比，而非与延迟过量的平方根成正比。安全操作包络显示，失控上浮由调节器饱和触发，而非线性稳定性丧失，因此稳定潜水员与不稳定潜水员可共享同一逃逸阈值。当接近 onset 时，一阶自相关和方差上升，拟合恢复率下降并与谱横坐标匹配，提供了精确到特征值的转变早期预警。
英文摘要
A scuba diver holding constant depth balances on an unstable equilibrium: the gas carried in the suit and buoyancy compensator compresses with depth, so the buoyant force falls as the diver sinks and rises as the diver ascends. We represent the diver as a proportional-derivative controller that regulates this compressible-buoyancy saddle after a finite reaction delay, and we derive the governing delay differential equation from the vertical force balance and the isothermal gas law, reducing it to a damping ratio, two control gains, and a dimensionless delay. The characteristic spectrum, obtained by pseudospectral collocation of the semigroup generator and checked against a direct Newton solution of the characteristic equation, locates the Hopf boundary that separates stable hovering from sustained porpoising; for the baseline diver the critical reaction delay is 3.36 s and the onset period is 28.8 s. The bifurcation is supercritical, and because the saturating force is the quadratic hydrodynamic drag, the limit-cycle amplitude grows in proportion to the delay excess rather than as its square root. The safe-operating envelope shows that runaway ascent is triggered by saturation of the compensator, not by loss of linear stability, so a stable and an unstable diver can share the same escape threshold. As onset is approached, the lag-one autocorrelation and variance rise while the fitted recovery rate falls and matches the spectral abscissa, giving an eigenvalue-exact early warning of the transition.
