New construction of phase space analysis that conforms to low-dimensional dynamical systems

为了明确判断低维线性微分系统、半线性微分系统或拟线性微分系统的平衡点（或零解）是否渐近稳定、全局渐近稳定或一致全局渐近稳定的判据，对相空间（或平面）分析进行了改进和构造。将所得准则应用于某小型船舶的自由横摇运动分析。众所周知，小型渔船自由滚动运动的阻力主要分为三种：摩擦阻力；造涡阻力（或粘压阻力）；兴波阻力。虽然造波阻力与角速度成正比，但摩擦阻力和造涡阻力与角速度的平方成正比。因此，滚动运动方程需要用阻尼超线性振荡器来描述。本文给出了阻尼超线性振荡器平衡点全局渐近稳定的充分必要条件。上述准则也适用于猎物（或捕食者）接受环境时变的Lotka-Volterra捕食者-食饵模型。这个模型作为一个生态学模型是非常著名的，具有独特的内部平衡。在本研究中，得到了内部平衡全局渐近稳定的充分条件。此外，还讨论了考虑猎物迁移的Rosenzweig - MacArthur捕食-食饵模型的极限环唯一性问题。

Phase space (or plane) analysis was improved and constructed in order to clarify criteria which judge whether the equilibrium (or the zero solution) of low- dimensional linear differential systems, half-linear differential systems, or quasi-linear differential systems is asymptotically stable, globally asymptotically stable, or uniformly globally asymptotically stable.The obtained criteria were applied to the analysis of the free rolling motion of a small ship. It is well- known that the resistance of free rolling motion of a small fishing vessel is mainly classified into three types: frictional resistance; eddy-making resistance (or viscous pressure resistance); wave-making resistance. Although wave-making resistance is proportional to the angular velocity, frictional resistance and eddy-making resistance are proportional to the square of the angular velocity. For this reason, the equation of rolling motion need be described by a damped super-linear oscillator. In this research, a necessary and sufficient condition for the equilibrium of the damped super-linear oscillator to be globally asymptotically stable was found out.The above-mentioned criteria were also applied to a Lotka-Volterra predator-prey model with prey (or predator) receiving time-variation of the environment. This model is very famous as an ecology model and have a unique interior equilibrium. In this research, sufficient conditions for the interior equilibrium to be globally asymptotically stable were obtained. Moreover, the uniqueness problem of limit cycles about a Rosenzweig MacArthur predator-prey model in consideration of immigration of prey was discussed.