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

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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.