As we know ordinary differential equations, systems of ordinary differential
equations, in particular, two dimensional nonlinear differential systems have significant and
various applications in qualitative theory of ordinary differential equations. In some real world
applications, it is needed to have information in relation to the qualitative concepts called
stability, boundedness, convergence, etc. of solutions of that kind of mathematical models. Most
of time, exact solutions of that kind of equations cannot be obtained explicitly, except
numerically. In the pertinent literature, one of the famous method is known the Lyapunov's
second method, which allows to have information about qualitative behaviors of solutions
without solving the equation understudy. In this study, we deal with a nonlinear a two
dimensional nonlinear differential system. We examine uniform asymptotic stability,
boundedness, uniform boundedness and uniform-ultimate boundedness of solutions of that two
dimensional nonlinear differential system. We will prove three new theorems on the mentioned
qualitative concepts by using the Lyapunov's second method. We provide two examples to
demonstrate how the results of the study can be applied. The results of this study generalize some
recent results, which can be found in the present literature.