Some new qualitative results for two dimensional nonlinear differential systems

As we know ordinary differential equations, systems of ordinary differentialequations, in particular, two dimensional nonlinear differential systems have significant andvarious applications in qualitative theory of ordinary differential equations. In some real worldapplications, it is needed to have information in relation to the qualitative concepts calledstability, boundedness, convergence, etc. of solutions of that kind of mathematical models. Mostof time, exact solutions of that kind of equations cannot be obtained explicitly, exceptnumerically. In the pertinent literature, one of the famous method is known the Lyapunov’ssecond method, which allows to have information about qualitative behaviors of solutionswithout solving the equation understudy. In this study, we deal with a nonlinear a twodimensional nonlinear differential system. We examine uniform asymptotic stability,boundedness, uniform boundedness and uniform-ultimate boundedness of solutions of that twodimensional nonlinear differential system. We will prove three new theorems on the mentionedqualitative concepts by using the Lyapunov’s second method. We provide two examples todemonstrate how the results of the study can be applied. The results of this study generalize somerecent results, which can be found in the present literature.