To the study of the behaviour of trajectories of point mapping between planes in remote parts of the phase plane

The question of 'infinity stability-instability' is an important issue in the study of the dynamics of a system in the plane. In this work, to solve this question, a methodology for analysing the behaviour of trajectories of a point mapping between planes is proposed, using a substitution of variables that translates the plane into the interior of a circle of unit radius (i.e., an infinitely distant part of the plane into a finite one). On the example of studying specific quasilinear systems, the expediency of using the suggested methodology to obtain a clear picture of the behaviour of the trajectories of the point mapping on the entire phase plane, including its remote parts, is shown. It is noted that the use of the indicated replacement allows to confirm the earlier research of the influence of the character of nonlinearity on the results of the qualitative study of systems with small parameter at nonlinear terms by asymptotic methods.