The paper is devoted to the analysis of behavior of equilibrium trajectories in dynamic bimatrix games. We consider a system of differential equations which describe evolutionary dynamics of behavior of two players on the infinite time horizon. In the first case, we examine an approach based on the idea of guaranteed strategies in the sense of N.N. Krasovskii, for which the construction of the dynamic Nash equilibrium is implemented. In the second case, we consider a dynamic system based on the strategies of players' best replies. In this case, the equilibrium trajectory converges to the point of the static Nash equilibrium. For both cases, equilibrium trajectories are constructed and the comparison is carried out for the values of players' payoff functionals at the points of equilibrium. It is shown that characteristics of trajectories of the dynamic Nash equilibrium are better than properties of trajectories of the best reply dynamics. For demonstration of the behavior of equilibrium trajectories in the dynamic bimatrix games the model is presented in which we consider payoff matrices of two players on the financial markets of stocks and bonds. © 2023 American Institute of Physics Inc.. All rights reserved.